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Analogs

Sure, anything provided hereafter could be found already in the individual incidence matrix files, and sometimes also in some of the explanatory pages as well.None the less a missing link is that of dimensional analogy of the various members of a family of polytopes. Esp. for those generally existing cases.

In the followings some general dimensional series of polytopes get detailed.

Symmetry A_n
- Regular Simplex S_n = x3o...o3o
- Rectified Simplex rS_n = o3x3o...o3o
- Facetorectified Simplex frS_n = hemi(x3o...o3o3/2x )
- Birectified Simplex brS_n = o3o3x3o...o3o
- Truncated Simplex tS_n = x3x3o...o3o (= SimplexialTutsatope)
- Bitruncated Simplex btS_n = o3x3x3o...o3o
- Mid-rectified Simplex mrS_n = o3o...o3x3o...o3o
- Mid-truncated Simplex mtS_n = o3o...o3x3x3o...o3o
- Maximal Expanded Simplex eS_n = x3o...o3x
  - more general class of mostly hyperbolic honeycombsxPo3o...o3oPxQ*a
- Retroexpanded Simplex reS_n = o3x3x3/2*a3o...o3o
- Omnitruncated Simplex otS_n = x3x...x3x
Symmetry BC_n
- Regular Orthoplex O_n = x3o...o3o4o
  - more general class of mostly hyperbolic honeycombsxPo3o...o3o4o
- Regular Hypercube C_n = o3o...o3o4x
- Rectified Orthoplex rO_n = o3x3o...o3o4o
- Rectified Hypercube rC_n = o3o...o3x4o
- Facetorectified Hypercube frC_n = x3x3/2o3o...o3*a
- Birectified Orthoplex brO_n = o3o3x3o...o3o4o
- Birectified Hypercube brC_n = o3o...o3x3o4o (= Rectified Demihypercube rD_n)
- Truncated Orthoplex tO_n = x3x3o...o3o4o
- Truncated Hypercube tC_n = o3o...o3x4x
- Quasitruncated Hypercube qtC_n = o3o...o3x4/3x
- Bitruncated Hypercube btC_n = o3o...o3x3x4o
- Rhombated Hypercube rbC_n = o3o...o3x3o4/3x
- Quasirhombated Hypercube qrbC_n = o3o...o3x3o4/3x
- Maximal Expanded Hypercube eC_n = x3o...o3o4x (= eO_n)
- Quasiexpanded Hypercube qeC_n = x3o...o3o4/3x (= qeO_n)
- Retroexpanded Hypercube reC_n = o3x4x4/3*a3o...o3o (a.k.a. Socco Series)
- Quasiretroexpanded Hypercube qreC_n = o3x4/3x4*a3o...o3o (a.k.a. Gocco Series)
- Omnitruncated Hypercube otC_n = x3x...x3x4x (= otO_n)
Symmetry D_n
- Demihypercube D_n = x3o3o *b3o...o3o (= DemihypercubicAlterprism)
- Demicross hO_n = hemi(o3o3/2o3o3*a3o3o...o3o3x )
- Truncated Demihypercube tD_n = x3x3o *b3o...o3o
- Maximal Expanded Demihypercube eD_n = x3o3o *b3o...o3x
Symmetry E_n
- Gossetic n_2,1 = o3o...o3x *c3o
- Gossetic 2_n,1 = x3o...o3o *c3o
- Gossetic 1_n,2 = o3o...o3o *c3x
- Rectified Gossetic r(n_2,1) = o3o...o3x3o *c3o
- Rectified Gossetic r(2_n,1) = o3x3o...o3o *c3o
- Rectified Gossetic r(1_n,2) = o3o3x3o...o3o *c3o (= Birectified Gossetic br(2_n,1))
- Truncated Gossetic t(n_2,1) = o3o...o3x3x *c3o
- Truncated Gossetic t(2_n,1) = x3x3o...o3o *c3o
- Truncated Gossetic t(1_n,2) = o3o3x3o...o3o *c3x
- All-Ends Expanded Gossetic eE_n = x3o...o3x *c3x (= e(n_2,1), e(2_n,1), e(1_n,2) )
- Omnitruncated Gossetic otE_n = x3x...x3x3x *c3x (= ot(n_2,1), ot(2_n,1), ot(1_n,2) )
Some Axial Cases
- Simplexial Ursatope U_n = ofx3xoo3ooo...ooo3ooo&#xt
- Orthoplexial Ursatope oU_n = ofx3xoo3ooo...ooo3ooo4ooo&#xt
- Rectified Simplex Pyramid rS_n-py = oo3ox3oo...oo3oo&#x
Some Duoprismatic Cases
- Simplex Duoprism S_n×S_n = x3o...o3o x3o...o3o

Symmetry A_n

Regular Simplex S_n (up)

These polytopes generally are self-dual. Further they are closely related to thepyramid product.In fact S_n here is nothing but the S_n-1 pyramid. Thence, by means of thelace prism notation, S_n = x3o...o3o (n nodes) can be described as well asox3oo...oo3oo&#x (n-1 node positions).

Dimension	1D	2D	3D	4D	5D	nD
Dynkin diagram	x	x3o	x3o3o	x3o3o3o	x3o3o3o3o	x3o...o3o
Acronym	line	trig	tet	pen	hix	n-simplex
Vertex Count	2	3`line`	4`trig`	5`tet`	6`pen`	n+1
Facet Count	2	3`line`	4`trig`	5`tet`	6`pen`	n+1
Circumradius	1/2 0.5	1/sqrt(3) 0.577350	sqrt(3/8) 0.612372	sqrt(2/5) 0.632456	sqrt(5/12) 0.645497	sqrt(n)/sqrt[2(n+1)]
Inradius	1/2 0.5	1/sqrt(12) 0.288675	1/sqrt(24) 0.204124	1/sqrt(40) 0.158114	1/sqrt(60) 0.129099	1/sqrt[2n(n+1)]
Height 0-dim first	1/2 0.5	sqrt(3)/2 0.866025	sqrt(2/3) 0.816497	sqrt(5/8) 0.790569	sqrt(3/5) 0.774597	sqrt[(n+1)/(2n)]
Height 1-dim first		sqrt(3)/2 0.866025	1/sqrt(2) 0.707107	sqrt(5/12) 0.645497	sqrt(6)/4 0.612372	sqrt[(n+1)/(4(n-1))]
Height 2-dim first			sqrt(2/3) 0.816497	sqrt(5/12) 0.645497	1/sqrt(3) 0.577350	sqrt[(n+1)/(6(n-2))]
Height 3-dim first				sqrt(5/8) 0.790569	sqrt(6)/4 0.612372	sqrt[(n+1)/(8(n-3))]
Height 4-dim first				sqrt(5/8) 0.790569	sqrt(3/5) 0.774597	sqrt[(n+1)/(10(n-4))]
Volume	1	sqrt(3)/4 0.433013	sqrt(2)/12 0.117851	sqrt(5)/96 0.023292	sqrt(3)/480 0.0036084	sqrt[(n+1)/(2ⁿ)]/n!
Surface	2	3	sqrt(3) 1.732051	5 sqrt(2)/12 0.589256	sqrt(5)/16 0.139754	(n+1) sqrt[n/(2^n-1)]/(n-1)!
Dihedral angles	0°	60°	arccos(1/3) 70.528779°	arccos(1/4) 75.522488°	arccos(1/5) 78.463041°	arccos(1/n)
Dimension	6D	7D	8D	9D	10D	nD
Dynkin diagram	x3o3o3o3o3o	x3o3o3o3o3o3o	x3o3o3o3o3o3o3o	x3o3o3o3o3o3o3o3o	x3o3o3o3o3o3o3o3o3o	x3o...o3o
Acronym	hop	oca	ene	day	ux	n-simplex
Vertex Count	7`hix`	8`hop`	9`oca`	10`ene`	11`day`	n+1
Facet Count	7`hix`	8`hop`	9`oca`	10`ene`	11`day`	n+1
Circumradius	sqrt(3/7) 0.654654	sqrt(7)/4 0.661438	2/3 0.666667	sqrt(9/20) 0.670820	sqrt(5/11) 0.674200	sqrt(n)/sqrt[2(n+1)]
Inradius	1/sqrt(84) 0.109109	1/sqrt(112) 0.094491	1/12 0.083333	1/sqrt(180) 0.074536	1/sqrt(220) 0.067420	1/sqrt[2n(n+1)]
Height 0-dim first	sqrt(7/12) 0.763763	2/sqrt(7) 0.755929	3/4 0.75	sqrt(5)/3 0.745356	sqrt(11/20) 0.741620	sqrt[(n+1)/(2n)]
Height 1-dim first	sqrt(7/20) 0.591608	1/sqrt(3) 0.577350	3/sqrt(28) 0.566947	sqrt(5)/4 0.559017	sqrt(11)/6 0.552771	sqrt[(n+1)/(4(n-1))]
Height 2-dim first	sqrt(7/24) 0.540062	2/sqrt(15) 0.516398	1/2 0.5	sqrt(5/21) 0.487950	sqrt(11/48) 0.478714	sqrt[(n+1)/(6(n-2))]
Height 3-dim first	sqrt(7/24) 0.540062	1/2 0.5	3/sqrt(40) 0.474342	sqrt(5/24) 0.456435	sqrt(11/56) 0.443203	sqrt[(n+1)/(8(n-3))]
Height 4-dim first	sqrt(7/20) 0.591608	2/sqrt(15) 0.516398	3/sqrt(40) 0.474342	1/sqrt(5) 0.447214	sqrt(11/60) 0.428174	sqrt[(n+1)/(10(n-4))]
Height 5-dim first	sqrt(7/12) 0.763763	1/sqrt(3) 0.577350	1/2 0.5	sqrt(5/24) 0.456435	sqrt(11/60) 0.428174	sqrt[(n+1)/(12(n-5))]
Height 6-dim first		2/sqrt(7) 0.755929	3/sqrt(28) 0.566947	sqrt(5/21) 0.487950	sqrt(11/56) 0.443203	sqrt[(n+1)/(14(n-6))]
Height 7-dim first			3/4 0.75	sqrt(5)/4 0.559017	sqrt(11/48) 0.478714	sqrt[(n+1)/(16(n-7))]
Height 8-dim first				sqrt(5)/3 0.745356	sqrt(11)/6 0.552771	sqrt[(n+1)/(18(n-8))]
Height 9-dim first				sqrt(5)/3 0.745356	sqrt(11/20) 0.741620	sqrt[(n+1)/(20(n-9))]
Volume	sqrt(7)/5760 0.00045933	1/20160 0.000049603	1/215040 0.0000046503	sqrt(5)/5806080 0.00000038513	sqrt(11)/116121600 0.0000000028562	sqrt[(n+1)/(2ⁿ)]/n!
Surface	7 sqrt(3)/480 0.025259	sqrt(7)/720 0.0036747	1/2240 0.00044643	1/21504 0.000046503	11 sqrt(5)/5806080 0.0000042364	(n+1) sqrt[n/(2^n-1)]/(n-1)!
Dihedral angles	arccos(1/6) 80.405932°	arccos(1/7) 81.786789°	arccos(1/8) 82.819244°	arccos(1/9) 83.620630°	arccos(1/10) 84.260830°	arccos(1/n)

Rectified Simplex rS_n (up)

Within these polytopes rS_n generally can be described as thesegmentotope oftheregular simplex S_n-1 atop the rectified simplex rS_n-1.Thence, by means of thelace prism notation,rS_n = o3x3o...o3o (n nodes) can be described as well asxo3ox3oo...oo3oo&#x (n-1 node positions).

Furthermore are rectified simplices special cases of theCoxeter-Elte-Gosset polytopes k_m,n, in fact those generally are clearly the ones of the form 0_(n-1),1.

Dimension	1D	2D	3D	4D	5D	nD
Dynkin diagram		o3x	o3x3o	o3x3o3o	o3x3o3o3o	o3x3o...o3o
Acronym		trig	oct	rap	rix	rect. n-simplex
Vertex Count		3`line`	6`square`	10`trip`	15`tepe`	n(n+1)/2
Facet Count rect. facets		3`line`	4`trig`	5`oct`	6`rap`	n+1
Facet Count verf facets		3`line`	4`trig`	5`tet`	6`pen`	n+1
Circumradius		1/sqrt(3) 0.577350	1/sqrt(2) 0.707107	sqrt(3/5) 0.774597	sqrt(2/3) 0.816497	sqrt[(n-1)/(n+1)]
Inradius wrt. rect. facets		1/sqrt(3) 0.577350	1/sqrt(6) 0.408248	1/sqrt(10) 0.316228	1/sqrt(15) 0.258199	sqrt(2)/sqrt[n(n+1)]
Inradius wrt. verf facets		1/sqrt(12) 0.288675	1/sqrt(6) 0.408248	3/sqrt(40) 0.474342	2/sqrt(15) 0.516398	(n-1)/sqrt[2n(n+1)]
Volume		sqrt(3)/4 0.433013	sqrt(2)/3 0.471405	11 sqrt(5)/96 0.256216	13 sqrt(3)/240 0.093819	(2ⁿ-n-1) sqrt[(n+1)/(2ⁿ)]/n!
Surface		3	2 sqrt(3) 3.464102	25 sqrt(2)/12 2.946278	3 sqrt(5)/4 1.677051	(n+1)(2^n-1-n+1) sqrt[n/(2^n-1)]/(n-1)!
Dihedral angles rect - rect		60° verf - verf	2 sqrt(3) 3.464102	arccos(1/4) 75.522488°	arccos(1/5) 78.463041°	arccos(1/n)
Dihedral angles verf - rect		60° verf - verf	arccos(-1/3) 109.471221°	arccos(-1/4) 104.477512°	arccos(-1/5) 101.536959°	arccos(-1/n)
Dimension	6D	7D	8D	9D	10D	nD
Dynkin diagram	o3x3o3o3o3o	o3x3o3o3o3o3o	o3x3o3o3o3o3o3o	o3x3o3o3o3o3o3o3o	o3x3o3o3o3o3o3o3o3o	o3x3o...o3o
Acronym	ril	roc	rene	reday	ru	rect. n-simplex
Vertex Count	21`penp`	28`hixip`	36`hopip`	45`ocpe`	55`enep`	n(n+1)/2
Facet Count rect. facets	7`rix`	8`ril`	9`roc`	10`rene`	11`reday`	n+1
Facet Count verf facets	7`hix`	8`hop`	9`oca`	10`ene`	11`day`	n+1
Circumradius	sqrt(5/7) 0.845154	sqrt(3)/2 0.866025	sqrt(7)/3 0.881917	2/sqrt(5) 0.894427	3/sqrt(11) 0.904534	sqrt[(n-1)/(n+1)]
Inradius wrt. rect. facets	1/sqrt(21) 0.218218	1/sqrt(28) 0.188982	1/6 0.166667	1/sqrt(45) 0.149071	1/sqrt(55) 0.134840	sqrt(2)/sqrt[n(n+1)]
Inradius wrt. verf facets	5/sqrt(84) 0.545545	3/sqrt(28) 0.566947	7/12 0.583333	4/sqrt(45) 0.596285	9/sqrt(220) 0.606780	(n-1)/sqrt[2n(n+1)]
Volume	19 sqrt(7)/1920 0.026182	1/168 0.0059524	247/215040 0.0011486	251 sqrt(5)/2903040 0.00019333	1013 sqrt(11)/116121600 0.000028933	(2ⁿ-n-1) sqrt[(n+1)/(2ⁿ)]/n!
Surface	63 sqrt(3)/160 0.681995	29 sqrt(7)/360 0.213130	121/2240 0.054018	31/2688 0.011533	5533 sqrt(5)/5806080 0.0021309	(n+1)(2^n-1-n+1) sqrt[n/(2^n-1)]/(n-1)!
Dihedral angles rect - rect	arccos(1/6) 80.405932°	arccos(1/7) 81.786789°	arccos(1/8) 82.819244°	arccos(1/9) 83.620630°	arccos(1/10) 84.260830°	arccos(1/n)
Dihedral angles verf - rect	arccos(-1/6) 99.594068°	arccos(-1/7) 98.213211°	arccos(-1/8) 97.180756°	arccos(-1/9) 96.379370°	arccos(-1/10) 95.739170°	arccos(-1/n)

Facetorectified Simplex frS_n (up)

These non-convex polytopes frS_n generally are facetings of therectified simplex rS_n.

According to the fact that the mere Wythoffian construction providesGrünbaumian polytopes only,it is the secondary operation of replacing those double covered facets by single covers instead, which breaks down their orientability for all odd dimensions.Thence there avolume cannot be calculated.For the even dimensional cases we observe that no hemifacets occur and that the facet types alternate between prograde and retrograde wrt. the increasing absolute values of their inradii.

From the above shownsegmentotope representation of therectified simplex rS_n,it becomes obvious that the polytopes frS_n likewise can be given as such, though non-covex for sure, being generally the stack of thesimplex S_n-1 atop the facetorectified simplex frS_n-1.

Dimension	3D	4D	5D	6D	nD
Dynkin diagram	hemi(x3o3/2x )	hemi(x3o3o3/2x )	hemi(x3o3o3o3/2x )	hemi(x3o3o3o3o3/2x )	hemi(x3o...o3o3/2x )
Acronym	thah	firp	firx	firl	facetorect. n-simplex
Vertex Count	6	10	15	21	n(n+1)/2
Facet Count simplex	4`trig`	5`tet`	6`pen`	7`hix`	n+1
Facet Count prism	3`square` (hemi)	10`trip`	15`tepe`	21`penp`	n(n+1)/2
Facet Count duoprism	3`square` (hemi)	10`trip`	10`triddip` (hemi)	35`tratet`	(n-1)n(n+1)/6
Circumradius	1/sqrt(2) 0.707107	sqrt(3/5) 0.774597	sqrt(2/3) 0.816497	sqrt(5/7) 0.845154	sqrt[(n-1)/(n+1)]
Inradius wrt. simplex	1/sqrt(6) 0.408248	− 3/sqrt(40) 0.474342	2/sqrt(15) 0.516398	+ 5/sqrt(84) 0.545545	(n-1)/sqrt[2n(n+1)]
Inradius wrt. prism	0	+ 1/sqrt(60) 0.129099	1/sqrt(24) 0.204124	− 3/sqrt(140) 0.253546	(n-3)/sqrt[4(n-1)(n+1)]
Inradius wrt. duoprism	0	+ 1/sqrt(60) 0.129099	0	+ 1/sqrt(168) 0.077152	(n-5)/sqrt[6(n-2)(n+1)]
Volume	-	sqrt(5)/32 0.069877	-	sqrt(7)/576 0.0045933	- / sqrt[(n+1)/2ⁿ⁺²]/((n/2)!)²
Surface	3+sqrt(3) 4.732051	[5 sqrt(2)+30 sqrt(3)]/12 4.919383	[30+20 sqrt(2)+sqrt(5)]/16 3.782521	[7 sqrt(3)+105 sqrt(5) +350 sqrt(6)]/480 2.300485	?
Dihedral angles simp. - (next)	arccos[1/sqrt(3)] 54.735610°	arccos[sqrt(3/8)] 52.238756°	arccos[sqrt(2/5)] 50.768480°	arccos[sqrt(5/12)] 49.797034°	arccos[sqrt((n-1)/2n)]
Dihedral angles prism - (next)		arccos(2/3) 48.189685° (prism - prism)	45°	?	?
Dihedral angles duopr. - (next)		arccos(2/3) 48.189685° (prism - prism)	45°	? (duopr. - duopr.)	?
Dimension	7D	8D	9D	10D	nD
Dynkin diagram	hemi(x3o3o3o3o3o3/2x )	hemi(x3o3o3o3o3o3o3/2x )	hemi(x3o3o3o3o3o3o3o3/2x )	hemi(x3o3o3o3o3o3o3o3o3/2x )	hemi(x3o...o3o3/2x )
Acronym	froc	?	?	?	facetorect. n-simplex
Vertex Count	28	36	45	55	n(n+1)/2
Facet Count simplex	8`hop`	9`oca`	10`ene`	11`day`	n+1
Facet Count prism	28`hixip`	36`hopip`	45`ocpe`	55`enep`	n(n+1)/2
Facet Count duoprism I	56`trapen`	84`trahix`	120`trihop`	165`trioc`	(n-1)n(n+1)/6
Facet Count duoprism II	35`tetdip` (hemi)	126`tetpen`	210`tethix`	330`tethop`	(n-2)(n-1)n(n+1)/24
Facet Count duoprism III	35`tetdip` (hemi)	126`tetpen`	126`pendip` (hemi)	462`penhix`	(n-3)(n-2)(n-1)n(n+1)/120
Circumradius	sqrt(3)/2 0.866025	sqrt(7)/3 0.881917	2/sqrt(5) 0.894427	3/sqrt(11) 0.904534	sqrt[(n-1)/(n+1)]
Inradius wrt. simplex	3/sqrt(28) 0.566947	− 7/12 0.583333	4/sqrt(45) 0.596285	+ 9/sqrt(220) 0.606780	(n-1)/sqrt[2n(n+1)]
Inradius wrt. prism	1/sqrt(12) 0.288675	+ 5/sqrt(252) 0.314970	3/sqrt(80) 0.335410	− 7/sqrt(396) 0.351763	(n-3)/sqrt[4(n-1)(n+1)]
Inradius wrt. duoprism I	1/sqrt(60) 0.129099	− 1/6 0.166667	2/sqrt(105) 0.195180	+ 5/sqrt(528) 0.217597	(n-5)/sqrt[6(n-2)(n+1)]
Inradius wrt. douprism II	0	+ 1/sqrt(360) 0.052705	1/sqrt(120) 0.091287	− 3/sqrt(616) 0.120873	(n-7)/sqrt[8(n-3)(n+1)]
Inradius wrt. douprism III	0	+ 1/sqrt(360) 0.052705	0	+ 1/sqrt(660) 0.038925	(n-9)/sqrt[10(n-4)(n+1)]
Volume	-	1/6144 0.00016276	-	sqrt(11)/921600 0.0000035988	- / sqrt[(n+1)/2ⁿ⁺²]/((n/2)!)²
Surface	?	?	?	?	?
Dihedral angles simp. - (next)	arccos[sqrt(3/7)] 49.106605°	arccos[sqrt(7)/4] 48.590378°	arccos(2/3) 48.189685°	arccos[3/sqrt(20)] 47.869585°	arccos[sqrt((n-1)/2n)]
Dihedral angles prism - (next)	?	?	?	?	?
Dihedral angles duopr. I - (next)	?	?	?	?	?
Dihedral angles duopr. II - (next)		? (duopr. II - duopr. II)	?	?	?
Dihedral angles duopr. III - (next)		? (duopr. II - duopr. II)	?	? (duopr. III - duopr. III)	?

Birectified Simplex brS_n (up)

Within these polytopes brS_n generally can be described as thesegmentotope oftherectified simplex rS_n-1 atop the birectified simplex brS_n-1.Thence, by means of thelace prism notation,brS_n = o3o3x3o...o3o (n nodes) can be described as well asoo3xo3ox3oo...oo3oo&#x (n-1 node positions).

Furthermore are birectified simplices special cases of theCoxeter-Elte-Gosset polytopes k_m,n, in fact those generally are clearly the ones of the form 0_(n-2),2.

Dimension	3D	4D	5D	6D	nD
Dynkin diagram	o3o3x	o3o3x3o	o3o3x3o3o	o3o3x3o3o3o	o3o3x3o...o3o
Acronym	tet	rap	dot	bril	birect. n-simplex
Vertex Count	4`trig`	10`trip`	20`triddip`	35`tratet`	(n-1)n(n+1)/6
Facet Count rect. fac.	4`trig`	5`oct`	6`rap`	7`rix`	n+1
Facet Count birect. fac.	4`trig`	5`tet`	6`rap`	7`dot`	n+1
Circumradius	sqrt(3/8) 0.612372	sqrt(3/5) 0.774597	sqrt(3)/2 0.866025	sqrt(6/7) 0.925820	sqrt[(3n-6)/(2n+2)]
Inradius wrt. rect. facets	1/sqrt(24) 0.204124	1/sqrt(10) 0.316228	sqrt(3/20) 0.387298	2/sqrt(21) 0.436436	(n-2)/sqrt[2n(n+1)]
Inradius wrt. birect. facets	1/sqrt(24) 0.204124	3/sqrt(40) 0.474342	sqrt(3/20) 0.387298	sqrt(3/28) 0.327327	3/sqrt[2n(n+1)]
Volume	sqrt(2)/12 0.117851	11 sqrt(5)/96 0.256216	11 sqrt(3)/80 0.238157	151 sqrt(7)/2880 0.138718	[3ⁿ-(n+1) 2ⁿ+n(n+1)/2] sqrt[(n+1)/2ⁿ]/n!
Surface	sqrt(3) 1.732051	25 sqrt(2)/12 2.946278	11 sqrt(5)/8 3.074593	161 sqrt(3)/120 2.323835	(n+1) [3^n-1-(n-1) 2^n-1+n(n-3)/2] sqrt[n/2^n-1]/(n-1)!
Dihedral angles rect. - rect.		arccos(1/4) 75.522488°	arccos(1/5) 78.463041°	arccos(1/6) 80.405932°	arccos(1/n)
Dihedral angles rect. - birect.		arccos(-1/4) 104.477512°	arccos(-1/5) 101.536959°	arccos(-1/6) 99.594068°	arccos(-1/n)
Dihedral angles birect. - birect.	arccos(1/3) 70.528779°	arccos(1/4) 75.522488°	arccos(1/5) 78.463041°	arccos(1/6) 80.405932°	arccos(1/n)
Dimension	7D	8D	9D	10D	nD
Dynkin diagram	o3o3x3o3o3o3o	o3o3x3o3o3o3o3o	o3o3x3o3o3o3o3o3o	o3o3x3o3o3o3o3o3o3o	o3o3x3o...o3o
Acronym	broc	brene	breday	bru	birect. n-simplex
Vertex Count	56`trapen`	84`trahix`	120`trahop`	165`traoc`	(n-1)n(n+1)/6
Facet Count rect. facets	8`ril`	9`roc`	10`rene`	11`reday`	n+1
Facet Count birect. facets	8`bril`	9`broc`	10`brene`	11`breday`	n+1
Circumradius	sqrt(15)/4 0.968246	1	sqrt(21/20) 1.024695	sqrt(12/11) 1.044466	sqrt[(3n-6)/(2n+2)]
Inradius wrt. rect. facets	5/sqrt(112) 0.472456	1/2 0.5	7/sqrt(180) 0.521749	4/sqrt(55) 0.539360	(n-2)/sqrt[2n(n+1)]
Inradius wrt. birect. facets	3/sqrt(112) 0.283473	1/4 0.25	1/sqrt(20) 0.223607	3/sqrt(220) 0.202260	3/sqrt[2n(n+1)]
Volume	397/6720 0.059077	1431/71680 0.019964	913 sqrt(5)/362880 0.0056259	299 sqrt(11)/725760 0.0013664	[3ⁿ-(n+1) 2ⁿ+n(n+1)/2] sqrt[(n+1)/2ⁿ]/n!
Surface	359 sqrt(7)/720 1.319201	1311/2240 0.585268	1135/5376 0.211124	16621 sqrt(5)/580608 0.064012	(n+1) [3^n-1-(n-1) 2^n-1+n(n-3)/2] sqrt[n/2^n-1]/(n-1)!
Dihedral angles rect. - rect.	arccos(1/7) 81.786789°	arccos(1/8) 82.819244°	arccos(1/9) 83.620630°	arccos(1/10) 84.260830°	arccos(1/n)
Dihedral angles rect. - birect.	arccos(-1/7) 98.213211°	arccos(-1/8) 97.180756°	arccos(-1/9) 96.379370°	arccos(-1/10) 95.739170°	arccos(-1/n)
Dihedral angles birect. - birect.	arccos(1/7) 81.786789°	arccos(1/8) 82.819244°	arccos(1/9) 83.620630°	arccos(1/10) 84.260830°	arccos(1/n)

Truncated Simplex tS_n (up)

Within these polytopes tS_n generally can be described as thebistratic lace tower oftheregular simplex S_n-1 atop anu-scaled S_n-1 atop the truncated simplex tS_n-1.Thence, by means of thelace tower notation,tS_n = x3x3o...o3o (n nodes) can be described as well asxux3oox3ooo...ooo3ooo&#xt (n-1 node positions).As such those also could be referred to as simplexialtutsatopes: in fact tutsatopes are quite similarily defined as theursatopes, just that the part that there was played (within 4D) by the lacingteddies here now is taken by accordingtuts.

Dimension	1D	2D	3D	4D	5D	nD
Dynkin diagram		x3x	x3x3o	x3x3o3o	x3x3o3o3o	x3x3o...o3o
Acronym		hig	tut	tip	tix	trunc. n-simplex
Vertex Count		6	12	20	30	n(n+1)
Facet Count trunc. facets		3`line`	4`hig`	5`tut`	6`tip`	n+1
Facet Count verf facets		3`line`	4`trig`	5`tet`	6`pen`	n+1
Circumradius		1	sqrt(11/8) 1.172604	sqrt(8/5) 1.264911	sqrt(7)/2 1.322876	sqrt[(5n-4)/(2n+2)]
Inradius wrt. trunc. facets		sqrt(3)/2 0.866025	sqrt(3/8) 0.612372	3/sqrt(40) 0.474342	sqrt(3/20) 0.387298	3/sqrt[2n(n+1)]
Inradius wrt. verf facets		sqrt(3)/2 0.866025	5/sqrt(24) 1.020621	7/sqrt(40) 1.106797	sqrt(27/20) 1.161895	(2n-1)/sqrt[2n(n+1)]
Volume		3 sqrt(3)/2 2.598076	23 sqrt(2)/12 2.710576	19 sqrt(5)/24 1.770220	79 sqrt(3)/160 0.855200	(3ⁿ-n-1) sqrt[(n+1)/(2ⁿ)]/n!
Surface		6	7 sqrt(3) 12.124356	10 sqrt(2) 14.142136	77 sqrt(5)/16 10.761077	(n+1)(3^n-1-n+1) sqrt[n/(2^n-1)]/(n-1)!
Dihedral angles trunc - trunc		6	7 sqrt(3) 12.124356	arccos(1/4) 75.522488°	arccos(1/5) 78.463041°	arccos(1/n)
Dihedral angles verf - trunc		120°	arccos(-1/3) 109.471221°	arccos(-1/4) 104.477512°	arccos(-1/5) 101.536959°	arccos(-1/n)
Dimension	6D	7D	8D	9D	10D	nD
Dynkin diagram	x3x3o3o3o3o	x3x3o3o3o3o3o	x3x3o3o3o3o3o3o	x3x3o3o3o3o3o3o3o	x3x3o3o3o3o3o3o3o3o	x3x3o...o3o
Acronym	til	toc	tene	teday	tu	trunc. n-simplex
Vertex Count	42	56	72	90	110	n(n+1)
Facet Count trunc. facets	7`tix`	8`til`	9`toc`	10`tene`	11`teday`	n+1
Facet Count verf facets	7`hix`	8`hop`	9`oca`	10`ene`	11`day`	n+1
Circumradius	sqrt(13/7) 1.362771	sqrt(31)/4 1.391941	sqrt(2) 1.414214	sqrt(41/20) 1.431782	sqrt(23/11) 1.445998	sqrt[(5n-4)/(2n+2)]
Inradius wrt. trunc. facets	sqrt(3/28) 0.327327	3/sqrt(112) 0.283473	1/4 0.25	1/sqrt(20) 0.223607	3/sqrt(220) 0.202260	3/sqrt[2n(n+1)]
Inradius wrt. verf facets	11/sqrt(84) 1.200198	13/sqrt(112) 1.228385	5/4 1.25	17/sqrt(180) 1.267105	19/sqrt(220) 1.280980	(2n-1)/sqrt[2n(n+1)]
Volume	361 sqrt(7)/2880 0.331638	2179/20160 0.108085	39/1280 0.030469	19673 sqrt(5)/5806080 0.0075766	4217 sqrt(11)/8294400 0.0016862	(3ⁿ-n-1) sqrt[(n+1)/(2ⁿ)]/n!
Surface	833 sqrt(3)/240 6.011660	241 sqrt(7)/240 2.656775	109/112 0.973214	6553/21504 0.304734	12023 sqrt(5)/322560 0.083346	(n+1)(3^n-1-n+1) sqrt[n/(2^n-1)]/(n-1)!
Dihedral angles trunc - trunc	arccos(1/6) 80.405932°	arccos(1/7) 81.786789°	arccos(1/8) 82.819244°	arccos(1/9) 83.620630°	arccos(1/10) 84.260830°	arccos(1/n)
Dihedral angles verf - trunc	arccos(-1/6) 99.594068°	arccos(-1/7) 98.213211°	arccos(-1/8) 97.180756°	arccos(-1/9) 96.379370°	arccos(-1/10) 95.739170°	arccos(-1/n)

Bitruncated Simplex btS_n (up)

Within these polytopes btS_n for n>3 can be described as thebistratic lace towerof thetruncated simplex tS_n-1 atop anu-scaledrectified simplex rS_n-1 atop the bitruncated simplex btS_n-1.Thence, by means of thelace tower notation, btSn =o3x3x3o...o3o (n nodes) can be described aswell asxoo3xux3oox3ooo...ooo3ooo&#xt (n-1 node positions).A posteriori that latter lace tower then applies even for n=3 too, thereby quite similarily just reducing to its first 2 node positions.

Dimension	3D	4D	5D	6D	nD
Dynkin diagram	o3x3x	o3x3x3o	o3x3x3o3o	o3x3x3o3o3o	o3x3x3o...o3o
Acronym	tut	deca	bittix	batal	bitrunc. n-simplex
Vertex Count	12	30	60	105	(n+1)n(n-1)/2
Facet Count bitrunc. fac.	4`trig`	5`tut`	6`deca`	7`bittix`	n+1
Facet Count trunc. fac.	4`hig`	5`tut`	6`tip`	7`tix`	n+1
Circumradius	sqrt(11/8) 1.172604	sqrt(2) 1.414214	sqrt(29/12) 1.554563	sqrt(19/7) 1.647509	sqrt[(9n-16)/(2n+2)]
Inradius bitrunc. fac.	5/sqrt(24) 1.020621	sqrt(5/8) 0.790569	sqrt(5/12) 0.645497	5/sqrt(84) 0.545545	5/sqrt[2n(n+1)]
Inradius trunc. fac.	sqrt(3/8) 0.612372	sqrt(5/8) 0.790569	7/sqrt(60) 0.903696	sqrt(27/28) 0.981981	(2n-3)/sqrt[2n(n+1)]
Volume	23 sqrt(2)/12 2.710576	115 sqrt(5)/48 5.357246	841 sqrt(3)/240 6.069395	?	?
Surface	7 sqrt(3) 12.124356	115 sqrt(2)/6 27.105760	153 sqrt(5)/8 42.764800	?	?
Dihedral angles bitrunc - bitrunc	7 sqrt(3) 12.124356	arccos(1/4) 75.522488°	arccos(1/5) 78.463041°	arccos(1/6) 80.405932°	arccos(1/n)
Dihedral angles bitrunc - trunc	arccos(-1/3) 109.471221°	arccos(-1/4) 104.477512°	arccos(-1/5) 101.536959°	arccos(-1/6) 99.594068°	arccos(-1/n)
Dihedral angles trunc - trunc	arccos(1/3) 70.528779°	arccos(1/4) 75.522488°	arccos(1/5) 78.463041°	arccos(1/6) 80.405932°	arccos(1/n)
Dimension	7D	8D	9D	10D	nD
Dynkin diagram	o3x3x3o3o3o3o	o3x3x3o3o3o3o3o	o3x3x3o3o3o3o3o3o	o3x3x3o3o3o3o3o3o3o	o3x3x3o...o3o
Acronym	bittoc	batene	biteday	?	bitrunc. n-simplex
Vertex Count	168	252	360	495	(n+1)n(n-1)/2
Facet Count bitrunc. fac.	8`batal`	9`bittoc`	10`batene`	11`biteday`	n+1
Facet Count trunc. fac.	8`til`	9`toc`	10`tene`	11`teday`	n+1
Circumradius	sqrt(47)/4 1.713914	sqrt(29)/3 1.795055	sqrt(13)/2 1.802776	sqrt(37/11) 1.834022	sqrt[(9n-16)/(2n+2)]
Inradius bitrunc. fac.	5/sqrt(112) 0.472456	5/12 0.416667	sqrt(5)/6 0.372678	sqrt(5/44) 0.337100	5/sqrt[2n(n+1)]
Inradius trunc. fac.	11/sqrt(112) 1.039402	13/12 1.083333	sqrt(45)/6 1.118034	17/sqrt(220) 1.146140	(2n-3)/sqrt[2n(n+1)]
Volume	?	?	?	?	?
Surface	?	?	?	?	?
Dihedral angles bitrunc - bitrunc	arccos(1/7) 81.786789°	arccos(1/8) 82.819244°	arccos(1/9) 83.620630°	arccos(1/10) 84.260830°	arccos(1/n)
Dihedral angles bitrunc - trunc	arccos(-1/7) 98.213211°	arccos(-1/8) 97.180756°	arccos(-1/9) 96.379370°	arccos(-1/10) 95.739170°	arccos(-1/n)
Dihedral angles trunc - trunc	arccos(1/7) 81.786789°	arccos(1/8) 82.819244°	arccos(1/9) 83.620630°	arccos(1/10) 84.260830°	arccos(1/n)

Mid-rectified Simplex mrS_n (up)

This case applies toodd dimensions only. These also occur (scaled down) as intersection kernels of facet-regularbi-simplex compounds.Further they occur (again scaled) as equatorial mid-sections of the vertex-first oriented (then even-dimensional)hypercube C_n+1.

Note that those can be generally provided too as next-to-center rectified simplex alterprismsoo3oo3...xo3ox...3oo3oo&#x (n-1 node positions).

Dimension	1D	3D	5D	7D	9D	nD (2k+1)D
Dynkin diagram	x	o3x3o	o3o3x3o3o	o3o3o3x3o3o3o	o3o3o3o3x3o3o3o3o	o3o...o3x3o...o3o
Acronym	line	oct	dot	he	icoy	mid-rect. n-simplex
Vertex Count	2	6`square`	20`triddip`	70`tetdip`	252`pendip`	(n+1)!/[((n+1)/2)!]² (2(k+1))!/((k+1)!)²
Facet Count	2	4+4`trig`	6+6`rap`	8+8`bril`	10+10`trene`	2(n+1) 4(k+1)
Circumradius	1/2 0.5	1/sqrt(2) 0.707107	sqrt(3)/2 0.866025	1	sqrt(5)/2 1.118034	sqrt[(n+1)/8] sqrt(k+1)/2
Inradius	1/2 0.5	1/sqrt(6) 0.408248	sqrt(3/20) 0.387298	1/sqrt(7) 0.377964	sqrt(5)/6 0.372678	sqrt[(n+1)/(8n)] sqrt[(k+1)/(8k+4)]
Volume	1	sqrt(2)/3 0.471405	11 sqrt(3)/80 0.238157	151/1260 0.119841	15619 sqrt(5)/580608 0.060153	?
Surface	2	2 sqrt(3) 3.464102	11 sqrt(5)/8 3.074593	151 sqrt(7)/180 2.219491	?	?
Dihedral angles wrt. mid-rect margin	0°	arccos(-1/3) 109.471221°	arccos(-1/5) 101.536959°	arccos(-1/7) 98.213211°	arccos(-1/9) 96.379370°	arccos(-1/n)
Dihedral angles wrt. offset margin	0°	arccos(-1/3) 109.471221°	arccos(1/5) 78.463041°	arccos(1/7) 81.786789°	arccos(1/9) 83.620630°	arccos(1/n)

Mid-truncated Simplex mtS_n (up)

This case applies toeven dimensions only. These also occur (scaled down) as intersection kernels of facet-regularbi-simplex compounds.Further they occur (again scaled) as equatorial mid-sections of the vertex-first oriented (then odd-dimensional)hypercube C_n+1.

Dimension	2D	4D	6D	8D	10D	nD (2k)D
Dynkin diagram	x3x	o3x3x3o	o3o3x3x3o3o	o3o3o3x3x3o3o3o	o3o3o3o3x3x3o3o3o3o	o3o...o3x3x3o...o3o
Acronym	hig	deca	fe	be	?	mid-trunc. n-simplex
Vertex Count	6	30	140	630	2772	(n+1)!/((n/2)!)² (2k+1)!/(k!)²
Facet Count	3+3`line`	5+5`tut`	7+7`bittix`	9+9`tattoc`	11+11`?`	2(n+1) 2(2k+1)
Circumradius	1	sqrt(2) 1.414214	sqrt(3) 1.732051	2	sqrt(5) 2.236068	sqrt(n/2) sqrt(k)
Inradius	sqrt(3)/2 0.866025	sqrt(5/8) 0.790569	sqrt(7/12) 0.763763	3/4 0.75	sqrt(11/20) 0.741620	sqrt[(n+1)/(2n)] sqrt[(2k+1)/(4k)]
Volume	3 sqrt(3)/2 2.598076	115 sqrt(5)/48 5.357246	5887 sqrt(7)/1440 10.816346	?	?	?
Surface	6	115 sqrt(2)/6 27.105760	5887 sqrt(3)/120 84.971526	?	?	?
Dihedral angles wrt. mid-trunc margin	120°	arccos(-1/4) 104.477512°	arccos(-1/6) 99.594068°	arccos(-1/8) 97.180756°	arccos(-1/10) 95.739170°	arccos(-1/n)
Dihedral angles wrt. offset margin	120°	arccos(1/4) 75.522488°	arccos(1/6) 80.405932°	arccos(1/8) 82.819244°	arccos(1/10) 84.260830°	arccos(1/n)

Maximal Expanded Simplex eS_n (up)

The common unit circumradius of all these shows that they occur asvertex figure of an according dimensionalhoneycomb. In fact they are the hull-of-roots polytopes of the according dimensionalroot lattice A_n.Furthermore it forces that the facet-to-bodycenter pyramids all areCRF, i.e. that all these polytopes can bedecomposed accordingly.

Within these polytopes eS_n generally can be described as thebistratic lace tower oftheregular simplex S_n-1 atop the maximal expanded simplex eS_n-1 atop the dual regular simplex -S_n-1.Thence, by means of thelace tower notation,eS_n = x3o...o3x (n nodes) can be described as well asxxo3ooo...ooo3oxx&#xt (n-1 node positions).Note that the midsection here is of the very same form eS_n-1, just one dimension less. Therefore that mentioned unit circumradius property here simply follows by dimensional induction.

Dimension	1D	2D	3D	4D	5D	nD
Dynkin diagram		x3x	x3o3x	x3o3o3x	x3o3o3o3x	x3o3o...o3o3x
Acronym		hig	co	spid	scad	max-exp. n-simplex
Vertex Count		6	12	20	30	n(n+1)
Facet Count simplex		3+3`line`	4+4`trig`	5+5`tet`	6+6`pen`	n+1 per type
Facet Count prism			6`square`	10+10`trip`	15+15`tepe`	n(n+1)/2 per type
Facet Count duoprism I			6`square`	10+10`trip`	20`triddip`	(n+1)n(n-1)/6 per type
Circumradius		1	1	1	1	1
Inradius wrt. simplex facets		sqrt(3)/2 0.866025	sqrt(2/3) 0.816497	sqrt(5/8) 0.790569	sqrt(3/5) 0.774597	sqrt[(n+1)/2n]
Inradius wrt. prism facets			1/sqrt(2) 0.707107	sqrt(5/12) 0.645497	sqrt(3/8) 0.612372	sqrt[(n+1)/(4n-4)]
Inradius wrt. d.pr. I fac.			1/sqrt(2) 0.707107	sqrt(5/12) 0.645497	1/sqrt(3) 0.577350	sqrt[(n+1)/(6n-12)]
Volume		3 sqrt(3)/2 2.598076	5 sqrt(2)/3 2.357023	35 sqrt(5)/48 1.630466	21 sqrt(3)/40 0.909327	(2n)! sqrt[(n+1)/(2ⁿ)]/(n!)³
Surface		6	6+2 sqrt(3) 9.464102	5 sqrt(2)/6+5 sqrt(3) 9.838765	(30+20 sqrt(2)+sqrt(5))/8 7.565042	?
Dihedral angles simplex - (next)		120°	arccos[-1/sqrt(3)] 125.264390°	arccos(-sqrt(3/8)) 127.761244°	arccos[-sqrt(2/5)] 129.231520°	arccos[-sqrt((n-1)/2n)]
Dihedral angles prism - (next)		120°	arccos[-1/sqrt(3)] 125.264390°	arccos(-2/3) 131.810315°	135°	arccos[-sqrt((2n-4)/(3n-3)]
Dimension	6D	7D	8D	9D	10D	nD
Dynkin diagram	x3o3o3o3o3x	x3o3o3o3o3o3x	x3o3o3o3o3o3o3x	x3o3o3o3o3o3o3o3x	x3o3o3o3o3o3o3o3o3x	x3o3o...o3o3x
Acronym	staf	suph	soxeb	?	?	max-exp. n-simplex
Vertex Count	42	56	72	90	110	n(n+1)
Facet Count simplex	7+7`hix`	8+8`hop`	9+9`oca`	10+10`ene`	11+11`day`	n+1 per type
Facet Count prism	21+21`penp`	28+28`hixip`	36+36`hopip`	45+45`ocpe`	55+55`enep`	(n+1)n/2 per type
Facet Count duoprism I	35+35`tratet`	56+56`trapen`	84+84`trahix`	120+120`trihop`	165+165`trioc`	(n+1)n(n-1)/6 per type
Facet Count duoprism II		70`tetdip`	126+126`tetpen`	210+210`tethix`	330+330`tethop`	(n+1)n(n-1)(n-2)/24 per type
Facet Count duoprism III		70`tetdip`	126+126`tetpen`	252`pendip`	462+462`penhix`	(n+1)n(n-1)(n-2)(n-3)/120 per type
Circumradius	1	1	1	1	1	1
Inradius wrt. simplex facets	sqrt(7/12) 0.763763	2/sqrt(7) 0.755929	3/4 0.75	sqrt(5)/3 0.745356	sqrt(11/20) 0.741620	sqrt[(n+1)/2n]
Inradius wrt. prism facets	sqrt(7/20) 0.591608	1/sqrt(3) 0.577350	3/sqrt(28) 0.566947	sqrt(5)/4 0.559017	sqrt(11)/6 0.552771	sqrt[(n+1)/(4n-4)]
Inradius wrt. d.pr. I fac.	sqrt(7/24) 0.540062	2/sqrt(15) 0.516398	1/2 0.5	sqrt(5/21) 0.487950	sqrt(11/48) 0.478714	sqrt[(n+1)/(6n-12)]
Inradius wrt. d.pr. II fac.		1/2 0.5	3/sqrt(40) 0.474342	sqrt(5/24) 0.456435	sqrt(11/56) 0.443203	sqrt[(n+1)/(8n-24)]
Inradius wrt. d.pr. III fac.		1/2 0.5	3/sqrt(40) 0.474342	1/sqrt(5) 0.447214	sqrt(11/60) 0.428174	sqrt[(n+1)/(10n-40)]
Volume	77 sqrt(7)/480 0.424423	143/840 0.170238	429/7168 0.059849	2431 sqrt(5)/290304 0.018725	46189 sqrt(11)/29030400 0.0052769	(2n)! sqrt[(n+1)/(2ⁿ)]/(n!)³
Surface	7[sqrt(3)+15 sqrt(5)+50 sqrt(6)]/240 4.600970	[350+42 sqrt(3)+sqrt(7)+105 sqrt(15)]/360 2.311264	?	?	?	?
Dihedral angles simplex - (next)	arccos[-sqrt(5/12)] 130.202966°	arccos[-sqrt(3/7)] 130.893395°	arccos[-sqrt(7)/4] 131.409622°	arccos(-2/3) 131.810315°	arccos[-3/sqrt(20)] 132.130415°	arccos[-sqrt((n-1)/2n)]
Dihedral angles prism - (next)	arccos[-sqrt(8/15)] 136.911277°	arccos[-sqrt(5)/3] 138.189685°	arccos[-2/sqrt(7)] 139.106605°	arccos[-sqrt(7/12)] 139.797034°	arccos[-4/sqrt(27)] 140.335965°	arccos[-sqrt((2n-4)/(3n-3))]
Dihedral angles d.pr. I - (next)	?	?	?	?	?	?
Dihedral angles d.pr. II - (next)			?	?	?	?
Dihedral angles d.pr. III - (next)			?	?	?	?

Interestingly this class belongs to an even wider class of (then mostlyhyperbolic) polytopes which all have that common property that the nD (or rather: rank n) representant occurs asridge faceting midsection within the (n+1)D case (for the finite cases) resp. as aridge faceting subspace within the rank n+1 case (for the infinite cases). This then is the general maximal-expanded classxPo3o...o3oPx, or, even more general, the class ofcyclotruncatedxPo3o...o3oPxQ*a. Below is a small enlisting thereof.

	P = 3	P = 4	P = 5	P = 6
	xPo3o...o3oPxQ*a
Q = 2	r = 1 x3o3x -cox3o3o3x -spidx3o3o3o3x -scadx3o3o3o3o3x -stafx3o3o3o3o3o3x -suphx3o3o3o3o3o3o3x -soxeb...	r = ∞ x4o4x -squatx4o3o4x -chonx4o3o3o4x -testx4o3o3o3o4x -penthx4o3o3o3o3o4x -axhx4o3o3o3o3o3o4x -hepth...	r = sqrt[-(1+sqrt(5))/2] = 1.272020i x5o5x -tepetx5o3o5x -spidded x5o3o3o5x...	r = sqrt(-1) = 1i x6o6x -tehatx6o3o6x -spiddihexah...
Q = 3	r = ∞ x3o3x3a -thatx3o3o3x3a -batatohx3o3o3o3x3a -cytopitx3o3o3o3o3x3a -cytaxhx3o3o3o3o3o3x3*a -cytloh...	r = sqrt(-1) = 1i x4o4x3a -tehatx4o3o4x3a -cytochx4o3o3o4x3*a...	r = sqrt[-(sqrt(5)-1)/2] = 0.786151i x5o5x3a -phat x5o3o5x3a...	r = 1/sqrt(-2) = 0.707107i x6o6x3*a -shexat...
Q = 4	r = sqrt(-1) = 1i x3o3x4a -tehatx3o3o3x4a -cyticthx3o3o3o3x4*a...	r = 1/sqrt[-sqrt(2)] = 0.840896i x4o4x4a -teoctx4o3o4x4a...	r = sqrt[(3+sqrt(2)-sqrt(5)-sqrt(10))/2] = 0.701474i x5o5x4*a...	r = sqrt[-(sqrt(2)-1)] = 0.643594i x6o6x4*a...

Retroexpanded Simplex reS_n (up)

These non-convex polytopes reS_n generally are facetings of themaximal expanded simplexeS_n.

While one third of the facets always are hemifacets and the second third could be considered to be prograde throughout,the remaining one would alternate wrt. its retrogradeness.Thence for the odd dimensional series members thevolume always results in zero.

Dimension	3D	4D	5D	6D	nD
Dynkin diagram	o3x3x3/2*a	o3x3x3/2*a3o	o3x3x3/2*a3o3o	o3x3x3/2*a3o3o3o	o3x3x3/2*a3o...o3o
Acronym	oho	duhd	dehad	fohaf	retroexp. n-simplex
Vertex Count	12	20	30	42	n(n+1)
Facet Count simplex	4+4`trig`	5+5`tet`	6+6`pen`	7+7`hix`	n+1 (each)
Facet Count retroexp. simp.	4`hig`	5`oho`	6`duhd`	7`dehad`	n+1
Circumradius	1	1	1	1	1
Inradius wrt. simplex facets	sqrt(2/3) 0.816497	sqrt(5/8) 0.790569	sqrt(3/5) 0.774597	sqrt(7/12) 0.763763	sqrt[(n+1)/2n]
Inradius wrt. retroexp. simp.	0	0	0	0	0
Volume	0	5 sqrt(5)/8 0.232924	0	7 sqrt(7)/2880 0.0064306	0 / sqrt[(n+1)³/2^n-2]/n!
Surface	8 sqrt(3) 13.856406	5 sqrt(2)/6 1.178511	31 sqrt(5)/8 8.664763	7 sqrt(3)/240 0.050518	?
Dihedral angles sim. - r.exp. sim.	arccos(1/3) 70.528779°	arccos(1/4) 75.522488°	arccos(1/5) 78.463041°	arccos(1/6) 80.405932°	arccos(1/n)
Dihedral angles r.exp. s. - r.exp. s.	arccos(1/3) 70.528779°	arccos(1/4) 75.522488°	arccos(1/5) 78.463041°	arccos(1/6) 80.405932°	arccos(1/n)
Dimension	7D	8D	9D	10D	nD
Dynkin diagram	o3x3x3/2*a3o3o3o3o	o3x3x3/2*a3o3o3o3o3o	o3x3x3/2*a3o3o3o3o3o3o	o3x3x3/2*a3o3o3o3o3o3o3o	o3x3x3/2*a3o...o3o
Acronym	hehah	?	?	?	retroexp. n-simplex
Vertex Count	56	72	90	110	n(n+1)
Facet Count simplex	8+8`hop`	9+9`oca`	10+10`ene`	11+11`day`	n+1 per type
Facet Count prism	8`fohaf`	9`hehah`	10`?`	11`?`	n+1
Circumradius	1	1	1	1	1
Inradius wrt. simplex	2/sqrt(7) 0.755929	3/4 0.75	sqrt(5)/3 0.745356	sqrt(11/20) 0.741620	sqrt[(n+1)/2n]
Inradius wrt. retroexp. simp.	0	0	0	0	0
Volume	0	3/35840 0.000083705	0	11 sqrt(11)/58060800 0.00000062836	0 / sqrt[(n+1)³/2^n-2]/n!
Surface	?	?	?	?	?
Dihedral angles sim. - r.exp. sim.	arccos(1/7) 81.786789°	arccos(1/8) 82.819244°	arccos(1/9) 83.620630°	arccos(1/10) 84.260830°	arccos(1/n)
Dihedral angles r.exp. sim. - r.exp. sim.	arccos(1/7) 81.786789°	arccos(1/8) 82.819244°	arccos(1/9) 83.620630°	arccos(1/10) 84.260830°	arccos(1/n)

Omnitruncated Simplex otS_n (up)

These polytopes otS_n also are known aspermutotopes P_n+1 and in fact the set of their vertices each can be found to be in one-to-one correspondence, that is being mapped from and therefore being labeled by the permutations of the first n+1 natural numbersin such a way that the edges will represent the set oftranspositions (permutations of any 2 elements only).

This very labeling moreover shows that each otS_n also can be represented within an n-dimensional subspace ofthe (n+1)-dimensional space, where that labeling just is given by the respective all-integer coordinates.In fact this representation then is nothing but a sqrt(2)-scaled version of otS_n when being used asone of the facets of the also sqrt(2)-scaled otS_n+1.

It further should be mentioned that otS_n generally is also the Voronoi cell of theroot lattice A_n^*. It therefore always allows for anoble periodic continuation as aeuclidean honeycomb, the Voronoi complex V(A_n^*) =x3x3x3...x3*a.

Dimension	1D	2D	3D	4D	5D	nD
Dynkin diagram	x	x3x	x3x3x	x3x3x3x	x3x3x3x3x	x3x...x3x
Acronym	dyad	hig	toe	gippid	gocad	omnitr. n-simplex
Vertex Count	2	6	24	120	720	(n+1)!
Facet Count wrt. type 1	2`vertex`	3`line`	4`hig`	5`toe`	6`gippid`	n+1 (n+1)!/[n! 1!]
Facet Count wrt. type 2		3`line`	6`square`	10`hip`	15`tope`	n(n+1)/2 (n+1)!/[(n-1)! 2!]
Facet Count wrt. type 3			4`hig`	10`hip`	20`hiddip`	(n+1)!/[(n-2)! 3!]
Facet Count wrt. type 4				5`toe`	15`tope`	(n+1)!/[(n-3)! 4!]
Facet Count wrt. type 5				5`toe`	6`gippid`	(n+1)!/[(n-4)! 5!]
Circumradius	1/2 0.5	1	sqrt(5/2) 1.581139	sqrt(5) 2.236068	sqrt(35)/2 2.958040	sqrt[(n+2)!/((n-1)! 4!)]
Inradius wrt. facet type 1	1/2 0.5	sqrt(3)/2 0.866025	sqrt(3/2) 1.224745	sqrt(5/2) 1.581139	sqrt(15)/2 1.936492	sqrt[(n²+n)/8]
Inradius wrt. facet type 2		sqrt(3)/2 0.866025	sqrt(2) 1.414214	sqrt(15)/2 1.936492	sqrt(6) 2.449490	sqrt(n²-1)/2
Inradius wrt. facet type 3			sqrt(3/2) 1.224745	sqrt(15)/2 1.936492	sqrt(27)/2 2.598076	sqrt[3(n²-n-2)/8]
Inradius wrt. facet type 4				sqrt(5/2) 1.581139	sqrt(6) 2.449490	sqrt[(n²-2n-3)/2]
Inradius wrt. facet type 5				sqrt(5/2) 1.581139	sqrt(15)/2 1.936492	sqrt[5(n²-3n-4)/8]
Volume	1	3 sqrt(3)/2 2.598076	8 sqrt(2) 11.313708	125 sqrt(5)/4 69.877124	324 sqrt(3) 561.184462	(n+1)^n-1 sqrt[(n+1)/(2ⁿ)]
Surface	2	6	6+12 sqrt(3) 26.784610	?	?	?
Dihedral angles types 1 - 2		120°	arccos[-1/sqrt(3)] 125.264390°	arccos[-sqrt(3/8)] 127.761244°	arccos[-sqrt(2/5)] 129.231520°	arccos[-sqrt((n-1)/2n)]
Dihedral angles types 1 - 3			arccos(-1/3) 109.471221°	arccos[-1/sqrt(6)] 114.094843°	arccos[-1/sqrt(5)] 116.565051°	?
Dihedral angles types 1 - 4				arccos(-1/4) 104.477512°	arccos[-1/sqrt(10)] 108.434949°	?
Dihedral angles types 1 - 5				arccos(-1/4) 104.477512°	arccos(-1/5) 101.536959°	?
Dihedral angles types 2 - 3			arccos[-1/sqrt(3)] 125.264390°	arccos(-2/3) 131.810315°	135°	arccos[-sqrt([2(n-2)]/[3(n-1)])]
Dihedral angles types 2 - 4				arccos[-1/sqrt(6)] 114.094843°	120°	?
Dihedral angles types 2 - 5				arccos[-1/sqrt(6)] 114.094843°	arccos[-1/sqrt(10)] 108.434949°	?
Dihedral angles types 3 - 4				arccos[-sqrt(3/8)] 127.761244°	135°	arccos[-sqrt([3(n-3)]/[4(n-2)])]
Dihedral angles types 3 - 5					arccos[-1/sqrt(5)] 116.565051°	?
Dihedral angles types 4 - 5					arccos[-sqrt(2/5)] 129.231520°	arccos[-sqrt([4(n-4)]/[5(n-3)])]
Dimension	6D	7D	8D	9D	10D	nD
Dynkin diagram	x3x3x3x3x3x	x3x3x3x3x3x3x	x3x3x3x3x3x3x3x	x3x3x3x3x3x3x3x3x	x3x3x3x3x3x3x3x3x3x	x3x...x3x
Acronym	gotaf	guph	goxeb	?	?	omnitr. n-simplex
Vertex Count	5040	40320	362880	3628800	39916800	(n+1)!
Facet Count wrt. type 1	7`gocad`	8`gotaf`	9`guph`	10`goxeb`	11`?`	n+1 (n+1)!/[n! 1!]
Facet Count wrt. type 2	21`gippiddip`	28`gocadip`	36`gotafip`	45`guphip`	55`?`	n(n+1)/2 (n+1)!/[(n-1)! 2!]
Facet Count wrt. type 3	35`hatoe`	56`hagippid`	84`hagocad`	120`hagotaf`	165`haguph`	(n+1)!/[(n-2)! 3!]
Facet Count wrt. type 4	35`hatoe`	70`toedip`	126`toegippid`	210`toegocad`	330`toegotaf`	(n+1)!/[(n-3)! 4!]
Facet Count wrt. type 5	21`gippiddip`	56`hagippid`	126`toegippid`	252`?`	462`?`	(n+1)!/[(n-4)! 5!]
Facet Count wrt. type 6	7`gocad`	28`gocadip`	84`hagocad`	210`toegocad`	462`?`	(n+1)!/[(n-5)! 6!]
Facet Count wrt. type 7		8`gotaf`	36`gotafip`	120`hagotaf`	330`toegotaf`	(n+1)!/[(n-6)! 7!]
Facet Count wrt. type 8			9`guph`	45`guphip`	165`haguph`	(n+1)!/[(n-7)! 8!]
Facet Count wrt. type 9				10`goxeb`	55`?`	(n+1)!/[(n-8)! 9!]
Facet Count wrt. type 10				10`goxeb`	11`?`	(n+1)!/[(n-9)! 10!]
Circumradius	sqrt(14) 3.741657	sqrt(21) 4.582576	sqrt(30) 5.477226	sqrt(165)/2 6.422616	sqrt(55) 7.416198	sqrt[(n+2)!/((n-1)! 4!)]
Inradius wrt. facet type 1	sqrt(21)/2 2.291288	sqrt(7) 2.645751	3	3 sqrt(5)/2 3.354102	sqrt(55)/2 3.708099	sqrt[(n²+n)/8]
Inradius wrt. facet type 2	sqrt(35)/2 2.958040	2 sqrt(3) 3.464102	sqrt(63)/2 3.968627	2 sqrt(5) 4.472136	3 sqrt(11)/2 4.974937	sqrt(n²-1)/2
Inradius wrt. facet type 3	sqrt(21/2) 3.240370	sqrt(15) 3.872983	9/2 4.5	sqrt(105)/2 5.123475	sqrt(33) 5.744563	sqrt[3(n²-n-2)/8]
Inradius wrt. facet type 4	sqrt(21/2) 3.240370	4	3 sqrt(5/2) 4.743416	sqrt(30) 5.477226	sqrt(77/2) 6.204837	sqrt[(n²-2n-3)/2]
Inradius wrt. facet type 5	sqrt(35)/2 2.958040	sqrt(15) 3.872983	3 sqrt(5/2) 4.743416	5 sqrt(5)/2 5.590170	sqrt(165)/2 6.422616	sqrt[5(n²-3n-4)/8]
Inradius wrt. facet type 6	sqrt(21)/2 2.291288	2 sqrt(3) 3.464102	9/2 4.5	sqrt(30) 5.477226	sqrt(165)/2 6.422616	sqrt[3(n²-4n-5)]/2
Inradius wrt. facet type 7		sqrt(7) 2.645751	sqrt(63)/2 3.968627	sqrt(105)/2 5.123475	sqrt(77/2) 6.204837	sqrt[7(n²-5n-6)/8]
Inradius wrt. facet type 8			3	2 sqrt(5) 4.472136	sqrt(33) 5.744563	sqrt(n²-6n-7)
Inradius wrt. facet type 9				3 sqrt(5)/2 3.354102	3 sqrt(11)/2 4.974937	3 sqrt[(n²-7n-8)/8]
Inradius wrt. facet type 10				3 sqrt(5)/2 3.354102	sqrt(55)/2 3.708099	sqrt[5(n²-8n-9)]/2
Volume	16807 sqrt(7)/8 5558.392786	65536	14348907/16 896806.6875	?	?	(n+1)^n-1 sqrt[(n+1)/(2ⁿ)]
Surface	?	?	?	?	?	?
Dihedral angles types 1 - 2	arccos[-sqrt(5/12)] 130.202966°	arccos[-sqrt(3/7)] 130.893395°	arccos[-sqrt(7)/4] 131.409622°	arccos(-2/3) 131.810315°	arccos[-3/sqrt(20)] 132.130415°	arccos[-sqrt((n-1)/2n)]
Dihedral angles types 1 - 3	?	?	?	?	?	?
Dihedral angles types 1 - 4	?	?	?	?	?	?
Dihedral angles types 1 - 5	?	?	?	?	?	?
Dihedral angles types 1 - 6	?	?	?	?	?	?
Dihedral angles types 1 - 7		?	?	?	?	?
Dihedral angles types 1 - 8			?	?	?	?
Dihedral angles types 1 - 9				?	?	?
Dihedral angles types 1 - 10				?	?	?
Dihedral angles types 2 - 3	arccos[-sqrt(8/15)] 136.911277°	arccos[-sqrt(5)/3] 138.189685°	arccos[-2/sqrt(7)] 139.106605°	arccos[-sqrt(7/12)] 139.797034°	arccos[-4/sqrt(27)] 140.335965°	arccos[-sqrt([2(n-2)]/[3(n-1)])]
Dihedral angles types 2 - 4	?	?	?	?	?	?
Dihedral angles types 2 - 5	?	?	?	?	?	?
Dihedral angles types 2 - 6	?	?	?	?	?	?
Dihedral angles types 2 - 7		?	?	?	?	?
Dihedral angles types 2 - 8			?	?	?	?
Dihedral angles types 2 - 9				?	?	?
Dihedral angles types 2 - 10				?	?	?
Dihedral angles types 3 - 4	arccos(-3/4) 138.590378°	arccos[-sqrt(3/5)] 140.768480°	arccos[-sqrt(5/8)] 142.238756°	arccos[-3/sqrt(14)] 143.300775°	arccos[-sqrt(21/32)] 144.104978°	arccos[-sqrt([3(n-3)]/[4(n-2)])]
Dihedral angles types 3 - 5	?	?	?	?	?	?
Dihedral angles types 3 - 6	?	?	?	?	?	?
Dihedral angles types 3 - 7		?	?	?	?	?
Dihedral angles types 3 - 8			?	?	?	?
Dihedral angles types 3 - 9				?	?	?
Dihedral angles types 3 - 10				?	?	?
Dihedral angles types 4 - 5	arccos[-sqrt(8/15)] 136.911277°	arccos[-sqrt(3/5)] 140.768480°	arccos(-4/5) 143.130102°	arccos[-sqrt(2/3)] 144.735610°	arccos[-sqrt(24/35)] 145.901874°	arccos[-sqrt([4(n-4)]/[5(n-3)])]
Dihedral angles types 4 - 6	?	?	?	?	?	?
Dihedral angles types 4 - 7		?	?	?	?	?
Dihedral angles types 4 - 8			?	?	?	?
Dihedral angles types 4 - 9				?	?	?
Dihedral angles types 4 - 10				?	?	?
Dihedral angles types 5 - 6	arccos[-sqrt(5/12)] 130.202966°	arccos[-sqrt(5)/3] 138.189685°	arccos[-sqrt(5/8)] 142.238756°	arccos[-sqrt(2/3)] 144.735610°	arccos(-5/6) 146.442690°	arccos[-sqrt([5(n-5)]/[6(n-4)])]
Dihedral angles types 5 - 7		?	?	?	?	?
Dihedral angles types 5 - 8			?	?	?	?
Dihedral angles types 5 - 9				?	?	?
Dihedral angles types 5 - 10				?	?	?
Dihedral angles types 6 - 7		arccos[-sqrt(3/7)] 130.893395°	arccos[-2/sqrt(7)] 139.106605°	arccos[-3/sqrt(14)] 143.300775°	arccos[-sqrt(24/35)] 145.901874°	arccos[-sqrt([6(n-6)]/[7(n-5)])]
Dihedral angles types 6 - 8			?	?	?	?
Dihedral angles types 6 - 9				?	?	?
Dihedral angles types 6 - 10				?	?	?
Dihedral angles types 7 - 8			arccos[-sqrt(7)/4] 131.409622°	arccos[-sqrt(7/12)] 139.797034°	arccos[-sqrt(21/32)] 144.104978°	arccos[-sqrt([7(n-7)]/[8(n-6)])]
Dihedral angles types 7 - 9				?	?	?
Dihedral angles types 7 - 10				?	?	?
Dihedral angles types 8 - 9				arccos(-2/3) 131.810315°	arccos[-4/sqrt(27)] 140.335965°	arccos[-sqrt([8(n-8)]/[9(n-7)])]
Dihedral angles types 8 - 10					?	?
Dihedral angles types 9 - 10					arccos[-3/sqrt(20)] 132.130415°	arccos[-sqrt([9(n-9)]/[10(n-8)])]

Symmetry BC_n

Regular Orthoplex O_n (up)

These polytopes are closely related to thetegum product.In fact O_n here is nothing but the O_n-1bipyramid. Thence, by means of thetegum sum notation,O_n = x3o...o3o4o (n nodes) can be described as well asqo ox3oo...oo3oo4oo&#zx (n node positions).

On the other hand these polytopes O_n generally can also be described as thesegmentotope oftheregular simplex S_n-1 atop the dual simplex -S_n-1.Thence, by means of thelace prism notation,O_n = x3o...o3o4o (n nodes) can be described as well asxo3oo...oo3ox&#x (n-1 node positions).

The regular Orthoplex O_n generally is the dual of theregular hypercube C_n.

Dimension	1D	2D	3D	4D	5D	nD
Dynkin diagram	q	x4o	x3o4o	x3o3o4o	x3o3o3o4o	x3o...o3o4o
Acronym	q-line	square	oct	hex	tac	n-orthoplex
Vertex Count	2	4`q-line`	6`square`	8`oct`	10`hex`	2n
Facet Count	2	4`line`	8`trig`	16`tet`	32`pen`	2ⁿ
Circumradius	1/sqrt(2) 0.707107	1/sqrt(2) 0.707107	1/sqrt(2) 0.707107	1/sqrt(2) 0.707107	1/sqrt(2) 0.707107	1/sqrt(2) 0.707107
Inradius	1/sqrt(2) 0.707107	1/2 0.5	1/sqrt(6) 0.408248	1/sqrt(8) 0.353553	1/sqrt(10) 0.316228	1/sqrt(2n)
Volume	sqrt(2) 1.414214	1	sqrt(2)/3 0.471405	1/6 0.166667	sqrt(2)/30 0.047140	sqrt(2ⁿ)/n!
Surface	2	4	2 sqrt(3) 3.464102	4 sqrt(2)/3 1.885618	sqrt(5)/3 0.745356	2 sqrt[2^n-1 n]/(n-1)!
Dihedral angles	0°	90°	arccos(-1/3) 109.471221°	120°	arccos(-3/5) 126.869898°	arccos(2/n - 1)
Dimension	6D	7D	8D	9D	10D	nD
Dynkin diagram	x3o3o3o3o4o	x3o3o3o3o3o4o	x3o3o3o3o3o3o4o	x3o3o3o3o3o3o3o4o	x3o3o3o3o3o3o3o3o4o	x3o...o3o4o
Acronym	gee	zee	ek	vee	ka	n-simplex
Vertex Count	12`tac`	14`gee`	16`zee`	18`ek`	20`vee`	2n
Facet Count	64`hix`	128`hop`	256`oca`	512`ene`	1024`day`	2ⁿ
Circumradius	1/sqrt(2) 0.707107	1/sqrt(2) 0.707107	1/sqrt(2) 0.707107	1/sqrt(2) 0.707107	1/sqrt(2) 0.707107	1/sqrt(2) 0.707107
Inradius	1/sqrt(12) 0.288675	1/sqrt(14) 0.267261	1/4 0.25	1/sqrt(18) 0.235702	1/sqrt(20) 0.223607	1/sqrt(2n)
Volume	1/90 0.011111	sqrt(2)/630 0.0022448	1/2520 0.00039683	sqrt(2)/22680 0.000062355	1/113400 0.0000088183	sqrt(2ⁿ)/n!
Surface	2 sqrt(3)/15 0.230940	sqrt(7)/45 0.058794	4/315 0.012698	1/420 0.0023810	sqrt(5)/5670 0.00039437	2 sqrt[2^n-1 n]/(n-1)!
Dihedral angles	arccos(-2/3) 131.810315°	arccos(-5/7) 135.584691°	arccos(-3/4) 138.590378°	arccos(-7/9) 141.057559°	arccos(-4/5) 143.130102°	arccos(2/n - 1)

Interestingly this class belongs to an even wider class of (then mostlyhyperbolic) polytopes which all have that common property that the nD (or rather: rank n) representant occurs asridge faceting midsection within the (n+1)D case (for the finite cases) resp. as aridge faceting subspace within the rank n+1 case (for the infinite cases). This then is the general regular class of theregularxPo3o...o3o4o. Within this class it happens moreover generally that this subspace additionally acts as a true mirror of symmetry.Below is a small enlisting thereof.

xPo3o...o3o4o
P = 3	P = 4	P = 5	P = 6
r = 1/sqrt(2) = 0.707107 x3o4o -octx3o3o4o -hexx3o3o3o4o -tacx3o3o3o3o4o -geex3o3o3o3o3o4o -zeex3o3o3o3o3o3o4o -ekx3o3o3o3o3o3o3o4o -veex3o3o3o3o3o3o3o3o4o -ka...	r = ∞ x4o4o -squatx4o3o4o -chonx4o3o3o4o -testx4o3o3o3o4o -penthx4o3o3o3o3o4o -axhx4o3o3o3o3o3o4o -hepth...	r = sqrt[-1-sqrt(5)]/2 = 0.899454i x5o4o -peatx5o3o4o -doehonx5o3o3o4o -shitte...	r = 1/sqrt(-2) = 0.707107i x6o4o -shexatx6o3o4o -shexah...

xPo3o...o3o4o

P = 3

P = 4

P = 5

P = 6

r = 1/sqrt(2) = 0.707107

x3o4o -octx3o3o4o -hexx3o3o3o4o -tacx3o3o3o3o4o -geex3o3o3o3o3o4o -zeex3o3o3o3o3o3o4o -ekx3o3o3o3o3o3o3o4o -veex3o3o3o3o3o3o3o3o4o -ka...

r = ∞

x4o4o -squatx4o3o4o -chonx4o3o3o4o -testx4o3o3o3o4o -penthx4o3o3o3o3o4o -axhx4o3o3o3o3o3o4o -hepth...

r = sqrt[-1-sqrt(5)]/2 = 0.899454i

x5o4o -peatx5o3o4o -doehonx5o3o3o4o -shitte...

r = 1/sqrt(-2) = 0.707107i

x6o4o -shexatx6o3o4o -shexah...

Regular Hypercube C_n (up)

These polytopes are closely related to theprism product.In fact C_n generally can be described as the C_n-1-prism, i.e. thesegmentotope ofthe regular hypercube C_n-1 atop the (identical) hypercube C_n-1.Thence, by means of thelace prism notation,C_n = o3o...o3o4x (n nodes) can be described as well asoo3oo...oo3oo4xx&#x (n-1 node positions).

The regular hypercube C_n generally is the dual of theregular orthoplex O_n.

Dimension	1D	2D	3D	4D	5D	nD
Dynkin diagram	x	o4x	o3o4x	o3o3o4x	o3o3o3o4x	o3o...o3o4x
Acronym	line	square	cube	tes	pent	n-hypercube
Vertex Count	2	4`q-line`	8`q-trig`	16`q-tet`	32`q-pen`	2ⁿ
Facet Count	2	4`line`	6`square`	8`cube`	10`tes`	2n
Circumradius	1/2 0.5	1/sqrt(2) 0.707107	sqrt(3)/2 0.866025	1	sqrt(5)/2 1.118034	sqrt(n)/2
Inradius	1/2 0.5	1/2 0.5	1/2 0.5	1/2 0.5	1/2 0.5	1/2 0.5
Volume	1	1	1	1	1	1
Surface	2	4	6	8	10	2n
Dihedral angles	0°	90°	90°	90°	90°	90°
Dimension	6D	7D	8D	9D	10D	nD
Dynkin diagram	o3o3o3o3o4x	o3o3o3o3o3o4x	o3o3o3o3o3o3o4x	o3o3o3o3o3o3o3o4x	o3o3o3o3o3o3o3o3o4x	o3o...o3o4x
Acronym	ax	hept	octo	enne	deker	n-hypercube
Vertex Count	64`q-hix`	128`q-hop`	256`q-oca`	512`q-ene`	1024`q-day`	2ⁿ
Facet Count	12`pent`	14`ax`	16`hept`	18`octo`	20`enne`	2n
Circumradius	sqrt(3/2) 1.224745	sqrt(7)/2 1.322876	sqrt(2) 1.414214	3/2 1.5	sqrt(5/2) 1.581139	sqrt(n)/2
Inradius	1/2 0.5	1/2 0.5	1/2 0.5	1/2 0.5	1/2 0.5	1/2 0.5
Volume	1	1	1	1	1	1
Surface	12	14	16	18	20	2n
Dihedral angles	90°	90°	90°	90°	90°	90°

Rectified Orthoplex rO_n (up)

The common unit circumradius of all these shows that they occur asvertex figure of an according dimensionalhoneycomb. In fact they are the hull-of-large-roots polytopes of the according dimensionalroot lattice C_n(or equivalently the hull-of-small-roots polytopes of the according dimensionalroot lattice B_n).Furthermore it forces that the facet-to-bodycenter pyramids all areCRF, i.e. that all these polytopes can bedecomposed accordingly.

Within these polytopes rO_n generally can be described as thebistratic lace towerof theregular orthoplex O_n-1 atop the rectified orthoplex rO_n-1 atop the regular orthoplex O_n-1.Thence, by means of thelace tower notation,rO_n = o3x3o...o3o4o (n nodes) can be described as well asxox3oxo3ooo...ooo3ooo4ooo&#xt (n-1 node positions).

On the other hand these polytopes rO_n generally can also be described within a different orientation as thebistratic lace tower of therectified simplex rS_n-1 atop themaximal-expanded simplexeS_n-1 atop the inverted rectified simplex -rS_n-1.Thence, by means of thelace tower notation,rO_n = o3x3o...o3o4o (n nodes) can be described as well asoxo3xoo3ooo...ooo3oox3oxo&#xt (n-1 node positions).As the according midsection therefore generally is eS_n-1, and those polytopes already where mentioned to have this unit circumradius property,it becomes apparent that this property here applies as well.

Dimension	3D	4D	5D	6D	nD
Dynkin diagram	o3x4o	o3x3o4o	o3x3o3o4o	o3x3o3o3o4o	o3x3o...o3o4o
Acronym	co	ico	rat	rag	rect. n-orthoplex
Vertex Count	12`x2q`	24`cube`	40`ope`	60`hexip`	2n(n-1)
Facet Count rect. facets	8`trig`	16`oct`	32`rap`	64`rix`	2ⁿ
Facet Count verf facets	6`square`	8`oct`	10`hex`	12`tac`	2n
Circumradius	1	1	1	1	1
Inradius wrt. rect. facets	sqrt(2/3) 0.816497	1/sqrt(2) 0.707107	sqrt(2/5) 0.632456	1/sqrt(3) 0.577350	sqrt(2/n)
Inradius wrt. verf facets	1/sqrt(2) 0.707107	1/sqrt(2) 0.707107	1/sqrt(2) 0.707107	1/sqrt(2) 0.707107	1/sqrt(2) 0.707107
Volume	5 sqrt(2)/3 2.357023	2	9 sqrt(2)/10 1.272792	29/45 0.644444	(2ⁿ-n) sqrt(2ⁿ)/n!
Surface	6+2 sqrt(3) 9.464102	8 sqrt(2) 11.313708	(5+11 sqrt(5))/3 9.865583	(6 sqrt(2)+52 sqrt(3))/15 6.570128	2 [n+(2^n-1-n) sqrt(n)] sqrt(2^n-1)/(n-1)!
Dihedral angles rect. - orthopl.	arccos[-1/sqrt(3)] 125.264390°	120°	arccos[-1/sqrt(5)] 116.565051°	arccos[-1/sqrt(6)] 114.094843°	arccos[-1/sqrt(n)]
Dihedral angles rect. - rect.	arccos[-1/sqrt(3)] 125.264390°	120°	arccos(-3/5) 126.869898°	arccos(-2/3) 131.810315°	arccos(2/n - 1)
Dimension	7D	8D	9D	10D	nD
Dynkin diagram	o3x3o3o3o3o4o	o3x3o3o3o3o3o4o	o3x3o3o3o3o3o3o4o	o3x3o3o3o3o3o3o3o4o	o3x3o...o3o4o
Acronym	rez	rek	riv	rake	rect. n-orthoplex
Vertex Count	84`taccup`	112`geep`	144`zeep`	180`ekip`	2n(n-1)
Facet Count rect. facets	128`ril`	256`roc`	512`rene`	1024`reday`	2ⁿ
Facet Count verf facets	14`gee`	16`zee`	18`ek`	20`vee`	2n
Circumradius	1	1	1	1	1
Inradius wrt. rect. facets	sqrt(2/7) 0.534522	1/2 0.5	sqrt(2)/3 0.471405	1/sqrt(5) 0.447214	sqrt(2/n)
Inradius wrt. verf facets	1/sqrt(2) 0.707107	1/sqrt(2) 0.707107	1/sqrt(2) 0.707107	1/sqrt(2) 0.707107	1/sqrt(2) 0.707107
Volume	121 sqrt(2)/630 0.271619	31/315 0.098413	503 sqrt(2)/22680 0.031365	169/18900 0.0089418	(2ⁿ-n) sqrt(2ⁿ)/n!
Surface	(7+57 sqrt(7))/45 3.506841	(480+8 sqrt(2))/315 1.559726	25/42 0.595238	[5 sqrt(2)+502 sqrt(5)]/5760 0.199220	2 [n+(2^n-1-n) sqrt(n)] sqrt(2^n-1)/(n-1)!
Dihedral angles rect. - orthopl.	arccos[-1/sqrt(7)] 112.207654°	arccos[-1/sqrt(8)] 110.704811°	arccos(-1/3) 109.471221°	arccos[-1/sqrt(10)] 108.434949°	arccos[-1/sqrt(n)]
Dihedral angles rect. - rect.	arccos(-5/7) 135.584691°	arccos(-3/4) 138.590378°	arccos(-7/9) 141.057559°	arccos(-4/5) 143.130102°	arccos(2/n - 1)

Rectified Hypercube rC_n (up)

Within these polytopes rC_n generally can be described as thebistratic lace towerof the rectified hypercube rC_n-1 atop the q-scaledhypercube C_n-1 atop the (alike oriented) rectified hypercube rC_n-1.Thence, by means of thelace tower notation,rC_n = o3o...o3x4o (n nodes) can be described as well asooo3ooo...ooo3xox4oqo&#xt (n-1 node positions).

Dimension	3D	4D	5D	6D	nD
Dynkin diagram	o3x4o	o3o3x4o	o3o3o3x4o	o3o3o3o3x4o	o3o...o3x4o
Acronym	co	rit	rin	rax	rect. n-hypercube
Vertex Count	12`x q`	32`o3x q`	80`o3o3x q`	192`o3o3o3x q`	n 2^n-1
Facet Count rect. facets	6`square`	8`co`	10`rit`	12`rin`	2n
Facet Count verf facets	8`trig`	16`tet`	32`pen`	64`hix`	2ⁿ
Circumradius	1	sqrt(3/2) 1.224745	sqrt(2) 1.414214	sqrt(5/2) 1.581139	sqrt[(n-1)/2]
Inradius wrt. rect. facets	1/sqrt(2) 0.707107	1/sqrt(2) 0.707107	1/sqrt(2) 0.707107	1/sqrt(2) 0.707107	1/sqrt(2) 0.707107
Inradius wrt. verf facets	sqrt(2/3) 0.816497	3/sqrt(8) 1.060660	sqrt(8/5) 1.264911	5/sqrt(12) 1.443376	(n-1)/sqrt(2n)
Volume	5 sqrt(2)/3 2.357023	23/6 3.833333	119 sqrt(2)/30 5.609714	719/90 7.988889	(n!-1) sqrt(2ⁿ)/n!
Surface	6+2 sqrt(3) 9.464102	44 sqrt(2)/3 20.741799	(115+sqrt(5))/3 39.078689	(714 sqrt(2)+2 sqrt(3))/15 67.547506	[n!-n+sqrt(n)] sqrt(2ⁿ⁺¹)/(n-1)!
Dihedral angles rect. - simplex	arccos[-1/sqrt(3)] 125.264390°	120°	arccos[-1/sqrt(5)] 116.565051°	arccos[-1/sqrt(6)] 114.094843°	arccos[-1/sqrt(n)]
Dihedral angles rect. - rect.	arccos[-1/sqrt(3)] 125.264390°	90°	90°	90°	90°
Dimension	7D	8D	9D	10D	nD
Dynkin diagram	o3o3o3o3o3x4o	o3o3o3o3o3o3x4o	o3o3o3o3o3o3o3x4o	o3o3o3o3o3o3o3o3x4o	o3o...o3x4o
Acronym	rasa	recto	ren	rade	rect. n-hypercube
Vertex Count	448	1024	2304	5120	n 2^n-1
Facet Count rect. facets	14`rax`	16`rasa`	18`recto`	20`ren`	2n
Facet Count verf facets	128`hop`	256`oca`	512`ene`	1024`day`	2ⁿ
Circumradius	sqrt(3) 1.732051	sqrt(7/2) 1.870829	2	3/sqrt(2) 2.121320	sqrt[(n-1)/2]
Inradius wrt. rect. facets	1/sqrt(2) 0.707107	1/sqrt(2) 0.707107	1/sqrt(2) 0.707107	1/sqrt(2) 0.707107	1/sqrt(2) 0.707107
Inradius wrt. verf facets	6/sqrt(14) 1.603567	7/4 1.75	8/sqrt(18) 1.885618	9/sqrt(20) 2.012461	(n-1)/sqrt(2n)
Volume	5039 sqrt(2)/630 11.311464	40319/2520 15.999603	362879 sqrt(2)/22680 22.627355	3628799/113400 31.999991	(n!-1) sqrt(2ⁿ)/n!
Surface	(5033+sqrt(7)/45 111.903239	(4+40312 sqrt(2))/315 180.996118	60479/210 287.995238	(1814395 sqrt(2)+sqrt(5))/5670 452.547487	[n!-n+sqrt(n)] sqrt(2ⁿ⁺¹)/(n-1)!
Dihedral angles rect. - orthopl.	arccos[-1/sqrt(7)] 112.207654°	arccos[-1/sqrt(8)] 110.704811°	arccos(-1/3) 109.471221°	arccos[-1/sqrt(10)] 108.434949°	arccos[-1/sqrt(n)]
Dihedral angles rect. - rect.	90°	90°	90°	90°	90°

Facetorectified Hypercube frC_n (up)

These non-convex polytopes frC_n generally are facetings of therectified hypercube rC_n.

Facets here always come within pairs – except for the hemifacets, which occur for the odd dimensional series members. Subsequent ones always alternate between prograde and retrograde. Thence for these odd dimensional series members thevolume always results in zero,as the facet pyramids of those hemifacets clearly are degenerate, while the other ones cancel out by means of those pairings, then using a prograde and a retrograde base respectively. For the even dimensional series members however, due to the missing hemifacets,those pairings will be either both pro- or both retrograde.

Dimension	3D	4D	5D	6D	nD
Dynkin diagram	x3x3/2o3*a	x3x3/2o3o3*a	x3x3/2o3o3o3*a	x3x3/2o3o3o3o3*a	x3x3/2o3o...o3*a
Acronym	oho	firt	firn	forx	facetorect. n-hyp.c.
Vertex Count	12	32	80	192	n 2^n-1
Facet Count simplex	4+4`trig`	8+8`tet`	16+16`pen`	32+32`hix`	2^n-1 (each)
Facet Count trunc. simp.	4`hig`	8+8`tut`	16+16`tip`	32+32`tix`	2^n-1 (each)
Facet Count bitrunc. simp.	4`hig`	8+8`tut`	16`deca`	32+32`bittix`	2^n-1 (each)
Circumradius	1	sqrt(3/2) 1.224745	sqrt(2) 1.414214	sqrt(5/2) 1.581139	sqrt[(n-1)/2]
Inradius wrt. simplex	+/− sqrt(2/3) 0.816497	−/− 3/sqrt(8) 1.060660	+/− sqrt(8/5) 1.264911	+/+ 5/sqrt(12) 1.443376	(n-1)/sqrt(2n)
Inradius wrt. trunc. simp.	0	+/+ 1/sqrt(8) 0.353553	−/+ sqrt(2/5) 0.632456	−/− sqrt(3)/2 0.866025	(n-3)/sqrt(2n)
Inradius wrt. bitrunc. simp.	0	+/+ 1/sqrt(8) 0.353553	0	+/+ 1/sqrt(12) 0.288675	(n-5)/sqrt(2n)
Volume	0	10/3 3.333333	0	488/45 10.844444	0 / ?
Surface	8 sqrt(3) 13.856406	32 sqrt(2) 45.254834	64 sqrt(5) 143.108351	256 sqrt(3) 443.405007	sqrt(n 8^n-1)
Dihedral angles sim. - trunc.sim.	arccos(1/3) 70.528779°	60°	arccos(3/5) 53.130102°	arccos(2/3) 48.189685°	arccos[(n-2)/n]
Dihedral angles tr.sim. - bitr.sim.		60°	arccos(3/5) 53.130102°	arccos(2/3) 48.189685°	arccos[(n-2)/n]
Dihedral angles `k-`tr.s. -`k-`tr.s.		60°	arccos(3/5) 53.130102°	arccos(2/3) 48.189685°	arccos[(n-2)/n] n 2(k+1)
Dimension	7D	8D	9D	10D	nD
Dynkin diagram	x3x3/2o3o3o3o3o3*a	x3x3/2o3o3o3o3o3o3*a	x3x3/2o3o3o3o3o3o3o3*a	x3x3/2o3o3o3o3o3o3o3o3*a	x3x3/2o3o...o3*a
Acronym	frasa	fro	fren	frade	facetorect. n-hyp.c.
Vertex Count	448	1024	2304	5120	n 2^n-1
Facet Count simplex	64+64`hop`	128+128`oca`	256+256`ene`	512+512`day`	2^n-1 (each)
Facet Count trunc. simp.	64+64`til`	128+128`toc`	256+256`tene`	512+512`teday`	2^n-1 (each)
Facet Count bitrunc. simp.	64+64`batal`	128+128`bittoc`	256+256`batene`	512+512`biteday`	2^n-1 (each)
Facet Count tritrunc. simp.	64`fe`	128+128`tattoc`	256+256`tatene`	512+512`tatday` (?)	2^n-1 (each)
Facet Count quadritr. simp.	64`fe`	128+128`tattoc`	256`be`	512+512`quatday` (?)	2^n-1 (each)
Circumradius	sqrt(3) 1.732051	sqrt(7/2) 1.870829	2	3/sqrt(2) 2.121320	sqrt[(n-1)/2]
Inradius wrt. simplex	+/− sqrt(18/7) 1.603567	−/− 7/4 1.75	+/− sqrt(32)/3 1.885618	+/+ 9/sqrt(20) 2.012461	(n-1)/sqrt(2n)
Inradius wrt. trunc. simpl.	−/+ sqrt(8/7) 1.069045	+/+ 5/4 1.25	−/+ sqrt(2) 1.414214	−/− 7/sqrt(20) 1.565248	(n-3)/sqrt(2n)
Inradius wrt. bitrunc. simpl.	+/− sqrt(2/7) 0.534522	−/− 3/4 0.75	+/− sqrt(8)/3 0.942809	+/+ sqrt(5)/2 1.118034	(n-5)/sqrt(2n)
Inradius wrt. tritrunc. simpl.	0	+/+ 1/4 0.25	−/+ sqrt(2)/3 0.471405	−/− 3/sqrt(20) 0.670820	(n-7)/sqrt(2n)
Inradius wrt. quadritr. simpl.	0	+/+ 1/4 0.25	0	+/+ 1/sqrt(20) 0.223607	(n-9)/sqrt(2n)
Volume	0	?	0	?	0 / ?
Surface	512 sqrt(7) 1354.624671	4096	12288	16384 sqrt(5) 36635.737743	sqrt(n 8^n-1)
Dihedral angles sim. - trunc.sim.	arccos(5/7) 44.415309°	arccos(3/4) 41.409622°	arccos(7/9) 38.942441°	arccos(4/5) 36.869898°	arccos[(n-2)/n]
Dihedral angles tr.sim. - bitr.sim.	arccos(5/7) 44.415309°	arccos(3/4) 41.409622°	arccos(7/9) 38.942441°	arccos(4/5) 36.869898°	arccos[(n-2)/n]
Dihedral angles bitr.sim. - tritr.sim.	arccos(5/7) 44.415309°	arccos(3/4) 41.409622°	arccos(7/9) 38.942441°	arccos(4/5) 36.869898°	arccos[(n-2)/n]
Dihedral angles tritr.s. - quadrit.s.		arccos(3/4) 41.409622°	arccos(7/9) 38.942441°	arccos(4/5) 36.869898°	arccos[(n-2)/n]
Dihedral angles `k-`tr.s. -`k-`tr.s.		arccos(3/4) 41.409622°	arccos(7/9) 38.942441°	arccos(4/5) 36.869898°	arccos[(n-2)/n] n 2(k+1)

Birectified Orthoplex brO_n (up)

Within these polytopes brO_n generally can be described as thebistratic lace towerof therectified orthoplex rO_n-1 atop the birectified orthoplex brO_n-1 atop the rectified orthoplex rO_n-1.Thence, by means of thelace tower notation,brO_n = o3o3x3o...o3o4o (n nodes) can be described as well asooo3xox3oxo3ooo...ooo3ooo4ooo&#xt (n-1 node positions).

On the other hand these polytopes brO_n generally can also be described within a different orientation as a tristratic lace toweroooo3oxoo3ooxo3ooox3oooo...oooo3xooo3oxoo3ooxo3oooo&#xt (n-1 node positions),where the right hand decorations and the lefthand decorations for the smaller dimensions well might interlace, or in the extremal 3D case even overlay and run out of the other end:ouoo3oouo&#xt.

Dimension	3D	4D	5D	6D	nD
Dynkin diagram	o3o4q	o3o3x4o	o3o3x3o4o	o3o3x3o3o4o	o3o3x3o...o3o4o
Acronym	q-cube	rit	nit	brag	birect. n-orthoplex
Vertex Count	8`u-trig`	32`o3x q`	80`tisdip`	160`troct`	4n(n-1)(n-2)/3
Facet Count birect. facets	8`u-trig`	16`tet`	32`rap`	64`dot`	2ⁿ
Facet Count rect. facets	6`q-square`	8`co`	10`ico`	12`rat`	2n
Circumradius	sqrt(3/2) 1.224745	sqrt(3/2) 1.224745	sqrt(3/2) 1.224745	sqrt(3/2) 1.224745	sqrt(3/2) 1.224745
Inradius wrt. birect. facets	sqrt(3/2) 1.224745	3/sqrt(8) 1.060660	3/sqrt(10) 0.948683	sqrt(3)/2 0.866025	3/sqrt(2n)
Inradius wrt. rect. facets	1/sqrt(2) 0.707107	1/sqrt(2) 0.707107	1/sqrt(2) 0.707107	1/sqrt(2) 0.707107	1/sqrt(2) 0.707107
Volume	sqrt(8) 2.828427	23/6 3.833333	31 sqrt(2)/10 4.384062	4	(3ⁿ-n 2ⁿ+n(n-1)/2) sqrt(2ⁿ)/n!
Surface	12	44 sqrt(2)/3 20.741799	[60+11 sqrt(5)]/3 28.198916	(54 sqrt(2)+44 sqrt(3))/5 30.515554	(3^n-1 sqrt(n 2ⁿ⁺¹)-n(sqrt(n)-1) sqrt(2^3n-1)+n(n-1)(sqrt(n)-2) sqrt(2^n-1))/(n-1)!
Dihedral angles birect. - birect.		44 sqrt(2)/3 20.741799	arccos(-3/5) 126.869898°	arccos(-2/3) 131.810315°	arccos(2/n - 1)
Dihedral angles birect. - rect.		120°	arccos[-1/sqrt(5)] 116.565051°	arccos[-1/sqrt(6)] 114.094843°	arccos[-1/sqrt(n)]
Dihedral angles rect. - rect.	90°	90°	90°	90°	90°
Dimension	7D	8D	9D	10D	nD
Dynkin diagram	o3o3x3o3o3o4o	o3o3x3o3o3o3o4o	o3o3x3o3o3o3o3o4o	o3o3x3o3o3o3o3o3o4o	o3o3x3o...o3o4o
Acronym	barz	bark	brav	brake	birect. n-orthoplex
Vertex Count	280`trahex`	448`tratac`	672`trigee`	960`trizee`	4n(n-1)(n-2)/3
Facet Count rect. facets	128`bril`	256`broc`	512`brene`	1024`breday`	2ⁿ
Facet Count verf facets	14`rag`	16`rez`	18`rek`	20`riv`	2n
Circumradius	sqrt(3/2) 1.224745	sqrt(3/2) 1.224745	sqrt(3/2) 1.224745	sqrt(3/2) 1.224745	sqrt(3/2) 1.224745
Inradius wrt. birect. facets	3/sqrt(14) 0.801784	3/4 0.75	1/sqrt(2) 0.707107	3/sqrt(20) 0.670820	3/sqrt(2n)
Inradius wrt. rect. facets	1/sqrt(2) 0.707107	1/sqrt(2) 0.707107	1/sqrt(2) 0.707107	1/sqrt(2) 0.707107	1/sqrt(2) 0.707107
Volume	656 sqrt(2)/315 2.945156	4541/2520 1.801984	1679 sqrt(2)/2520 0.942248	24427/56700 0.430811	(3ⁿ-n 2ⁿ+n(n-1)/2) sqrt(2ⁿ)/n!
Surface	(406+302 sqrt(7))/45 26.778153	(4764+968 sqrt(2))/315 19.469710	1679/140 11.992857	(2515 sqrt(2)+14608 sqrt(5))/5670 6.388224	(3^n-1 sqrt(n 2ⁿ⁺¹)-n(sqrt(n)-1) sqrt(2^3n-1)+n(n-1)(sqrt(n)-2) sqrt(2^n-1))/(n-1)!
Dihedral angles birect. - birect.	arccos(-5/7) 135.584691°	arccos(-3/4) 138.590378°	arccos(-7/9) 141.057559°	arccos(-4/5) 143.130102°	arccos(2/n - 1)
Dihedral angles birect. - rect.	arccos[-1/sqrt(7)] 112.207654°	arccos[-1/sqrt(8)] 110.704811°	arccos(-1/3) 109.471221°	arccos[-1/sqrt(10)] 108.434949°	arccos[-1/sqrt(n)]
Dihedral angles rect. - rect.	90°	90°	90°	90°	90°

Birectified Hypercube brC_n (up)

Within these polytopes brC_n generally can be described as thebistratic lace towerof the birectified hypercube brC_n-1 atop therectified hypercube rC_n-1 atop the birectified hypercube brC_n-1.Thence, by means of thelace tower notation,brC_n = o3o...o3x3o4o (n nodes) can be described as well asooo3ooo...ooo3xox3oxo4ooo&#xt (n-1 node positions).

Dimension	3D	4D	5D	6D	nD
Dynkin diagram	x3o4o	o3x3o4o	o3o3x3o4o	o3o3o3x3o4o	o3o...o3x3o4o
Acronym	oct	ico	nit	brox	birect. n-hypercube
Vertex Count	6`square`	24`cube`	80`tisdip`	240`squatet`	n(n-2) 2^n-3
Facet Count rect. simplex	8`trig`	16`oct`	32`rap`	64`rix`	2ⁿ
Facet Count birect. h.cube	8`trig`	8`oct`	10`ico`	12`nit`	2n
Circumradius	1/sqrt(2) 0.707107	1	sqrt(3/2) 1.224745	sqrt(2) 1.414214	sqrt[(n-2)/2]
Inradius wrt. rect. simplex	1/sqrt(6) 0.408248	1/sqrt(2) 0.707107	3/sqrt(10) 0.948683	2/sqrt(3) 1.154701	(n-2)/sqrt(2n)
Inradius wrt. birect. h.cube	1/sqrt(6) 0.408248	1/sqrt(2) 0.707107	1/sqrt(2) 0.707107	1/sqrt(2) 0.707107	1/sqrt(2) 0.707107
Volume	sqrt(2)/3 0.471405	2	31 sqrt(2)/10 4.384062	331/45 14.711111	?
Surface	2 sqrt(3) 3.464102	8 sqrt(2) 11.313708	[60+11 sqrt(5)]/3 28.198916	122 sqrt(2)/3 57.511352	?
Dihedral angles rect. - rect.	arccos(-1/3) 109.471221°	120°	arccos(-3/5) 126.869898°	arccos(-2/3) 131.810315°	arccos(2/n-1)
Dihedral angles rect. - birect.		120°	arccos[-1/sqrt(5)] 116.565051°	arccos[-1/sqrt(6)] 114.094843°	arccos[-1/sqrt(n)]
Dihedral angles birect. - birect.		120°	90°	90°	90°
Dimension	7D	8D	9D	10D	nD
Dynkin diagram	o3o3o3o3x3o4o	o3o3o3o3o3x3o4o	o3o3o3o3o3o3x3o4o	o3o3o3o3o3o3o3x3o4o	o3o...o3x3o4o
Acronym	bersa	bro	barn	brade	birect. n-hypercube
Vertex Count	672`squapen`	1792`squahix`	4608`squahop`	11520`squoc`	n(n-2) 2^n-3
Facet Count rect. simplex	128`ril`	256`roc`	512`rene`	1024`reday`	2ⁿ
Facet Count birect. h.cube	14`brox`	16`bersa`	18`bro`	20`barn`	2n
Circumradius	sqrt(5/2) 1.581139	sqrt(3) 1.732051	sqrt(7/2) 1.870829	2	sqrt[(n-2)/2]
Inradius wrt. rect. simplex	5/sqrt(14) 1.336306	3/2 1.5	7/sqrt(18) 1.649916	4/sqrt(5) 1.788854	(n-2)/sqrt(2n)
Inradius wrt. birect. h.cube	1/sqrt(2) 0.707107	1/sqrt(2) 0.707107	1/sqrt(2) 0.707107	1/sqrt(2) 0.707107	1/sqrt(2) 0.707107
Volume	4919 sqrt(2)/630 11.042090	?	?	?	?
Surface	?	?	?	?	?
Dihedral angles rect. - rect.	arccos(-5/7) 135.584691°	arccos(-3/4) 138.590378°	arccos(-7/9) 141.057559°	arccos(-4/5) 143.130102°	arccos(2/n - 1)
Dihedral angles rect. - birect.	arccos[-1/sqrt(7)] 112.207654°	arccos[-1/sqrt(8)] 110.704811°	arccos(-1/3) 109.471221°	arccos[-1/sqrt(10)] 108.434949°	arccos[-1/sqrt(n)]
Dihedral angles birect. - birect.	90°	90°	90°	90°	90°

Truncated Orthoplex tO_n (up)

Within these polytopes tO_n generally can be described as the tetrastratic lace tower of theregular orthoplex O_n-1 atop the u-scaled regular orthoplex O_n-1 atop the truncated orthoplex tO_n-1 atop the u-scaled regular orthoplex O_n-1 atop the regular orthoplex O_n-1.Thence, by means of thelace tower notation,tO_n = x3x3o...o3o4o (n nodes) can be described as well asxuxux3ooxoo3ooooo...ooooo3ooooo4ooooo&#xt (n-1 node positions).

Dimension	3D	4D	5D	6D	nD
Dynkin diagram	x3x4o	x3x3o4o	x3x3o3o4o	x3x3o3o3o4o	x3x3o...o3o4o
Acronym	toe	thex	tot	tag	trunc. n-orthoplex
Vertex Count	24	48	80	120	4n(n-1)
Facet Count trunc. facets	8`hig`	16`tut`	32`tip`	64`tix`	2ⁿ
Facet Count verf facets	6`square`	8`oct`	10`hex`	12`tac`	2n
Circumradius	sqrt(5/2) 1.581139	sqrt(5/2) 1.581139	sqrt(5/2) 1.581139	sqrt(5/2) 1.581139	sqrt(5/2) 1.581139
Inradius wrt. trunc. facets	sqrt(3/2) 1.224745	3/sqrt(8) 1.060660	3/sqrt(10) 0.948683	sqrt(3)/2 0.866025	3/sqrt(2n)
Inradius wrt. verf facets	sqrt(2) 1.414214	sqrt(2) 1.414214	sqrt(2) 1.414214	sqrt(2) 1.414214	sqrt(2) 1.414214
Volume	8 sqrt(2) 11.313708	77/6 12.833333	119 sqrt(2)/15 11.219428	241/30 8.033333	(3ⁿ-n) sqrt(2ⁿ)/n!
Surface	6+12 sqrt(3) 26.784610	100 sqrt(2)/3 47.140452	(5+76 sqrt(5))/3 58.313722	(2 sqrt(2)+158 sqrt(3))/5 55.298491	2[n+(3^n-1-n) sqrt(n)] sqrt(2^n-1)/(n-1)!
Dihedral angles trunc. - orthopl.	arccos[-1/sqrt(3)] 125.264390°	120°	arccos[-1/sqrt(5)] 116.565051°	arccos[-1/sqrt(6)] 114.094843°	arccos[-1/sqrt(n)]
Dihedral angles trunc. - trunc.	arccos(-1/3) 109.471221°	120°	arccos(-3/5) 126.869898°	arccos(-2/3) 131.810315°	arccos(2/n - 1)
Dimension	7D	8D	9D	10D	nD
Dynkin diagram	x3x3o3o3o3o4o	x3x3o3o3o3o3o4o	x3x3o3o3o3o3o3o4o	x3x3o3o3o3o3o3o3o4o	x3x3o...o3o4o
Acronym	taz	tek	tiv	take	trunc. n-orthoplex
Vertex Count	168	224	288	360	4n(n-1)
Facet Count trunc. facets	128`til`	256`toc`	512`tene`	1024`teday`	2ⁿ
Facet Count verf facets	14`gee`	16`zee`	18`ek`	20`vee`	2n
Circumradius	sqrt(5/2) 1.581139	sqrt(5/2) 1.581139	sqrt(5/2) 1.581139	sqrt(5/2) 1.581139	sqrt(5/2) 1.581139
Inradius wrt. trunc. facets	3/sqrt(14) 0.801784	3/4 0.75	1/sqrt(2) 0.707107	3/sqrt(20) 0.670820	3/sqrt(2n)
Inradius wrt. verf facets	sqrt(2) 1.414214	sqrt(2) 1.414214	sqrt(2) 1.414214	sqrt(2) 1.414214	sqrt(2) 1.414214
Volume	218 sqrt(2)/63 4.893628	6553/2520 2.600397	1093 sqrt(2)/1260 1.226774	59039/113400 0.520626	(3ⁿ-n) sqrt(2ⁿ)/n!
Surface	(7+722 sqrt(7))/45 42.605165	(8716+8 sqrt(2))/315 27.705758	437/28 15.607143	[5 sqrt(2)+19673 sqrt(5)]/5670 7.759654	2[n+(3^n-1-n) sqrt(n)] sqrt(2^n-1)/(n-1)!
Dihedral angles trunc. - orthopl.	arccos[-1/sqrt(7)] 112.207654°	arccos[-1/sqrt(8)] 110.704811°	arccos(-1/3) 109.471221°	arccos[-1/sqrt(10)] 108.434949°	arccos[-1/sqrt(n)]
Dihedral angles trunc. - trunc.	arccos(-5/7) 135.584691°	arccos(-3/4) 138.590378°	arccos(-7/9) 141.057559°	arccos(-4/5) 143.130102°	arccos(2/n - 1)

Truncated Hypercube tC_n (up)

Within these polytopes tC_n generally can be described as the tristratic lace towerof the truncated hypercube tC_n-1 atop the w-scaledregular hypercube C_n-1 atop a further w-scaled regular hypercube C_n-1 atop the truncated hypercube tC_n-1.Thence, by means of thelace tower notation,tC_n = o3o...o3x4x (n nodes) can be described as well asoooo3oooo...oooo3xoox4xwwx&#xt (n-1 node positions).

Dimension	3D	4D	5D	6D	nD
Dynkin diagram	o3x4x	o3o3x4x	o3o3o3x4x	o3o3o3o3x4x	o3o...o3x4x
Acronym	tic	tat	tan	tox	trunc. n-hypercube
Vertex Count	24	64	160	384	n 2ⁿ
Facet Count trunc. facets	6`oc`	8`tic`	10`tat`	12`tan`	2n
Facet Count verf facets	8`trig`	16`tet`	32`pen`	64`hix`	2ⁿ
Circumradius	sqrt[7+4 sqrt(2)]/2 1.778824	sqrt[(5+3 sqrt(2))/2] 2.149726	sqrt[13+8 sqrt(2)]/2 2.465447	sqrt[(8+5 sqrt(2))/2] 2.745093	sqrt[(3n-2)+(2n-2) sqrt(2)]/2
Inradius wrt. trunc. facets	[1+sqrt(2)]/2 1.207107	[1+sqrt(2)]/2 1.207107	[1+sqrt(2)]/2 1.207107	[1+sqrt(2)]/2 1.207107	[1+sqrt(2)]/2 1.207107
Inradius wrt. verf facets	(3+2 sqrt(2))/sqrt(12) 1.682522	(3+2 sqrt(2))/sqrt(8) 2.060660	(5+4 sqrt(2))/sqrt(20) 2.382945	(5+3 sqrt(2))/sqrt(12) 2.668121	[n+(n-1) sqrt(2)]/sqrt(4n)
Volume	(21+14 sqrt(2))/3 13.599663	(101+72 sqrt(2))/6 33.803896	(1230+869 sqrt(2))/30 81.965053	(8909+6300 sqrt(2))/90 197.983838	?
Surface	12+12 sqrt(2)+2 sqrt(3) 32.434664	(168+116 sqrt(2))/3 110.682924	(505+360 sqrt(2)+sqrt(5))/3 338.784317	(7380+5214 sqrt(2)+2 sqrt(3))/15 983.811574	?
Dihedral angles trunc. - simplex	arccos[-1/sqrt(3)] 125.264390°	120°	arccos[-1/sqrt(5)] 116.565051°	arccos[-1/sqrt(6)] 114.094843°	arccos[-1/sqrt(n)]
Dihedral angles trunc. - trunc.	90°	90°	90°	90°	90°
Dimension	7D	8D	9D	10D	nD
Dynkin diagram	o3o3o3o3o3x4x	o3o3o3o3o3o3x4x	o3o3o3o3o3o3o3x4x	o3o3o3o3o3o3o3o3x4x	o3o...o3x4x
Acronym	tasa	tocto	ten	tade	trunc. n-hypercube
Vertex Count	896	2048	4608	10240	n 2ⁿ
Facet Count trunc. facets	14`tox`	16`tasa`	18`tocto`	20`ten`	2n
Facet Count verf facets	128`hop`	256`oca`	512`ene`	1024`day`	2ⁿ
Circumradius	sqrt[19+12 sqrt(2)]/2 2.998773	sqrt[(11+7 sqrt(2))/2] 3.232607	sqrt[25+16 sqrt(2)]/2 3.450631	sqrt[(14+9 sqrt(2))/2] 3.655675	sqrt[(3n-2)+(2n-2) sqrt(2)]/2
Inradius wrt. trunc. facets	[1+sqrt(2)]/2 1.207107	[1+sqrt(2)]/2 1.207107	[1+sqrt(2)]/2 1.207107	[1+sqrt(2)]/2 1.207107	[1+sqrt(2)]/2 1.207107
Inradius wrt. verf facets	(7+6 sqrt(2))/sqrt(28) 2.926443	(7+4 sqrt(2))/4 3.164214	(9+8 sqrt(2))/6 3.385618	(9+5 sqrt(2))/sqrt(20) 3.593600	[n+(n-1) sqrt(2)]/sqrt(4n)
Volume	?	?	?	?	?
Surface	?	?	?	?	?
Dihedral angles trunc. - simplex	arccos[-1/sqrt(7)] 112.207654°	arccos[-1/sqrt(8)] 110.704811°	arccos(-1/3) 109.471221°	arccos[-1/sqrt(10)] 108.434949°	arccos[-1/sqrt(n)]
Dihedral angles trunc. - trunc.	90°	90°	90°	90°	90°

Quasitruncated Hypercube qtC_n (up)

These non-convex polytopes qtC_n generally are nothing but the conjugates of thetruncated hypercube tC_n.

Dimension	3D	4D	5D	6D	nD
Dynkin diagram	o3x4/3x	o3o3x4/3x	o3o3o3x4/3x	o3o3o3o3x4/3x	o3o...o3x4/3x
Acronym	quith	quitit	quittin	quotox	quasitrunc. n-hypercube
Vertex Count	24	64	160	384	n 2ⁿ
Facet Count quasitr. fac.	6`og`	8`quith`	10`quitit`	12`quittin`	2n
Facet Count verf facets	8`trig`	16`tet`	32`pen`	64`hix`	2ⁿ
Circumradius	sqrt[7-4 sqrt(2)]/2 0.579471	sqrt[(5-3 sqrt(2))/2] 0.615370	sqrt[13-8 sqrt(2)]/2 0.649286	sqrt[(8-5 sqrt(2))/2] 0.681517	sqrt[(3n-2)-(2n-2) sqrt(2)]/2
Inradius wrt. quasitr. fac.	[sqrt(2)-1]/2 0.207107	[sqrt(2)-1]/2 0.207107	[sqrt(2)-1]/2 0.207107	[sqrt(2)-1]/2 0.207107	[sqrt(2)-1]/2 0.207107
Inradius wrt. verf facets	(3-2 sqrt(2))/sqrt(12) +0.049529	(4-3 sqrt(2))/4 -0.060660	(5-4 sqrt(2))/sqrt(20) -0.146877	(3 sqrt(2)-5)/sqrt(12) -0.218631	[(n-1) sqrt(2)-n]/sqrt(4n)
Volume	(21-14 sqrt(2))/3 0.400337	(72 sqrt(2)-101)/6 0.137229	(1230-869 sqrt(2))/30 0.034947	(6300 sqrt(2)-8909)/90 0.0060605	?
Surface	-12+12 sqrt(2)+2 sqrt(3) 8.434664	56-36 sqrt(2) 5.088312	?	?	?
Dihedral angles quasitr. - simpl.	arccos[1/sqrt(3)] 54.735610°	60°	arccos[1/sqrt(5)] 63.434949°	arccos[1/sqrt(6)] 65.905157°	arccos[1/sqrt(n)]
Dihedral angles quasitr. - quasitr.	90°	90°	90°	90°	90°
Dimension	7D	8D	9D	10D	nD
Dynkin diagram	o3o3o3o3o3x4/3x	o3o3o3o3o3o3x4/3x	o3o3o3o3o3o3o3x4/3x	o3o3o3o3o3o3o3o3x4/3x	o3o...o3x4/3x
Acronym	quitasa	queto	quiten	quitade	quasitrunc. n-hypercube
Vertex Count	896	2048	4608	10240	n 2ⁿ
Facet Count quasitr. fac.	14`quotox`	16`quitasa`	18`queto`	20`quiten`	2n
Facet Count verf facets	128`hop`	256`oca`	512`ene`	1024`day`	2ⁿ
Circumradius	sqrt[19-12 sqrt(2)]/2 0.712292	sqrt[(11-7 sqrt(2))/2] 0.741790	sqrt[25-16 sqrt(2)]/2 0.770160	sqrt[(14-9 sqrt(2))/2] 0.797521	sqrt[(3n-2)-(2n-2) sqrt(2)]/2
Inradius wrt. quasitr. fac.	[sqrt(2)-1]/2 0.207107	[sqrt(2)-1]/2 0.207107	[sqrt(2)-1]/2 0.207107	[sqrt(2)-1]/2 0.207107	[sqrt(2)-1]/2 0.207107
Inradius wrt. verf facets	(7-6 sqrt(2))/sqrt(28) -0.280692	(4 sqrt(2)-7)/4 -0.335786	(9-8 sqrt(2))/6 -0.385618	(5 sqrt(2)-9)/sqrt(20) -0.431322	[n-(n-1) sqrt(2)]/sqrt(4n)
Volume	?	?	?	?	?
Surface	?	?	?	?	?
Dihedral angles quasitr. - simpl.	arccos[1/sqrt(7)] 67.792346°	arccos[1/sqrt(8)] 69.295189°	arccos(1/3) 70.528779°	arccos[1/sqrt(10)] 71.565051°	arccos[1/sqrt(n)]
Dihedral angles quasitr. - quasitr.	90°	90°	90°	90°	90°

Bitruncated Hypercube btC_n (up)

Within these polytopes btC_n generally can be described as a stack of the bitruncated hypercube btC_n-1 atop u-scaledrectified hypercube rC_n-1 atop an(x,q)-varianttruncated hypercube tC_n-1 atop u-scaled rectified hypercube rC_n-1 (again) atop the opposite bitruncated hypercube btC_n-1.Thence, by means of thelace tower notation,btC_n = o3o...o3x3x4o (n nodes) can be described as well asooooo3ooooo...ooooo3xooox3xuxux4ooqoo&#xt (n-1 node positions).This representation then shows up those right angles generally.

Dimension	3D	4D	5D	6D	nD
Dynkin diagram	x3x4o	o3x3x4o	o3o3x3x4o	o3o3o3x3x4o	o3o...o3x3x4o
Acronym	toe	tah	bittin	botox	bitrunc. n-hypercube
Vertex Count	24	96	320	960	n(n-1) 2^n-1
Facet Count bitrunc. fac.	6`square`	8`toe`	10`tah`	12`bittin`	2n
Facet Count trunc. simpl.	8`hig`	16`tut`	32`tip`	64`tix`	2ⁿ
Circumradius	sqrt(5/2) 1.581139	sqrt(9/2) 2.121320	sqrt(13/2) 2.549510	sqrt(17/2) 2.915476	sqrt[(4n-7)/2]
Inradius wrt. bitrunc. fac.	sqrt(2) 1.414214	sqrt(2) 1.414214	sqrt(2) 1.414214	sqrt(2) 1.414214	sqrt(2) 1.414214
Inradius wrt. trunc. simpl.	sqrt(3/2) 1.224745	5/sqrt(8) 1.767767	7/sqrt(10) 2.213594	sqrt(27)/2 2.598076	(2n-3)/sqrt(2n)
Volume	8 sqrt(2) 11.313708	307/6 51.166667	1801 sqrt(2)/15 169.799908	?	?
Surface	6+12 sqrt(3) 26.784610	?	?	?	?
Dihedral angles trunc. - trunc.	arccos(-1/3) 109.471221°	120°	arccos(-3/5) 126.869898°	arccos(-2/3) 131.810315°	arccos[-(n-2)/n]
Dihedral angles trunc. - bitrunc	arccos[-1/sqrt(3)] 125.264390°	120°	arccos[-1/sqrt(5)] 116.565051°	arccos[-1/sqrt(6)] 114.094843°	arccos[-1/sqrt(n)]
Dihedral angles bitrunc. - bitrunc.	arccos[-1/sqrt(3)] 125.264390°	90°	90°	90°	90°
Dimension	7D	8D	9D	10D	nD
Dynkin diagram	o3o3o3o3x3x4o	o3o3o3o3o3x3x4o	o3o3o3o3o3o3x3x4o	o3o3o3o3o3o3o3x3x4o	o3o...o3x3x4o
Acronym	betsa	bato	?	?	bitrunc. n-hypercube
Vertex Count	2688	7168	18432	46080	n(n-1) 2^n-1
Facet Count bitrunc. fac.	14`botox`	16`betsa`	18`bato`	20`?`	2n
Facet Count trunc. simpl.	128`til`	256`toc`	512`tene`	1024`teday`	2ⁿ
Circumradius	sqrt(21/2) 3.240370	5/sqrt(2) 3.535534	sqrt(29/2) 3.807887	sqrt(33/2) 4.062019	sqrt[(4n-7)/2]
Inradius wrt. bitrunc. fac.	sqrt(2) 1.414214	sqrt(2) 1.414214	sqrt(2) 1.414214	sqrt(2) 1.414214	sqrt(2) 1.414214
Inradius wrt. trunc. simpl.	11/sqrt(14) 2.939874	13/4 3.25	5/sqrt(2) 3.535534	17/sqrt(20) 3.801316	(2n-3)/sqrt(2n)
Volume	?	?	?	?	?
Surface	?	?	?	?	?
Dihedral angles trunc. - trunc.	arccos(-5/7) 135.584691°	arccos(-3/4) 138.590378°	arccos(-7/9) 141.057559°	arccos(-4/5) 143.130102°	arccos[-(n-2)/n]
Dihedral angles trunc. - bitrunc.	arccos[-1/sqrt(7)] 112.207654°	arccos[-1/sqrt(8)] 110.704811°	arccos(-1/3) 109.471221°	arccos[-1/sqrt(10)] 108.434949°	arccos[-1/sqrt(n)]
Dihedral angles bitrunc. - bitrunc.	90°	90°	90°	90°	90°

Rhombated Hypercube rbC_n (up)

Dimension	3D	4D	5D	6D	nD
Dynkin diagram	x3o4x	o3x3o4x	o3o3x3o4x	o3o3o3x3o4x	o3o...o3x3o4x
Acronym	sirco	srit	sirn	srox	rhomb. n-hypercube
Vertex Count	24	96	320	960	n(n-1) 2^n-1
Facet Count rect. simpl.	8`trig`	16`oct`	32`rap`	64`rix`	2ⁿ
Facet Count prism	12`square`	32`trip`	80`tepe`	192`penp`	n 2^n-1
Facet Count rhomb. hyp.cube	6`square`	8`sirco`	10`srit`	12`sirn`	2n
Circumradius	sqrt[5+2 sqrt(2)]/2 1.398966	sqrt[2+sqrt(2)] 1.847759	[3+sqrt(2)]/2 2.207107	sqrt[(7+4 sqrt(2))/2] 2.515637	sqrt[3n-4+2(n-2) sqrt(2)]/2
Inradius wrt. rect. simp. facets	[3+sqrt(2)]/sqrt(12) 1.274274	1+1/sqrt(2) 1.707107	sqrt[(43+30 sqrt(2))/20] 2.066717	sqrt[(17+12 sqrt(2))/6] 2.379445	sqrt[(3n²-8n+8+2n(n-2) sqrt(2))/4n]
Inradius wrt. prism facets	(1+sqrt(2))/2 1.207107	sqrt[(17+12 sqrt(2))/12] 1.682522	sqrt[(17+12 sqrt(2))/8] 2.060660	sqrt[(57+40 sqrt(2))/20] 2.382945	sqrt[(3n²-10n+9+2(n-1)(n-2) sqrt(2))/(4(n-1))]
Inradius wrt. rh. hyp.cub. facets	(1+sqrt(2))/2 1.207107	(1+sqrt(2))/2 1.207107	(1+sqrt(2))/2 1.207107	(1+sqrt(2))/2 1.207107	(1+sqrt(2))/2 1.207107
Volume	[12+10 sqrt(2)]/3 8.714045	[45+32 sqrt(2)]/3 30.084945	[1205+843 sqrt(2)]/30 79.906068	[4426+3141 sqrt(2)]/45 197.067662	?
Surface	18+2 sqrt(3) 21.464102	8[4+4 sqrt(2)+sqrt(3)] 91.111240	[450+340 sqrt(2)+11 sqrt(5)]/3 318.476453	?	?
Dihedral angles rect. - prism	arccos[-sqrt(2/3)] 144.735610°	150°	arccos[-2/sqrt(5)] 153.434949°	arccos[-sqrt(5/6)] 155.905157°	arccos[-sqrt((n-1)/n)]
Dihedral angles rect. - rhomb.	arccos[-sqrt(2/3)] 144.735610°	120°	arccos[-1/sqrt(5)] 116.565051°	arccos[-1/sqrt(6)] 114.094843°	arccos[-1/sqrt(n)]
Dihedral angles prism - rhomb.	135°	arccos[-1/sqrt(3)] 125.264390°	120°	arccos[-1/sqrt(5)] 116.565051°	arccos[-1/sqrt(n-1)]
Dihedral angles rhomb. - rhomb.	135°	90°	90°	90°	90°
Dimension	7D	8D	9D	10D	nD
Dynkin diagram	o3o3o3o3x3o4x	o3o3o3o3o3x3o4x	o3o3o3o3o3o3x3o4x	o3o3o3o3o3o3o3x3o4x	o3o...o3x3o4x
Acronym	sersa	soro	?	?	rhomb. n-hypercube
Vertex Count	2688	7168	18432	46080	n(n-1) 2^n-1
Facet Count rect. simpl.	128`ril`	256`roc`	512`rene`	1024`reday`	2ⁿ
Facet Count prism	448`hixip`	1024`hopip`	2304`ocpe`	5120`enep`	n 2^n-1
Facet Count rhomb. hyp.cube	14`srox`	16`sersa`	18`soro`	20`?`	2n
Circumradius	sqrt[17+10 sqrt(2)]/2 2.790257	sqrt[5+3 sqrt(2)] 3.040171	sqrt[23+14 sqrt(2)]/2 3.271047	sqrt[(13+8 sqrt(2))/2] 3.486668	sqrt[3n-4+2(n-2) sqrt(2)]/2
Inradius wrt. rect. simp. facets	sqrt[(99+70 sqrt(2))/28] 2.659182	sqrt[17+12 sqrt(2)]/2 2.914214	sqrt[179+126 sqrt(2)]/6 3.149916	sqrt[(57+40 sqrt(2))/10] 3.369993	sqrt[(3n²-8n+8+2n(n-2) sqrt(2))/4n]
Inradius wrt. prism facets	sqrt[(43+30 sqrt(2))/12] 2.668121	sqrt[(121+84 sqrt(2))/28] 2.926443	sqrt[81+56 sqrt(2)]/4 3.164214	sqrt[209+144 sqrt(2)]/6 3.385618	sqrt[(3n²-10n+9+2(n-1)(n-2) sqrt(2))/(4(n-1))]
Inradius wrt. rh. hyp.cub. facets	(1+sqrt(2))/2 1.207107	(1+sqrt(2))/2 1.207107	(1+sqrt(2))/2 1.207107	(1+sqrt(2))/2 1.207107	(1+sqrt(2))/2 1.207107
Volume	?	?	?	?	?
Surface	?	?	?	?	?
Dihedral angles rect. - prism	arccos[-sqrt(6/7)] 157.792346°	arccos[-sqrt(7/8)] 159.295189°	arccos[-sqrt(8/9)] 160.528779°	arccos[-sqrt(9/10)] 161.565051°	arccos[-sqrt((n-1)/n)]
Dihedral angles rect. - rhomb.	arccos[-1/sqrt(7)] 112.207654°	arccos[-1/sqrt(8)] 110.704811°	arccos[-1/3] 109.471221°	arccos[-1/sqrt(10)] 108.434949°	arccos[-1/sqrt(n)]
Dihedral angles prism - rhomb.	arccos[-1/sqrt(6)] 114.094843°	arccos[-1/sqrt(7)] 112.207654°	arccos[-1/sqrt(8)] 110.704811°	arccos[-1/3] 109.471221°	arccos[-1/sqrt(n-1)]
Dihedral angles rhomb. - rhomb.	90°	90°	90°	90°	90°

Quasihombateded Hypercube qrbC_n (up)

Dimension	3D	4D	5D	6D	nD
Dynkin diagram	x3o4/3x	o3x3o4/3x	o3o3x3o4/3x	o3o3o3x3o4/3x	o3o...o3x3o4/3x
Acronym	querco	qrit	quarn	qrax	quasirhomb. n-hypercube
Vertex Count	24	96	320	960	n(n-1) 2^n-1
Facet Count rect. simpl.	8`trig`	16`oct`	32`rap`	64`rix`	2ⁿ
Facet Count prism	12`square`	32`trip`	80`tepe`	192`penp`	n 2^n-1
Facet Count qu.rh. hyp.cube	6`square`	8`querco`	10`qrit`	12`quarn`	2n
Circumradius	sqrt[5-2 sqrt(2)]/2 0.736813	sqrt[2-sqrt(2)] 0.765367	[3-sqrt(2)]/2 0.792893	sqrt[(7-4 sqrt(2))/2] 0.819496	sqrt[3n-4-2(n-2) sqrt(2)]/2
Inradius wrt. rect. simp. facets	[3-sqrt(2)]/sqrt(12) 0.457777	1-1/sqrt(2) 0.292893	sqrt[(43-30 sqrt(2))/20] 0.169351	sqrt[(17-12 sqrt(2))/6] 0.0700443	sqrt[(3n²-8n+8-2n(n-2) sqrt(2))/4n]
Inradius wrt. prism facets	(sqrt(2)-1)/2 0.207107	sqrt[(17-12 sqrt(2))/12] 0.0495288	sqrt[(17-12 sqrt(2))/8] 0.0606602	sqrt[(57-40 sqrt(2))/20] 0.146877	sqrt[(3n²-10n+9-2(n-1)(n-2) sqrt(2))/(4(n-1))]
Inradius wrt. qrh. hyp.cub. facets	(sqrt(2)-1)/2 0.207107	(sqrt(2)-1)/2 0.207107	(sqrt(2)-1)/2 0.207107	(sqrt(2)-1)/2 0.207107	(sqrt(2)-1)/2 0.207107
Volume	[10 sqrt(2)-12]/3 0.714045	[32 sqrt(2)-45]/3 0.0849447	[1205-843 sqrt(2)]/30 0.427266	[3141 sqrt(2)-4426]/45 0.356551	?
Surface	18+2 sqrt(3) 21.464102	?	?	?	?
Dihedral angles rect. - prism	arccos[sqrt(2/3)] 35.264390°	30°	arccos[2/sqrt(5)] 26.565051°	arccos[sqrt(5/6)] 24.094843°	arccos[sqrt((n-1)/n)]
Dihedral angles rect. - qu.rh.	arccos[sqrt(2/3)] 35.264390°	60°	arccos[1/sqrt(5)] 63.434949°	arccos[1/sqrt(6)] 65.905157°	arccos[1/sqrt(n)]
Dihedral angles prism - qu.rh.	45°	arccos[1/sqrt(3)] 54.735610°	60°	arccos[1/sqrt(5)] 63.434949°	arccos[1/sqrt(n-1)]
Dihedral angles qu.rh. - qu.rh.	45°	90°	90°	90°	90°
Dimension	7D	8D	9D	10D	nD
Dynkin diagram	o3o3o3o3x3o4/3x	o3o3o3o3o3x3o4/3x	o3o3o3o3o3o3x3o4/3x	o3o3o3o3o3o3o3x3o4/3x	o3o...o3x3o4/3x
Acronym	quersa	qro	?	?	quasirhomb. n-hypercube
Vertex Count	2688	7168	18432	46080	n(n-1) 2^n-1
Facet Count rect. simpl.	128`ril`	256`roc`	512`rene`	1024`reday`	2ⁿ
Facet Count prism	448`hixip`	1024`hopip`	2304`ocpe`	5120`enep`	n 2^n-1
Facet Count qu.rh. hyp.cube	14`qrax`	16`quersa`	18`qro`	20`?`	2n
Circumradius	sqrt[17-10 sqrt(2)]/2 0.845261	sqrt[5-3 sqrt(2)] 0.870264	sqrt[23-14 sqrt(2)]/2 0.894568	sqrt[(13-8 sqrt(2))/2] 0.918230	sqrt[3n-4-2(n-2) sqrt(2)]/2
Inradius wrt. rect. simp. facets	sqrt[(99-70 sqrt(2))/28] 0.0134306	sqrt[17-12 sqrt(2)]/2 0.0857864	sqrt[179-126 sqrt(2)]/6 0.149916	sqrt[(57-40 sqrt(2))/10] 0.207716	sqrt[(3n²-8n+8-2n(n-2) sqrt(2))/4n]
Inradius wrt. prism facets	sqrt[(43-30 sqrt(2))/12] 0.218631	sqrt[(121-84 sqrt(2))/28] 0.280692	sqrt[81-56 sqrt(2)]/4 0.335786	sqrt[209-144 sqrt(2)]/6 0.385618	sqrt[(3n²-10n+9-2(n-1)(n-2) sqrt(2))/(4(n-1))]
Inradius wrt. qrh. hyp.cub. facets	(sqrt(2)-1)/2 0.207107	(sqrt(2)-1)/2 0.207107	(sqrt(2)-1)/2 0.207107	(sqrt(2)-1)/2 0.207107	(sqrt(2)-1)/2 0.207107
Volume	?	?	?	?	?
Surface	?	?	?	?	?
Dihedral angles rect. - prism	arccos[sqrt(6/7)] 22.207654°	arccos[sqrt(7/8)] 20.704811°	arccos[sqrt(8/9)] 19.471221°	arccos[sqrt(9/10)] 18.434949°	arccos[sqrt((n-1)/n)]
Dihedral angles rect. - qu.rh.	arccos[1/sqrt(7)] 67.792346°	arccos[1/sqrt(8)] 69.295189°	arccos[1/3] 70.528779°	arccos[1/sqrt(10)] 84.260830°	arccos[1/sqrt(n)]
Dihedral angles prism - qu.rh.	arccos[1/sqrt(6)] 65.905157°	arccos[1/sqrt(7)] 67.792346°	arccos[1/sqrt(8)] 69.295189°	arccos[1/3] 70.528779°	arccos[1/sqrt(n-1)]
Dihedral angles qu.rh. - qu.rh.	90°	90°	90°	90°	90°

Maximal Expanded Hypercube eC_n (up)

Within these polytopes eC_n generally can be described as the tristratic lace tower oftheregular hypercube C_n-1 atop the maximal expanded hypercube eC_n-1 atop a further maximal expanded hypercube eC_n-1 atop the regular hypercube C_n-1.Thence, by means of thelace tower notation,eC_n = x3o...o3o4x (n nodes) can be described as well asoxxo3oooo...oooo3oooo4xxxx&#xt (n-1 node positions).

Dimension	3D	4D	5D	6D	nD
Dynkin diagram	x3o4x	x3o3o4x	x3o3o3o4x	x3o3o3o3o4x	x3o...o3o4x
Acronym	sirco	sidpith	scant	stoxog	max-exp. n-hypercube
Vertex Count	24	64	160	384	n 2ⁿ
Facet Count simplex	8`trig`	16`tet`	32`pen`	64`hix`	2ⁿ n! 2^n-0/[(n-0)!0!]
Facet Count prism I	12`square`	32`trip`	80`tepe`	192`penp`	n 2^n-1 n! 2^n-1/[(n-1)!1!]
Facet Count duoprism I			80`tisdip`	240`squatet`	n(n-1) 2^n-3 n! 2^n-2/[(n-2)!2!]
Facet Count duoprism II			80`tisdip`	160`tracube`	n(n-1)(n-2) 2^n-4/3 n! 2^n-3/[(n-3)!3!]
Facet Count prism II		24`cube`	40`tes`	60`pent`	2n(n-1) n! 2²/[2!(n-2)!]
Facet Count hypercube	6`square`	8`cube`	10`tes`	12`pent`	2n n! 2¹/[1!(n-1)!]
Circumradius	sqrt[5+2 sqrt(2)]/2 1.398966	sqrt[(3+sqrt(2))/2] 1.485633	sqrt[7+2 sqrt(2)]/2 1.567516	sqrt[2+1/sqrt(2)] 1.645329	sqrt[(n+2)+sqrt(8)]/2
Inradius wrt. simplex facets	[3+sqrt(2)]/sqrt(12) 1.274274	[1+2 sqrt(2)]/sqrt(8) 1.353553	[5+sqrt(2)]/sqrt(20) 1.434262	[1+3 sqrt(2)]/sqrt(12) 1.513420	[n+sqrt(2)]/sqrt(4n)
Inradius wrt. prism I facets	(1+sqrt(2))/2 1.207107	[3+sqrt(2)]/sqrt(12) 1.274274	[1+2 sqrt(2)]/sqrt(8) 1.353553	[5+sqrt(2)]/sqrt(20) 1.434262	[(n-1)+sqrt(2)]/sqrt[4(n-1)]
Inradius wrt. d.pr. I fac.			[3+sqrt(2)]/sqrt(12) 1.274274	[1+2 sqrt(2)]/sqrt(8) 1.353553	[(n-2)+sqrt(2)]/sqrt[4(n-2)]
Inradius wrt. d.pr. II fac.			[3+sqrt(2)]/sqrt(12) 1.274274	[3+sqrt(2)]/sqrt(12) 1.274274	[(n-3)+sqrt(2)]/sqrt[4(n-3)]
Inradius wrt. prism II facets		(1+sqrt(2))/2 1.207107	(1+sqrt(2))/2 1.207107	(1+sqrt(2))/2 1.207107	(1+sqrt(2))/2 1.207107
Inradius wrt. hyp.cube fac.	(1+sqrt(2))/2 1.207107	(1+sqrt(2))/2 1.207107	(1+sqrt(2))/2 1.207107	(1+sqrt(2))/2 1.207107	(1+sqrt(2))/2 1.207107
Volume	[12+10 sqrt(2)]/3 8.714045	[43+32 sqrt(2)]/6 14.709139	[355+251 sqrt(2)]/30 23.665587	[833+579 sqrt(2)]/45 36.707326	?
Surface	18+2 sqrt(3) 21.464102	[96+4 sqrt(2)+24 sqrt(3)]/3 47.742025	[150+20 sqrt(2)+60 sqrt(3)+sqrt(5)]/3 94.814463	[1080+300 sqrt(2)+601 sqrt(3)+30 sqrt(5)]/15 174.153910	?
Dihedral angles simplex - (next)	arccos[-sqrt(2/3)] 144.735610°	150°	arccos[-2/sqrt(5)] 153.434949°	arccos[-sqrt(5/6)] 155.905157°	arccos[-sqrt((n-1)/n)]
Dihedral angles prism I - (next)	135°	arccos[-sqrt(2/3)] 144.735610°	150°	arccos[-2/sqrt(5)] 153.434949°	arccos[-sqrt((n-2)/(n-1))]
Dihedral angles d.pr. I - (next)			arccos[-sqrt(2/3)] 144.735610°	150°	arccos[-sqrt((n-3)/(n-2))]
Dihedral angles d.pr. II - (next)			arccos[-sqrt(2/3)] 144.735610°	arccos[-sqrt(2/3)] 144.735610°	arccos[-sqrt((n-4)/(n-3))]
Dihedral angles prism II - hyp.cube		135°	135°	135°	135°
Dimension	7D	8D	9D	10D	nD
Dynkin diagram	x3o3o3o3o3o4x	x3o3o3o3o3o3o4x	x3o3o3o3o3o3o3o4x	x3o3o3o3o3o3o3o3o4x	x3o...o3o4x
Acronym	suposaz	saxoke	?	?	max-exp. n-hypercube
Vertex Count	896	2048	4608	10240	n 2ⁿ
Facet Count simplex	128`hop`	256`oca`	512`ene`	1024`day`	2ⁿ n! 2^n-0/[(n-0)!0!]
Facet Count prism I	448`hixip`	1024`hopip`	2304`ocpe`	5120`enep`	n 2^n-1 n! 2^n-1/[(n-1)!1!]
Facet Count duoprism I	672`squapen`	1792`squahix`	4608`squahop`	11520`squoc`	n(n-1) 2^n-3 n! 2^n-2/[(n-2)!2!]
Facet Count duoprism II	560`tetcube`	1792`cubpen`	5376`cubhix`	15360`cubhop`	n(n-1)(n-2) 2^n-4/3 n! 2^n-3/[(n-3)!3!]
Facet Count duoprism III	280`tratess`	1120`tettes`	4032`pentes`	13440`teshix`	n! 2^n-4/[(n-4)!4!]
Facet Count duoprism IV		448`trapent`	2016`tetpent`	8064`penpent`	n! 2^n-5/[(n-5)!5!]
Facet Count duoprism V			672`triax`	3360`tetax`	n! 2^n-6/[(n-6)!6!]
Facet Count duoprism VI			672`triax`	960`tetax`	n! 2^n-7/[(n-7)!7!]
Facet Count prism II	84`ax`	112`hept`	144`octo`	180`enne`	2n(n-1) n! 2²/[2!(n-2)!]
Facet Count hypercube	14`ax`	16`hept`	18`octo`	20`enne`	2n n! 2¹/[1!(n-1)!]
Circumradius	sqrt[9+2 sqrt(2)]/2 1.719624	sqrt[(5+sqrt(2))/2] 1.790840	sqrt[11+2 sqrt(2)]/2 1.859330	sqrt[(6+sqrt(2))/2] 1.925385	sqrt[(n+2)+sqrt(8)]/2
Inradius wrt. simplex facets	[7+sqrt(2)]/sqrt(28) 1.590137	[1+4 sqrt(2)]/4 1.664214	[9+sqrt(2)]/6 1.735702	[1+5 sqrt(2)]/sqrt(20) 1.804746	[n+sqrt(2)]/sqrt(4n)
Inradius wrt. prism I facets	[1+3 sqrt(2)]/sqrt(12) 1.513420	[7+sqrt(2)]/sqrt(28) 1.590137	[1+4 sqrt(2)]/4 1.664214	[9+sqrt(2)]/6 1.735702	[(n-1)+sqrt(2)]/sqrt[4(n-1)]
Inradius wrt. d.pr. I fac.	[5+sqrt(2)]/sqrt(20) 1.434262	[1+3 sqrt(2)]/sqrt(12) 1.513420	[7+sqrt(2)]/sqrt(28) 1.590137	[1+4 sqrt(2)]/4 1.664214	[(n-2)+sqrt(2)]/sqrt[4(n-2)]
Inradius wrt. d.pr. II fac.	[1+2 sqrt(2)]/sqrt(8) 1.353553	[5+sqrt(2)]/sqrt(20) 1.434262	[1+3 sqrt(2)]/sqrt(12) 1.513420	[7+sqrt(2)]/sqrt(28) 1.590137	[(n-3)+sqrt(2)]/sqrt[4(n-3)]
Inradius wrt. d.pr. III fac.	[3+sqrt(2)]/sqrt(12) 1.274274	[1+2 sqrt(2)]/sqrt(8) 1.353553	[5+sqrt(2)]/sqrt(20) 1.434262	[1+3 sqrt(2)]/sqrt(12) 1.513420	[(n-4)+sqrt(2)]/sqrt[4(n-4)]
Inradius wrt. d.pr. IV fac.		[3+sqrt(2)]/sqrt(12) 1.274274	[1+2 sqrt(2)]/sqrt(8) 1.353553	[5+sqrt(2)]/sqrt(20) 1.434262	[(n-5)+sqrt(2)]/sqrt[4(n-5)]
Inradius wrt. d.pr. V fac.			[3+sqrt(2)]/sqrt(12) 1.274274	[1+2 sqrt(2)]/sqrt(8) 1.353553	[(n-6)+sqrt(2)]/sqrt[4(n-6)]
Inradius wrt. d.pr. VI fac.			[3+sqrt(2)]/sqrt(12) 1.274274	[3+sqrt(2)]/sqrt(12) 1.274274	[(n-7)+sqrt(2)]/sqrt[4(n-7)]
Inradius wrt. prism II facets	(1+sqrt(2))/2 1.207107	(1+sqrt(2))/2 1.207107	(1+sqrt(2))/2 1.207107	(1+sqrt(2))/2 1.207107	(1+sqrt(2))/2 1.207107
Inradius wrt. hyp.cube fac.	(1+sqrt(2))/2 1.207107	(1+sqrt(2))/2 1.207107	(1+sqrt(2))/2 1.207107	(1+sqrt(2))/2 1.207107	(1+sqrt(2))/2 1.207107
Volume	[8792+6101 sqrt(2)]/315 55.301959	?	?	?	?
Surface	?	?	?	?	?
Dihedral angles simplex - (next)	arccos[-sqrt(6/7)] 157.792346°	arccos[-sqrt(7/8)] 159.295189°	arccos[-sqrt(8)/3] 160.528779°	arccos[-3/sqrt(10)] 161.565051°	arccos[-sqrt((n-1)/n)]
Dihedral angles prism - (next)	arccos[-sqrt(5/6)] 155.905157°	arccos[-sqrt(6/7)] 157.792346°	arccos[-sqrt(7/8)] 159.295189°	arccos[-sqrt(8)/3] 160.528779°	arccos[-sqrt((n-2)/(n-1))]
Dihedral angles d.pr. I - (next)	arccos[-2/sqrt(5)] 153.434949°	arccos[-sqrt(5/6)] 155.905157°	arccos[-sqrt(6/7)] 157.792346°	arccos[-sqrt(7/8)] 159.295189°	arccos[-sqrt((n-3)/(n-2))]
Dihedral angles d.pr. II - (next)	150°	arccos[-2/sqrt(5)] 153.434949°	arccos[-sqrt(5/6)] 155.905157°	arccos[-sqrt(6/7)] 157.792346°	arccos[-sqrt((n-4)/(n-3))]
Dihedral angles d.pr. III - (next)	arccos[-sqrt(2/3)] 144.735610°	150°	arccos[-2/sqrt(5)] 153.434949°	arccos[-sqrt(5/6)] 155.905157°	arccos[-sqrt((n-5)/(n-4))]
Dihedral angles d.pr. IV - (next)		arccos[-sqrt(2/3)] 144.735610°	150°	arccos[-2/sqrt(5)] 153.434949°	arccos[-sqrt((n-6)/(n-5))]
Dihedral angles d.pr. V - (next)			arccos[-sqrt(2/3)] 144.735610°	150°	arccos[-sqrt((n-7)/(n-6))]
Dihedral angles d.pr. VI - (next)			arccos[-sqrt(2/3)] 144.735610°	arccos[-sqrt(2/3)] 144.735610°	arccos[-sqrt((n-8)/(n-7))]
Dihedral angles prism II - hyp.cube	135°	135°	135°	135°	135°

Quasiexpanded Hypercube qeC_n (up)

These non-convex polytopes qeC_n generally are nothing but the conjugates of themaximal expanded hypercube eC_n.

Note that the pattern of retrogradeness, which is required for the correct conjugacy of thevolume terms, has an interruption between the fourth and fifth dimension.This simply is because elsewise the volume values themselves would become negative, that is the choice of retrogradenessesjust got reversed thereafter.

Dimension	3D	4D	5D	6D	nD
Dynkin diagram	x3o4/3x	x3o3o4/3x	x3o3o3o4/3x	x3o3o3o3o4/3x	x3o...o3o4/3x
Acronym	querco	quidpith	quacant	quitoxog	quasiexp. n-hypercube
Vertex Count	24	64	160	384	n 2ⁿ
Facet Count simplex	8`trig`	16`tet`	32`pen`	64`hix`	2ⁿ n! 2^n-0/[(n-0)!0!]
Facet Count prism I	12`square`	32`trip`	80`tepe`	192`penp`	n 2^n-1 n! 2^n-1/[(n-1)!1!]
Facet Count duoprism I			80`tisdip`	240`squatet`	n(n-1) 2^n-3 n! 2^n-2/[(n-2)!2!]
Facet Count duoprism II			80`tisdip`	160`tracube`	n(n-1)(n-2) 2^n-4/3 n! 2^n-3/[(n-3)!3!]
Facet Count prism II		24`cube`	40`tes`	60`pent`	2n(n-1) n! 2²/[2!(n-2)!]
Facet Count hypercube	6`square`	8`cube`	10`tes`	12`pent`	2n n! 2¹/[1!(n-1)!]
Circumradius	sqrt[5-2 sqrt(2)]/2 0.736813	sqrt[(3-sqrt(2))/2] 0.890446	sqrt[7-2 sqrt(2)]/2 1.021221	sqrt[2-1/sqrt(2)] 1.137055	sqrt[(n+2)-sqrt(8)]/2
Inradius wrt. simplex facets	-[3-sqrt(2)]/sqrt(12) -0.457777	[2 sqrt(2)-1]/sqrt(8) 0.646447	[5-sqrt(2)]/sqrt(20) 0.801806	-[3 sqrt(2)-1]/sqrt(12) -0.936070	[n-sqrt(2)]/sqrt(4n)
Inradius wrt. prism I facets	(sqrt(2)-1)/2 0.207107	-[3-sqrt(2)]/sqrt(12) -0.457777	-[2 sqrt(2)-1]/sqrt(8) -0.646447	[5-sqrt(2)]/sqrt(20) 0.801806	[(n-1)-sqrt(2)]/sqrt[4(n-1)]
Inradius wrt. d.pr. I fac.			[3-sqrt(2)]/sqrt(12) 0.457777	-[2 sqrt(2)-1]/sqrt(8) -0.646447	[(n-2)-sqrt(2)]/sqrt[4(n-2)]
Inradius wrt. d.pr. II fac.			[3-sqrt(2)]/sqrt(12) 0.457777	[3-sqrt(2)]/sqrt(12) 0.457777	[(n-3)-sqrt(2)]/sqrt[4(n-3)]
Inradius wrt. prism II facets		(sqrt(2)-1)/2 0.207107	-(sqrt(2)-1)/2 -0.207107	-(sqrt(2)-1)/2 -0.207107	+/− (sqrt(2)-1)/2 0.207107
Inradius wrt. hyp.cube fac.	(sqrt(2)-1)/2 0.207107	(sqrt(2)-1)/2 0.207107	-(sqrt(2)-1)/2 -0.207107	-(sqrt(2)-1)/2 -0.207107	+/− (sqrt(2)-1)/2 0.207107
Volume	[10 sqrt(2)-12]/3 0.392837	[32 sqrt(2)-43]/6 0.375806	[355-251 sqrt(2)]/30 0.0010799	[833-579 sqrt(2)]/45 0.314897	?
Surface	18+2 sqrt(3) 21.464102	[96+4 sqrt(2)+24 sqrt(3)]/3 47.742025	[150+20 sqrt(2)+60 sqrt(3)+sqrt(5)]/3 94.814463	[1080+300 sqrt(2)+601 sqrt(3)+30 sqrt(5)]/15 174.153910	?
Dihedral angles simplex - (next)	arccos[sqrt(2/3)] 35.264390°	30°	arccos[2/sqrt(5)] 26.565051°	arccos[sqrt(5/6)] 24.094843°	arccos[sqrt((n-1)/n)]
Dihedral angles prism I - (next)	45°	arccos[sqrt(2/3)] 35.264390°	30°	arccos[2/sqrt(5)] 26.565051°	arccos[sqrt((n-2)/(n-1))]
Dihedral angles d.pr. I - (next)			arccos[sqrt(2/3)] 35.264390°	30°	arccos[sqrt((n-3)/(n-2))]
Dihedral angles d.pr. II - (next)			arccos[sqrt(2/3)] 35.264390°	arccos[sqrt(2/3)] 35.264390°	arccos[sqrt((n-4)/(n-3))]
Dihedral angles prism II - hyp.cube		45°	45°	45°	45°
Dimension	7D	8D	9D	10D	nD
Dynkin diagram	x3o3o3o3o3o4/3x	x3o3o3o3o3o3o4/3x	x3o3o3o3o3o3o3o4/3x	x3o3o3o3o3o3o3o3o4/3x	x3o...o3o4/3x
Acronym	quiposaz	quaxoke	?	?	quasiexp. n-hypercube
Vertex Count	896	2048	4608	10240	n 2ⁿ
Facet Count simplex	128`hop`	256`oca`	512`ene`	1024`day`	2ⁿ n! 2^n-0/[(n-0)!0!]
Facet Count prism I	448`hixip`	1024`hopip`	2304`ocpe`	5120`enep`	n 2^n-1 n! 2^n-1/[(n-1)!1!]
Facet Count duoprism I	672`squapen`	1792`squahix`	4608`squahop`	11520`squoc`	n(n-1) 2^n-3 n! 2^n-2/[(n-2)!2!]
Facet Count duoprism II	560`tetcube`	1792`cubpen`	5376`cubhix`	15360`cubhop`	n(n-1)(n-2) 2^n-4/3 n! 2^n-3/[(n-3)!3!]
Facet Count duoprism III	280`tratess`	1120`tettes`	4032`pentes`	13440`teshix`	n! 2^n-4/[(n-4)!4!]
Facet Count duoprism IV		448`trapent`	2016`tetpent`	8064`penpent`	n! 2^n-5/[(n-5)!5!]
Facet Count duoprism V			672`triax`	3360`tetax`	n! 2^n-6/[(n-6)!6!]
Facet Count duoprism VI			672`triax`	960`tetax`	n! 2^n-7/[(n-7)!7!]
Facet Count prism II	84`ax`	112`hept`	144`octo`	180`enne`	2n(n-1) n! 2²/[2!(n-2)!]
Facet Count hypercube	14`ax`	16`hept`	18`octo`	20`enne`	2n n! 2¹/[1!(n-1)!]
Circumradius	sqrt[9-2 sqrt(2)]/2 1.242133	sqrt[(5-sqrt(2))/2] 1.338990	sqrt[11-2 sqrt(2)]/2 1.429298	sqrt[(6-sqrt(2))/2] 1.514230	sqrt[(n+2)-sqrt(8)]/2
Inradius wrt. simplex facets	[7-sqrt(2)]/sqrt(28) 1.055614	-[4 sqrt(2)-1]/4 -1.164214	[9-sqrt(2)]/6 1.264298	-[5 sqrt(2)-1]/sqrt(20) -1.357532	[n-sqrt(2)]/sqrt(4n)
Inradius wrt. prism I facets	-[3 sqrt(2)-1]/sqrt(12) -0.936070	[7-sqrt(2)]/sqrt(28) 1.055614	-[4 sqrt(2)-1]/4 -1.164214	[9-sqrt(2)]/6 1.264298	[(n-1)-sqrt(2)]/sqrt[4(n-1)]
Inradius wrt. d.pr. I fac.	[5-sqrt(2)]/sqrt(20) 0.801806	-[3 sqrt(2)-1]/sqrt(12) -0.936070	[7-sqrt(2)]/sqrt(28) 1.055614	-[4 sqrt(2)-1]/4 -1.164214	[(n-2)-sqrt(2)]/sqrt[4(n-2)]
Inradius wrt. d.pr. II fac.	-[2 sqrt(2)-1]/sqrt(8) -0.646447	[5-sqrt(2)]/sqrt(20) 0.801806	-[3 sqrt(2)-1]/sqrt(12) -0.936070	[7-sqrt(2)]/sqrt(28) 1.055614	[(n-3)-sqrt(2)]/sqrt[4(n-3)]
Inradius wrt. d.pr. III fac.	[3-sqrt(2)]/sqrt(12) 0.457777	-[2 sqrt(2)-1]/sqrt(8) -0.646447	[5-sqrt(2)]/sqrt(20) 0.801806	-[3 sqrt(2)-1]/sqrt(12) -0.936070	[(n-4)-sqrt(2)]/sqrt[4(n-4)]
Inradius wrt. d.pr. IV fac.		[3-sqrt(2)]/sqrt(12) 0.457777	-[2 sqrt(2)-1]/sqrt(8) -0.646447	[5-sqrt(2)]/sqrt(20) 0.801806	[(n-5)-sqrt(2)]/sqrt[4(n-5)]
Inradius wrt. d.pr. V fac.			[3-sqrt(2)]/sqrt(12) 0.457777	-[2 sqrt(2)-1]/sqrt(8) -0.646447	[(n-6)-sqrt(2)]/sqrt[4(n-6)]
Inradius wrt. d.pr. VI fac.			[3-sqrt(2)]/sqrt(12) 0.457777	[3-sqrt(2)]/sqrt(12) 0.457777	[(n-7)-sqrt(2)]/sqrt[4(n-7)]
Inradius wrt. prism II facets	-(sqrt(2)-1)/2 -0.207107	-(sqrt(2)-1)/2 -0.207107	-(sqrt(2)-1)/2 -0.207107	-(sqrt(2)-1)/2 -0.207107	-(sqrt(2)-1)/2 -0.207107
Inradius wrt. hyp.cube fac.	-(sqrt(2)-1)/2 -0.207107	-(sqrt(2)-1)/2 -0.207107	-(sqrt(2)-1)/2 -0.207107	-(sqrt(2)-1)/2 -0.207107	-(sqrt(2)-1)/2 -0.207107
Volume	[8792-6101 sqrt(2)]/315 0.520264	?	?	?	?
Surface	?	?	?	?	?
Dihedral angles simplex - (next)	arccos[sqrt(6/7)] 22.207654°	arccos[sqrt(7/8)] 20.704811°	arccos[sqrt(8)/3] 19.471221°	arccos[3/sqrt(10)] 18.434949°	arccos[sqrt((n-1)/n)]
Dihedral angles prism - (next)	arccos[sqrt(5/6)] 24.094843°	arccos[sqrt(6/7)] 22.207654°	arccos[sqrt(7/8)] 20.704811°	arccos[sqrt(8)/3] 19.471221°	arccos[sqrt((n-2)/(n-1))]
Dihedral angles d.pr. I - (next)	arccos[2/sqrt(5)] 26.565051°	arccos[sqrt(5/6)] 24.094843°	arccos[sqrt(6/7)] 22.207654°	arccos[sqrt(7/8)] 20.704811°	arccos[sqrt((n-3)/(n-2))]
Dihedral angles d.pr. II - (next)	30°	arccos[2/sqrt(5)] 26.565051°	arccos[sqrt(5/6)] 24.094843°	arccos[sqrt(6/7)] 22.207654°	arccos[sqrt((n-4)/(n-3))]
Dihedral angles d.pr. III - (next)	arccos[sqrt(2/3)] 35.264390°	30°	arccos[2/sqrt(5)] 26.565051°	arccos[sqrt(5/6)] 24.094843°	arccos[sqrt((n-5)/(n-4))]
Dihedral angles d.pr. IV - (next)		arccos[sqrt(2/3)] 35.264390°	30°	arccos[2/sqrt(5)] 26.565051°	arccos[sqrt((n-6)/(n-5))]
Dihedral angles d.pr. V - (next)			arccos[sqrt(2/3)] 35.264390°	30°	arccos[sqrt((n-7)/(n-6))]
Dihedral angles d.pr. VI - (next)			arccos[sqrt(2/3)] 35.264390°	arccos[sqrt(2/3)] 35.264390°	arccos[sqrt((n-8)/(n-7))]
Dihedral angles prism II - hyp.cube	45°	45°	45°	45°	45°

Retroexpanded Hypercube reC_n (up)

These non-convex polytopes reC_n (a.k.a. socco series) generally are facetings of themaximal expanded hypercube eC_n.

All facets within each member of this series here are obviously prograde, except for the simplices. Those however alternate within their retrogradeness wrt. the dimensional n quite similarily as they did for thefacetorectified hypercubes frC_n.

Dimension	3D	4D	5D	6D	nD
Dynkin diagram	o3x4x4/3*a	o3x4x4/3*a3o	o3x4x4/3*a3o3o	o3x4x4/3*a3o3o3o	o3x4x4/3*a3o...o3o
Acronym	socco	steth	sinnont	soxaxog	retroexp. hypercube
Vertex Count	24	64	160	384	n 2ⁿ
Facet Count simplex	8`trig`	16`tet`	32`pen`	64`hix`	2ⁿ
Facet Count hypercube	6`square`	8`cube`	10`tes`	12`pent`	2n
Facet Count soc. ser. mem.	6`oc`	8`socco`	10`steth`	12`sinnont`	2n
Circumradius	sqrt[5+2 sqrt(2)]/2 1.398966	sqrt[(3+sqrt(2))/2] 1.485633	sqrt[7+2 sqrt(2)]/2 1.567516	sqrt[2+1/sqrt(2)] 1.645329	sqrt[(n+2)+sqrt(8)]/2
Inradius wrt. simplex facets	-[3+sqrt(2)]/sqrt(12) -1.274274	[1+2 sqrt(2)]/sqrt(8) 1.353553	-[5+sqrt(2)]/sqrt(20) -1.434262	[1+3 sqrt(2)]/sqrt(12) 1.513420	+/− [n+sqrt(2)]/sqrt(4n)
Inradius wrt. hyp.cube fac.	(1+sqrt(2))/2 1.207107	(1+sqrt(2))/2 1.207107	(1+sqrt(2))/2 1.207107	(1+sqrt(2))/2 1.207107	(1+sqrt(2))/2 1.207107
Inradius wrt. soc. ser. mem.	1/2 0.5	1/2 0.5	1/2 0.5	1/2 0.5	1/2 0.5
Volume	[6+8 sqrt(2)]/3 5.771236	[19+24 sqrt(2)]/6 8.823521	[120+149 sqrt(2)]/30 11.023927	[451+540 sqrt(2)]/90 13.496392	?
Surface	18+12 sqrt(2)+2 sqrt(3) 38.434664	[72+68 sqrt(2)]/3 56.055507	[125+120 sqrt(2)+sqrt(5)]/3 98.980565	[900+894 sqrt(2)+2 sqrt(3)]/15 144.518068	?
Dihedral angles simplex - soc.s.m.	arccos[1/sqrt(3)] 54.735610°	60°	arccos[1/sqrt(5)] 63.434949°	arccos[1/sqrt(6)] 65.905157°	arccos[1/sqrt(n)]
Dihedral angles hyp.cub. - soc.s.m.	90°	90°	90°	90°	90°
Dihedral angles soc.s.m. - soc.s.m.	90°	90°	90°	90°	90°
Dimension	7D	8D	9D	10D	nD
Dynkin diagram	o3x4x4/3*a3o3o3o3o	o3x4x4/3*a3o3o3o3o3o	o3x4x4/3*a3o3o3o3o3o3o	o3x4x4/3*a3o3o3o3o3o3o3o	o3x4x4/3*a3o...o3o
Acronym	sososaz	sook	?	?	retroexp. hypercube
Vertex Count	896	2048	4608	10240	n 2ⁿ
Facet Count simplex	128`hop`	256`oca`	512`ene`	1024`day`	2ⁿ
Facet Count hypercube	14`ax`	16`hept`	18`octo`	20`enne`	2n
Facet Count soc. ser. mem.	14`soxaxog`	16`sososaz`	18`sook`	20`?`	2n
Circumradius	sqrt[9+2 sqrt(2)]/2 1.719624	sqrt[(5+sqrt(2))/2] 1.790840	sqrt[11+2 sqrt(2)]/2 1.859330	sqrt[(6+sqrt(2))/2] 1.925385	sqrt[(n+2)+sqrt(8)]/2
Inradius wrt. simplex facets	-[7+sqrt(2)]/sqrt(28) -1.590137	[1+4 sqrt(2)]/4 1.664214	-[9+sqrt(2)]/6 -1.735702	[1+5 sqrt(2)]/sqrt(20) 1.804746	+/− [n+sqrt(2)]/sqrt(4n)
Inradius wrt. hyp.cube fac.	(1+sqrt(2))/2 1.207107	(1+sqrt(2))/2 1.207107	(1+sqrt(2))/2 1.207107	(1+sqrt(2))/2 1.207107	(1+sqrt(2))/2 1.207107
Inradius wrt. soc. ser. mem.	1/2 0.5	1/2 0.5	1/2 0.5	1/2 0.5	1/2 0.5
Volume	?	?	?	?	?
Surface	?	?	?	?	?
Dihedral angles simplex - soc.s.m.	arccos[1/sqrt(7)] 67.792346°	arccos[1/sqrt(8)] 69.295189°	arccos(1/3) 70.528779°	arccos[1/sqrt(10)] 71.565051°	arccos[1/sqrt(n)]
Dihedral angles hyp.cub. - soc.s.m.	90°	90°	90°	90°	90°
Dihedral angles soc.s.m. - soc.s.m.	90°	90°	90°	90°	90°

Quasiretroexpanded Hypercube qreC_n (up)

These non-convex polytopes qreC_n (a.k.a. gocco series) generally are memberwise conjugates of theretroexpanded hypercube reC_n.Thence they are related to thequasiexpanded hypercube qeC_nin a very similar way as the former had been to the non-quasi variants,i.e. themaximal expanded hypercubes eC_n.In fact the qeC_n are facetings of those.

Moreover it happens that all facets within each member of this series are fully prograde, without any exception. In fact their vertex figure always is convex. Thence those also are said to belocally convex.

Dimension	3D	4D	5D	6D	nD
Dynkin diagram	o3x4/3x4*a	o3x4/3x4*a3o	o3x4/3x4*a3o3o	o3x4/x43*a3o3o3o	o3x4/3x4*a3o...o3o
Acronym	gocco	gittith	ginnont	goxaxog	quasiretroexp. hyp.cube
Vertex Count	24	64	160	384	n 2ⁿ
Facet Count simplex	8`trig`	16`tet`	32`pen`	64`hix`	2ⁿ
Facet Count hypercube	6`square`	8`cube`	10`tes`	12`pent`	2n
Facet Count goc. ser. mem.	6`og`	8`gocco`	10`gittith`	12`ginnont`	2n
Circumradius	sqrt[5-2 sqrt(2)]/2 0.736813	sqrt[(3-sqrt(2))/2] 0.890446	sqrt[7-2 sqrt(2)]/2 1.021221	sqrt[(4-sqrt(2))/2] 1.137055	sqrt[(n+2)-sqrt(8)]/2
Inradius wrt. simplex facets	[3-sqrt(2)]/sqrt(12) 0.457777	[2 sqrt(2)-1]/sqrt(8) 0.646447	[5-sqrt(2)]/sqrt(20) 0.801806	[3 sqrt(2)-1]/sqrt(12) 0.936070	[n-sqrt(2)]/sqrt(4n)
Inradius wrt. hyp.cube fac.	(sqrt(2)-1)/2 0.207107	(sqrt(2)-1)/2 0.207107	(sqrt(2)-1)/2 0.207107	(sqrt(2)-1)/2 0.207107	(sqrt(2)-1)/2 0.207107
Inradius wrt. goc. ser. mem.	1/2 0.5	1/2 0.5	1/2 0.5	1/2 0.5	1/2 0.5
Volume	[8 sqrt(2)-6]/3 1.771236	[24 sqrt(2)-19]/6 2.490188	[149 sqrt(2)-120]/30 3.023927	[540 sqrt(2)-451]/90 3.474170	?
Surface	-6+12 sqrt(2)+2 sqrt(3) 14.434664	[68 sqrt(2)-24]/3 24.055507	[-65+120 sqrt(2)+sqrt(5)]/3 35.647232	[-540+894 sqrt(2)+2 sqrt(3)]/15 48.518068	?
Dihedral angles simplex - goc.s.m.	arccos[-1/sqrt(3)] 125.264390°	120°	arccos[-1/sqrt(5)] 116.565051°	arccos[-1/sqrt(6)] 114.094843°	arccos[-1/sqrt(n)]
Dihedral angles hyp.cub. - goc.s.m.	90°	90°	90°	90°	90°
Dihedral angles goc.s.m. - goc.s.m.	90°	90°	90°	90°	90°
Dimension	7D	8D	9D	10D	nD
Dynkin diagram	o3x4/3x4*a3o3o3o3o	o3x4/3x4*a3o3o3o3o3o	o3x4/3x4*a3o3o3o3o3o3o	o3x4/3x4*a3o3o3o3o3o3o3o	o3x4/3x4*a3o...o3o
Acronym	gososaz	gook	?	?	quasiretroexp. hyp.cube
Vertex Count	896	2048	4608	10240	n 2ⁿ
Facet Count simplex	128`hop`	256`oca`	512`ene`	1024`day`	2ⁿ
Facet Count hypercube	14`ax`	16`hept`	18`octo`	20`enne`	2n
Facet Count goc. ser. mem.	14`goxaxog`	16`gososaz`	18`gook`	20`?`	2n
Circumradius	sqrt[9-2 sqrt(2)]/2 1.242133	sqrt[(5-sqrt(2))/2] 1.338990	sqrt[11-2 sqrt(2)]/2 1.429298	sqrt[(6-sqrt(2))/2] 1.514230	sqrt[(n+2)-sqrt(8)]/2
Inradius wrt. simplex facets	[7-sqrt(2)]/sqrt(28) 1.055614	[4 sqrt(2)-1]/4 1.164214	[9-sqrt(2)]/6 1.264298	[5 sqrt(2)-1]/sqrt(20) 1.357532	[n-sqrt(2)]/sqrt(4n)
Inradius wrt. hyp.cube fac.	(sqrt(2)-1)/2 0.207107	(sqrt(2)-1)/2 0.207107	(sqrt(2)-1)/2 0.207107	(sqrt(2)-1)/2 0.207107	(sqrt(2)-1)/2 0.207107
Inradius wrt. goc. ser. mem.	1/2 0.5	1/2 0.5	1/2 0.5	1/2 0.5	1/2 0.5
Volume	?	?	?	?	?
Surface	?	?	?	?	?
Dihedral angles simplex - goc.s.m.	arccos[-1/sqrt(7)] 112.207654°	arccos[-1/sqrt(8)] 110.704811°	arccos(-1/3) 109.471221°	arccos[-1/sqrt(10)] 108.434949°	arccos[-1/sqrt(n)]
Dihedral angles hyp.cub. - goc.s.m.	90°	90°	90°	90°	90°
Dihedral angles goc.s.m. - goc.s.m.	90°	90°	90°	90°	90°

Omnitruncated Hypercube otC_n (up)

Dimension	2D	3D	4D	5D	nD
Dynkin diagram	x4x	x3x4x	x3x3x4x	x3x3x3x4x	x3x...x3x4x
Acronym	oc	girco	gidpith	gacnet	omnitr. n-hypercube
Vertex Count	8	48	384	3840	2ⁿ n!
Facet Count wrt. type 1	4`line`	8`hig`	16`toe`	32`gippid`	2ⁿ
Facet Count wrt. type 2	4`line`	12`square`	32`hip`	80`tope`	2^n-1 n
Facet Count wrt. type 3		6`oc`	24`op`	80`hodip`	2^n-2 n!/[(n-2)! 2!]
Facet Count wrt. type 4			8`girco`	40`gircope`	2^n-3 n!/[(n-3)! 3!]
Facet Count wrt. type 5			8`girco`	10`gidpith`	2^n-4 n!/[(n-4)! 4!]
Circumradius	sqrt[(2+sqrt(2))/2] 1.306563	sqrt[13+6 sqrt(2)]/2 2.317611	sqrt[8+3 sqrt(2)] 3.498949	sqrt[65+20 sqrt(2)]/2 4.829189	sqrt[n(2n²-3n+4)/3 + n(n-1) sqrt(2)]/2
Inradius wrt. facet type 1	(1+sqrt(2))/2 1.207107	sqrt[9+6 sqrt(2)]/2 2.090770	(2+3 sqrt(2))/2 3.121320	sqrt[45+20 sqrt(2)]/2 4.280312	sqrt[n(n²-2n+3)/2 + n(n-1) sqrt(2)]/2
Inradius wrt. facet type 2	(1+sqrt(2))/2 1.207107	(3+sqrt(2))/2 2.207107	sqrt[27+12 sqrt(2)]/2 3.315515	sqrt[(27+10 sqrt(2))/2] 4.535534	sqrt[(n-1)(n²+2)/2 + n(n-1) sqrt(2)]/2
Inradius wrt. facet type 3		(1+2 sqrt(2))/2 1.914214	(5+sqrt(2))/2 3.207107	sqrt[57+18 sqrt(2)]/2 4.540260	?
Inradius wrt. facet type 4			sqrt[19+6 sqrt(2)]/2 2.621320	(7+sqrt(2))/2 4.207107	?
Inradius wrt. facet type 5			sqrt[19+6 sqrt(2)]/2 2.621320	sqrt[33+8 sqrt(2)]/2 3.328427	?
Volume	2[1+sqrt(2)] 4.828427	2[11+7 sqrt(2)] 41.798990	2[131+92 sqrt(2)] 522.215295	2[2053+1564 sqrt(2)] 8529.660023	?
Surface	8	12[2+sqrt(2)+sqrt(3)] 61.755172	?	?	?
Dihedral angles types 1 - 2	135°	arccos[-sqrt(2/3)] 144.735610°	150°	arccos[-2/sqrt(5)] 153.434949°	arccos[-sqrt((n-1)/n)]
Dihedral angles types 1 - 3		arccos[-1/sqrt(3)] 125.264390°	135°	arccos[-sqrt(3/5)] 140.768480°	arccos[-sqrt((n-2)/n)]
Dihedral angles types 1 - 4			120°	arccos[-sqrt(2/5)] 129.231520°	arccos[-sqrt((n-3)/n)]
Dihedral angles types 1 - 5			120°	arccos[-1/sqrt(5)] 116.565051°	arccos[-sqrt((n-4)/n)]
Dihedral angles types 2 - 3		135°	arccos[-sqrt(2/3)] 144.735610°	150°	arccos[-sqrt((n-2)/(n-1))]
Dihedral angles types 2 - 4			arccos[-1/sqrt(3)] 125.264390°	135°	arccos[-sqrt((n-3)/(n-1))]
Dihedral angles types 2 - 5			arccos[-1/sqrt(3)] 125.264390°	120°	arccos[-sqrt((n-4)/(n-1))]
Dihedral angles types 3 - 4			135°	arccos[-sqrt(2/3)] 144.735610°	arccos[-sqrt((n-3)/(n-2))]
Dihedral angles types 3 - 5				arccos[-1/sqrt(3)] 125.264390°	arccos[-sqrt((n-4)/(n-2))]
Dihedral angles types 4 - 5				135°	arccos[-sqrt((n-4)/(n-3))]
Dimension	6D	7D	8D	9D	nD
Dynkin diagram	x3x3x3x3x4x	x3x3x3x3x3x4x	x3x3x3x3x3x3x4x	x3x3x3x3x3x3x3x4x	x3x...x3x4x
Acronym	gotaxog	guposaz	gaxoke	?	omnitr. n-hypercube
Vertex Count	46080	645120	10321920	185794560	2ⁿ n!
Facet Count wrt. type 1	64`gocad`	128`gotaf`	256`guph`	512`goxeb`	2ⁿ
Facet Count wrt. type 2	192`gippiddip`	448`gocadip`	1024`gotafip`	2304`guphip`	2^n-1 n
Facet Count wrt. type 3	240`otoe`	672`ogippid`	1792`ogocad`	4608`ogotaf`	2^n-2 n!/[(n-2)! 2!]
Facet Count wrt. type 4	160`hagirco`	560`toegirco`	1792`gircogippid`	5376`gircogocad`	2^n-3 n!/[(n-3)! 3!]
Facet Count wrt. type 5	60`gidpithip`	280`hagidpith`	1120`toegidpith`	4032`gippidgidpith`	2^n-4 n!/[(n-4)! 4!]
Facet Count wrt. type 6	12`gacnet`	84`gacnetip`	448`hagacnet`	2016`toegacnet`	2^n-5 n!/[(n-5)! 5!]
Facet Count wrt. type 7		14`gotaxog`	112`gotaxogip`	672`hagotaxog`	2^n-6 n!/[(n-6)! 6!]
Facet Count wrt. type 8			16`guposaz`	144`guposazip`	2^n-7 n!/[(n-7)! 7!]
Facet Count wrt. type 9			16`guposaz`	18`gaxoke`	2^n-8 n!/[(n-8)! 8!]
Circumradius	sqrt[(58+15 sqrt(2))/2] 6.293378	sqrt[189+42 sqrt(2)]/2 7.880307	sqrt[72+14 sqrt(2)] 9.581179	sqrt[417+72 sqrt(2)]/2 11.388847	sqrt[n(2n²-3n+4)/3 + n(n-1) sqrt(2)]/2
Inradius wrt. facet type 1	sqrt[81+30 sqrt(2)]/2 5.554872	sqrt[133+42 sqrt(2)]/2 6.935362	7+sqrt(2) 8.414214	sqrt[297+72 sqrt(2)]/2 9.985281	sqrt[n(n²-2n+3)/2 + n(n-1) sqrt(2)]/2
Inradius wrt. facet type 2	sqrt[95+30 sqrt(2)]/2 5.861450	sqrt[153+42 sqrt(2)]/2 7.286923	sqrt[231+56 sqrt(2)]/2 8.806190	9+sqrt(2) 10.414214	sqrt[(n-1)(n²+2)/2 + n(n-1) sqrt(2)]/2
Inradius wrt. facet type 3	sqrt[(51+14 sqrt(2))/2] 5.949747	sqrt[165+40 sqrt(2)]/2 7.442589	sqrt[249+54 sqrt(2)]/2 9.018974	?	?
Inradius wrt. facet type 4	sqrt[99+24 sqrt(2)]/2 5.765005	sqrt[(83+18 sqrt(2))/2] 7.363961	sqrt[255+50 sqrt(2)]/2 9.023728	?	?
Inradius wrt. facet type 5	(9+sqrt(2))/2 5.207107	sqrt[153+30 sqrt(2)]/2 6.989750	sqrt[(123+22 sqrt(2))/2] 8.778175	?	?
Inradius wrt. facet type 6	sqrt[51+10 sqrt(2)]/2 4.035534	(11+sqrt(2))/2 6.207107	sqrt[219+36 sqrt(2)]/2 8.214495	?	?
Inradius wrt. facet type 7		sqrt[73+12 sqrt(2)]/2 4.742641	(13+sqrt(2))/2 7.207107	?	?
Inradius wrt. facet type 8			sqrt[99+14 sqrt(2)]/2 5.449747	?	?
Inradius wrt. facet type 9			sqrt[99+14 sqrt(2)]/2 5.449747	?	?
Volume	?	?	?	?	?
Surface	?	?	?	?	?
Dihedral angles types 1 - 2	arccos[-sqrt(5/6)] 155.905157°	arccos[-sqrt(6/7)] 157.792346°	arccos[-sqrt(7/8)] 159.295189°	arccos[-sqrt(8)/3] 160.528779°	arccos[-sqrt((n-1)/n)]
Dihedral angles types 1 - 3	arccos[-sqrt(2/3)] 144.735610°	arccos[-sqrt(5/7)] 147.688467°	150°	arccos[-sqrt(7)/3] 151.874494°	arccos[-sqrt((n-2)/n)]
Dihedral angles types 1 - 4	135°	arccos[-sqrt(4/7)] 139.106605°	arccos[-sqrt(5/8)] 142.238756°	arccos[-sqrt(2/3)] 144.735610°	arccos[-sqrt((n-3)/n)]
Dihedral angles types 1 - 5	arccos[-1/sqrt(3)] 125.264390°	arccos[-sqrt(3/7)] 130.893395°	135°	arccos[-sqrt(5)/3] 138.189685°	arccos[-sqrt((n-4)/n)]
Dihedral angles types 1 - 6	arccos[-1/sqrt(6)] 114.094843°	arccos[-sqrt(2/7)] 122.311533°	arccos[-sqrt(3/8)] 127.761244°	arccos(-2/3) 131.810315°	arccos[-sqrt((n-5)/n)]
Dihedral angles types 1 - 7		arccos[-1/sqrt(7)] 112.207654°	120°	arccos[-1/sqrt(3)] 125.264390°	arccos[-sqrt((n-6)/n)]
Dihedral angles types 1 - 8			arccos[-1/sqrt(8)] 110.704811°	arccos[-sqrt(2)/3] 118.125506°	arccos[-sqrt((n-7)/n)]
Dihedral angles types 1 - 9			arccos[-1/sqrt(8)] 110.704811°	arccos(-1/3) 109.471221°	arccos[-sqrt((n-8)/n)]
Dihedral angles types 2 - 3	arccos[-2/sqrt(5)] 153.434949°	arccos[-sqrt(5/6)] 155.905157°	arccos[-sqrt(6/7)] 157.792346°	arccos[-sqrt(7/8)] 159.295189°	arccos[-sqrt((n-2)/(n-1))]
Dihedral angles types 2 - 4	arccos[-sqrt(3/5)] 140.768480°	arccos[-sqrt(2/3)] 144.735610°	arccos[-sqrt(5/7)] 147.688467°	150°	arccos[-sqrt((n-3)/(n-1))]
Dihedral angles types 2 - 5	arccos[-sqrt(2/5)] 129.231520°	135°	arccos[-sqrt(4/7)] 139.106605°	arccos[-sqrt(5/8)] 142.238756°	arccos[-sqrt((n-4)/(n-1))]
Dihedral angles types 2 - 6	arccos[-1/sqrt(5)] 116.565051°	arccos[-1/sqrt(3)] 125.264390°	arccos[-sqrt(3/7)] 130.893395°	135°	arccos[-sqrt((n-5)/(n-1))]
Dihedral angles types 2 - 7		arccos[-1/sqrt(6)] 114.094843°	arccos[-sqrt(2/7)] 122.311533°	arccos[-sqrt(3/8)] 127.761244°	arccos[-sqrt((n-6)/(n-1))]
Dihedral angles types 2 - 8			arccos[-1/sqrt(7)] 112.207654°	120°	arccos[-sqrt((n-7)/(n-1))]
Dihedral angles types 2 - 9			arccos[-1/sqrt(7)] 112.207654°	arccos[-1/sqrt(8)] 110.704811°	arccos[-sqrt((n-8)/(n-1))]
Dihedral angles types 3 - 4	150°	arccos[-2/sqrt(5)] 153.434949°	arccos[-sqrt(5/6)] 155.905157°	arccos[-sqrt(6/7)] 157.792346°	arccos[-sqrt((n-3)/(n-2))]
Dihedral angles types 3 - 5	135°	arccos[-sqrt(3/5)] 140.768480°	arccos[-sqrt(2/3)] 144.735610°	arccos[-sqrt(5/7)] 147.688467°	arccos[-sqrt((n-4)/(n-2))]
Dihedral angles types 3 - 6	120°	arccos[-sqrt(2/5)] 129.231520°	135°	arccos[-sqrt(4/7)] 139.106605°	arccos[-sqrt((n-5)/(n-2))]
Dihedral angles types 3 - 7		arccos[-1/sqrt(5)] 116.565051°	arccos[-1/sqrt(3)] 125.264390°	arccos[-sqrt(3/7)] 130.893395°	arccos[-sqrt((n-6)/(n-2))]
Dihedral angles types 3 - 8			arccos[-1/sqrt(6)] 114.094843°	arccos[-sqrt(2/7)] 122.311533°	arccos[-sqrt((n-7)/(n-2))]
Dihedral angles types 3 - 9			arccos[-1/sqrt(6)] 114.094843°	arccos[-1/sqrt(7)] 112.207654°	arccos[-sqrt((n-8)/(n-2))]
Dihedral angles types 4 - 5	arccos[-sqrt(2/3)] 144.735610°	150°	arccos[-2/sqrt(5)] 153.434949°	arccos[-sqrt(5/6)] 155.905157°	arccos[-sqrt((n-4)/(n-3))]
Dihedral angles types 4 - 6	arccos[-1/sqrt(3)] 125.264390°	135°	arccos[-sqrt(3/5)] 140.768480°	arccos[-sqrt(2/3)] 144.735610°	arccos[-sqrt((n-5)/(n-3))]
Dihedral angles types 4 - 7		120°	arccos[-sqrt(2/5)] 129.231520°	135°	arccos[-sqrt((n-6)/(n-3))]
Dihedral angles types 4 - 8			arccos[-1/sqrt(5)] 116.565051°	arccos[-1/sqrt(3)] 125.264390°	arccos[-sqrt((n-7)/(n-3))]
Dihedral angles types 4 - 9			arccos[-1/sqrt(5)] 116.565051°	arccos[-1/sqrt(6)] 114.094843°	arccos[-sqrt((n-8)/(n-3))]
Dihedral angles types 5 - 6	135°	arccos[-sqrt(2/3)] 144.735610°	150°	arccos[-2/sqrt(5)] 153.434949°	arccos[-sqrt((n-5)/(n-4))]
Dihedral angles types 5 - 7		arccos[-1/sqrt(3)] 125.264390°	135°	arccos[-sqrt(3/5)] 140.768480°	arccos[-sqrt((n-6)/(n-4))]
Dihedral angles types 5 - 8			120°	arccos[-sqrt(2/5)] 129.231520°	arccos[-sqrt((n-7)/(n-4))]
Dihedral angles types 5 - 9			120°	arccos[-1/sqrt(5)] 116.565051°	arccos[-sqrt((n-8)/(n-4))]
Dihedral angles types 6 - 7		135°	arccos[-sqrt(2/3)] 144.735610°	150°	arccos[-sqrt((n-6)/(n-5))]
Dihedral angles types 6 - 8			arccos[-1/sqrt(3)] 125.264390°	135°	arccos[-sqrt((n-7)/(n-5))]
Dihedral angles types 6 - 9			arccos[-1/sqrt(3)] 125.264390°	120°	arccos[-sqrt((n-8)/(n-5))]
Dihedral angles types 7 - 8			135°	arccos[-sqrt(2/3)] 144.735610°	arccos[-sqrt((n-7)/(n-6))]
Dihedral angles types 7 - 9				arccos[-1/sqrt(3)] 125.264390°	arccos[-sqrt((n-8)/(n-6))]
Dihedral angles types 8 - 9				135°	arccos[-sqrt((n-8)/(n-7))]

Symmetry D_n

Demihypercube D_n (up)

As these polytopes D_n generally are nothing but the alternation of theregular hypercube C_n,and C_n in turn is the prism of C_n-1 atop C_n-1, so D_n likewisecan be described as thesegmentotope of the demihypercube D_n-1 atop the alternate demihypercube ~D_n-1.Thence, by means of thelace prism notation, D_n = x3o3o *b3o...o3o (n nodes) can be described as well asxo3oo3ox *b3oo...oo3oo&#x (n-1 node positions),which as such is nothing else than the demihypercubicalterprism.

A short consideration of general demihypercubes already occuredhere as well. Furthermore are demihypercubes special cases of theCoxeter-Elte-Gosset polytopes k_m,n, in fact those generally are clearly the ones of the form 1_(n-2),1.

Dimension	3D	4D	5D	6D	nD
Dynkin diagram	x3o3o	x3o3o *b3o	x3o3o *b3o3o	x3o3o *b3o3o3o	x3o3o *b3o...o3o
Acronym	tet	hex	hin	hax	n-demihypercube
Vertex Count	4`trig`	8`oct`	16`rap`	32`rix`	2^n-1
Facet Count simplex	4`trig`	8`tet`	16`pen`	32`hix`	2^n-1
Facet Count demihyp.cube	4`trig`	8`tet`	10`hex`	12`hin`	2n
Circumradius	sqrt(3/8) 0.612372	1/sqrt(2) 0.707107	sqrt(5/8) 0.790569	sqrt(3)/2 0.866025	sqrt(n/8)
Inradius wrt. simplex	1/sqrt(24) 0.204124	1/sqrt(8) 0.353553	3/sqrt(40) 0.474342	1/sqrt(3) 0.577350	(n-2)/sqrt(8n)
Inradius wrt. demihyp.cube	1/sqrt(24) 0.204124	1/sqrt(8) 0.353553	1/sqrt(8) 0.353553	1/sqrt(8) 0.353553	1/sqrt(8) 0.353553
Volume	sqrt(2)/12 0.117851	1/6 0.166667	13 sqrt(2)/120 0.153206	43/360 0.119444	(1-2^n-1/n!)/sqrt(2ⁿ)
Surface	sqrt(3) 1.732051	4 sqrt(2)/3 1.885618	(10+sqrt(5))/6 2.039345	[39 sqrt(2)+2 sqrt(3)]/30 1.953948	[2 n!-2^n-1(n-sqrt(n))]/[(n-1)! sqrt(2^n-1)]
Dihedral angles simp. - demi.	arccos(1/3) 70.528779° (simp. - simp.)	120°	arccos[-1/sqrt(5)] 116.565051°	arccos[-1/sqrt(6)] 114.094843°	arccos[-1/sqrt(n)]
Dihedral angles demi. - demi.	arccos(1/3) 70.528779° (simp. - simp.)	120°	90°	90°	90°
Dimension	7D	8D	9D	10D	nD
Dynkin diagram	x3o3o *b3o3o3o3o	x3o3o *b3o3o3o3o3o	x3o3o *b3o3o3o3o3o3o	x3o3o *b3o3o3o3o3o3o3o	x3o3o *b3o...o3o
Acronym	hesa	hocto	henne	hede	n-demihypercube
Vertex Count	64`ril`	128`roc`	256`rene`	512`reday`	2^n-1
Facet Count simplex	64`hop`	128`oca`	256`ene`	512`day`	2^n-1
Facet Count demihyp.cube	14`hax`	16`hesa`	18`hocto`	20`henne`	2n
Circumradius	sqrt(7/8) 0.935414	1	3/sqrt(8) 1.060660	sqrt(5)/2 1.118034	sqrt(n/8)
Inradius wrt. simplex	5/sqrt(56) 0.668153	3/4 0.75	7/sqrt(72) 0.824958	2/sqrt(5) 0.894427	(n-2)/sqrt(8n)
Inradius wrt. demihyp.cube	1/sqrt(8) 0.353553	1/sqrt(8) 0.353553	1/sqrt(8) 0.353553	1/sqrt(8) 0.353553	1/sqrt(8) 0.353553
Volume	311 sqrt(2)/5040 0.087266	157/2520 0.062302	2833 sqrt(2)/90720 0.044163	14173/453600 0.031246	(1-2^n-1/n!)/sqrt(2ⁿ)
Surface	[301+2 sqrt(7)]/180 1.701619	[2+311 sqrt(2)]/315 1.402605	943/840 1.122619	[14165 sqrt(2)+2 sqrt(5)]/22680 0.883457	[2 n!-2^n-1(n-sqrt(n))]/[(n-1)! sqrt(2^n-1)]
Dihedral angles simp. - demi.	arccos[-1/sqrt(7)] 112.207654°	arccos[-1/sqrt(8)] 110.704811°	arccos(-1/3) 109.471221°	arccos[-1/sqrt(10)] 108.434949°	arccos[-1/sqrt(n)]
Dihedral angles demi. - demi.	90°	90°	90°	90°	90°

Demicross hO_n (up)

These non-convex polytopes hO_n generally are facetings of theregular orthoplex O_n using the maximal count of it's hemifacets, thereby reducing the facet simplices to the half of the former. Moreover it happens that generally hO_n-1 is thevertex figure of hO_n.As O_n could be seen as the S_n-1-antiprism thence too hO_ngenerally is the (non-convex)segmentotope oftheregular simplex S_n-1 atop the dual (pseudo?) simplex -(S_n-1).In fact the even dimensional demicrosses have inversion symmetry, i.e. the pseudo part does not apply,while for the odd dimensional ones the inversion would just result in the complementof the original demicross wrt. its convex hull (the orthoplex), i.e. here the pseudo part does apply.

These polytopes never are orientable. Accordingly novolume can be calculated either.

Dimension	3D	4D	5D	6D	nD
Dynkin diagram	hemi(x3/2o3x )	hemi(x3o3/2o3o3*a )	hemi(o3o3/2o3o3*a3x )	hemi(o3o3/2o3o3*a3o3x )	hemi(o3o3/2o3o3*a3o...o3x )
Acronym	thah	tho	hehad	thox	n-demicross
Vertex Count	6	8	10	12	2n
Facet Count simplex	4`trig`	8`tet`	16`pen`	32`hix`	2^n-1
Facet Count hemi facets	3`square`	4`oct`	5`hex`	6`tac`	n
Circumradius	1/sqrt(2) 0.707107	1/sqrt(2) 0.707107	1/sqrt(2) 0.707107	1/sqrt(2) 0.707107	1/sqrt(2) 0.707107
Inradius wrt. simplex	1/sqrt(6) 0.408248	1/sqrt(8) 0.353553	1/sqrt(10) 0.316228	1/sqrt(12) 0.288675	1/sqrt(2n)
Inradius wrt. hemi facets	0	0	0	0	0
Surface	3+sqrt(3) 4.732051	sqrt(8) 2.828427	[5+sqrt(5)]/6 1.206011	[3 sqrt(2)+sqrt(3)]/15 0.398313	(n+sqrt(n)) sqrt(2^n-1)/(n-1)!
Dihedral angles	arccos[1/sqrt(3)] 54.735610°	60°	arccos[1/sqrt(5)] 63.434949°	arccos[1/sqrt(6)] 65.905157°	arccos[1/sqrt(n)]
Dimension	7D	8D	9D	10D	nD
Dynkin diagram	hemi(o3o3/2o3o3*a3o3o3x )	hemi(o3o3/2o3o3*a3o3o3o3x )	hemi(o3o3/2o3o3*a3o3o3o3o3x )	hemi(o3o3/2o3o3*a3o3o3o3o3o3x )	hemi(o3o3/2o3o3*a3o...o3x )
Acronym	guhsa	zeho	ekhen	vehde	n-demicross
Vertex Count	14	16	18	20	2n
Facet Count simplex	64`hop`	128`oca`	256`ene`	512`day`	2^n-1
Facet Count hemi facets	7`gee`	8`zee`	9`ek`	10`vee`	n
Circumradius	1/sqrt(2) 0.707107	1/sqrt(2) 0.707107	1/sqrt(2) 0.707107	1/sqrt(2) 0.707107	1/sqrt(2) 0.707107
Inradius wrt. simplex	1/sqrt(14) 0.267261	1/4 0.25	1/sqrt(18) 0.235702	1/sqrt(20) 0.223607	1/sqrt(2n)
Inradius wrt. hemi facets	0	0	0	0	0
Surface	[7+sqrt(7)]/90 0.107175	[4+8 sqrt(2)]/630 0.0243075	1/210 0.00476190	[5 sqrt(2)+sqrt(5)]/11340 0.000820735	(n+sqrt(n)) sqrt(2^n-1)/(n-1)!
Dihedral angles	arccos[1/sqrt(7)] 67.792346°	arccos[1/sqrt(8)] 69.295189°	arccos(1/3) 70.528779°	arccos[1/sqrt(10)] 71.565051°	arccos[1/sqrt(n)]

Truncated Demihypercube tD_n (up)

As the non-truncateddemihypercubes D_n generally could be described as thesegmentotope of the demihypercube D_n-1 atop the alternate demihypercube ~D_n-1, their truncations tD_n become tristratic lace towers with the truncated demihypercube tD_n-1 at the top side and the alternate truncated demihypercube ~tD_n-1 at the bottom side. Inbetween there will be 2 vertex layers which happen to be non-uniform variants of therectified hypercube rC_n-1. In fact, by means of thelace tower notation, tD_n = x3o3o *b3o...o3o (n nodes) can be described as well asxuxo3xoox3oxux *b3oooo...oooo3oooo&#xt (n-1 node positions).

Dimension	3D	4D	5D	6D	nD
Dynkin diagram	x3x3o	x3x3o *b3o	x3x3o *b3o3o	x3x3o *b3o3o3o	x3x3o *b3o...o3o
Acronym	tut	thex	thin	thax	n-trunc. demihyp.cube
Vertex Count	12	48	160	480	2^n-2 n(n-1)
Facet Count trunc. simpl.	4`hig`	8`tut`	16`tip`	32`tix`	2^n-1
Facet Count rect. simpl.	4`trig`	8`oct`	16`rap`	32`rix`	2^n-1
Facet Count trunc. demi.	4`trig`	8`tut`	10`thex`	12`thin`	2n
Circumradius	sqrt(11/8) 1.172604	sqrt(5/2) 1.581139	sqrt(29/8) 1.903943	sqrt(19)/2 2.179449	sqrt[(9n-16)/8]
Inradius wrt. trunc. simpl.	sqrt(3/8) 0.612372	3/sqrt(8) 1.060660	9/sqrt(40) 1.423025	sqrt(3) 1.732051	3(n-2)/sqrt(8n)
Inradius wrt. rect. simpl.	5/sqrt(24) 1.020621	sqrt(2) 1.414214	11/sqrt(40) 1.739253	7/sqrt(12) 2.020726	(3n-4)/sqrt(8n)
Inradius wrt. trunc. demi.	5/sqrt(24) 1.020621	sqrt(2) 1.414214	3/sqrt(8) 1.060660	3/sqrt(8) 1.060660	3/sqrt(8) 1.060660
Volume	23 sqrt(2)/12 2.710576	77/6 12.833333	623 sqrt(2)/24 36.710627	31243/360 86.786111	?
Surface	7 sqrt(3) 12.124356	100 sqrt(2)/3 47.140452	(770+87 sqrt(5))/6 160.756319	[9345 sqrt(2)+526 sqrt(3)]/30 470.896149	?
Dihedral angles tr.simp. - re.simp.	arccos(-1/3) 109.471221°	120°	arccos(-3/5) 126.869898°	arccos(-2/3) 131.810315°	arccos[-(n-2)/n]
Dihedral angles tr.simp. - tr.demi.	arccos(1/3) 70.528779° (tr.simp. - tr.simp.)	120°	arccos[-1/sqrt(5)] 116.565051°	arccos[-1/sqrt(6)] 114.094843°	arccos[-1/sqrt(n)]
Dihedral angles re.simp. -tr.demi.		120°	arccos[-1/sqrt(5)] 116.565051°	arccos[-1/sqrt(6)] 114.094843°	arccos[-1/sqrt(n)]
Dihedral angles tr.demi. - tr.demi.		120°	90°	90°	90°
Dimension	7D	8D	9D	10D	nD
Dynkin diagram	x3x3o *b3o3o3o3o	x3x3o *b3o3o3o3o3o	x3x3o *b3o3o3o3o3o3o	x3x3o *b3o3o3o3o3o3o3o	x3x3o *b3o...o3o
Acronym	thesa	thocto	thenne	thede	n-trunc. demihyp.cube
Vertex Count	1344	3584	9216	23040	2^n-2 n(n-1)
Facet Count trunc. simpl.	64`til`	128`toc`	256`tene`	512`teday`	2^n-1
Facet Count rect. simpl.	64`ril`	128`roc`	256`rene`	512`reday`	2^n-1
Facet Count demihyp.cube	14`thax`	16`thesa`	18`thocto`	20`thenne`	2n
Circumradius	sqrt(47/8) 2.423840	sqrt(7) 2.645751	sqrt(65/8) 2.850439	sqrt(37)/2 3.041381	sqrt[(9n-16)/8]
Inradius wrt. trunc. simpl.	15/sqrt(56) 2.004459	9/4 2.25	7/sqrt(8) 2.474874	6/sqrt(5) 2.683282	3(n-2)/sqrt(8n)
Inradius wrt. rect. simpl.	17/sqrt(56) 2.271721	5/2 2.5	23/sqrt(72) 2.710576	13/sqrt(20) 2.906888	(3n-4)/sqrt(8n)
Inradius wrt. trunc. demi.	3/sqrt(8) 1.060660	3/sqrt(8) 1.060660	3/sqrt(8) 1.060660	3/sqrt(8) 1.060660	3/sqrt(8) 1.060660
Volume	34081 sqrt(2)/240 200.824218	?	?	?	?
Surface	?	?	?	?	?
Dihedral angles tr.simp. - re.simp.	arccos(-5/7) 135.584691°	arccos(-3/4) 138.590378°	arccos(-7/9) 141.057559°	arccos(-4/5) 143.130102°	arccos[-(n-2)/n]
Dihedral angles tr.simp. - tr.demi.	arccos[-1/sqrt(7)] 112.207654°	arccos[-1/sqrt(8)] 110.704811°	arccos(-1/3) 109.471221°	arccos[-1/sqrt(10)] 108.434949°	arccos[-1/sqrt(n)]
Dihedral angles re.simp. - tr.demi.	arccos[-1/sqrt(7)] 112.207654°	arccos[-1/sqrt(8)] 110.704811°	arccos(-1/3) 109.471221°	arccos[-1/sqrt(10)] 108.434949°	arccos[-1/sqrt(n)]
Dihedral angles tr.demi. - tr.demi.	90°	90°	90°	90°	90°

Maximal Expanded Demihypercube eD_n (up)

Within these polytopes eD_n generally can be described as the tristratic lace tower ofthedemihypercube D_n-1 atop the maximal expanded demihypercube eD_n-1 atop the maximal expanded alternate demihypercube ~eD_n-1 atop the alternate demihypercube ~D_n-1.Thence, by means of thelace tower notation,eD_n = x3o3o *b3o...o3x (n nodes) can be described as well asxxoo3oooo3ooxx *b3oooo...oooo3oxxo&#xt (n-1 node positions).

Dimension	3D	4D	5D	6D	nD
Dynkin diagram	x3x3o	x3o3o *b3x	x3o3o *b3o3x	x3o3o *b3o3o3x	x3o3o *b3o...o3x
Acronym	tut	rit	siphin	sochax	max-exp. n-demihyp.cube
Vertex Count	12	32	80	192	n 2^n-1
Facet Count simplex	4`trig`	8`tet`	16`pen`	32`hix`	2^n-1
Facet Count exp. simpl.	4`hig`	8`co`	16`spid`	32`scad`	2^n-1
Facet Count duoprism I			16`spid`	160`tratet`	4n(n-1)(n-2)/3
Facet Count prism			40`tepe`	60`hexip`	2n(n-1)
Facet Count demihyp.cube		8`tet`	10`hex`	12`hin`	2n
Circumradius	sqrt(11/8) 1.172604	sqrt(3/2) 1.224745	sqrt(13/8) 1.274755	sqrt(7)/2 1.322876	sqrt[(n+8)/8]
Inradius wrt. simplex facets	5/sqrt(24) 1.020621	3/sqrt(8) 1.060660	7/sqrt(40) 1.106797	2/sqrt(3) 1.154701	(n+2)/sqrt(8n)
Inradius wrt. exp. simpl. fac.	sqrt(3/8) 0.612372	1/sqrt(2) 0.707107	sqrt(5/8) 0.790569	sqrt(3)/2 0.866025	sqrt(n/8)
Inradius wrt. d.pr. I fac.			sqrt(5/8) 0.790569	5/sqrt(24) 1.020621	5/sqrt(24) 1.020621
Inradius wrt. prism facets			1	1	1
Inradius wrt. demihyp.c. fac.		3/sqrt(8) 1.060660	3/sqrt(8) 1.060660	3/sqrt(8) 1.060660	3/sqrt(8) 1.060660
Volume	23 sqrt(2)/12 2.710576	23/6 3.833333	467 sqrt(2)/120 5.503648	2737/360 7.602778	?
Surface	7 sqrt(3) 12.124356	44 sqrt(2)/3 20.741799	[10+20 sqrt(2)+71 sqrt(5)]/6 32.840850	[300+39 sqrt(2)+506 sqrt(3)+100 sqrt(6)]/30 49.217367	?
Dihedral angles simpl. - e.sim.	arccos(-1/3) 109.471221°	120°	arccos(-3/5) 126.869898°	arccos(-2/3) 131.810315°	arccos[-(n-2)/n]
Dihedral angles e.sim. - e.sim.	arccos(1/3) 70.528779°	90°	arccos(-1/5) 101.536959°	arccos(-1/3) 109.471221°	arccos[-(n-4)/n]
Dihedral angles e.sim. - d.pr. I			arccos(-1/5) 101.536959°	?	?
Dihedral angles e.sim. - prism			arccos[-sqrt(2/5)] 129.231520°	?	?
Dihedral angles e.sim. - demi.		120°	arccos[-1/sqrt(5)] 116.565051°	arccos[-1/sqrt(6)] 114.094843°	arccos[-1/sqrt(n)]
Dihedral angles prism - d.pr. I			arccos[-1/sqrt(5)] 116.565051°	?	?
Dihedral angles prism - demi.			135°	135°	135°
Dimension	7D	8D	9D	10D	nD
Dynkin diagram	x3o3o *b3o3o3o3x	x3o3o *b3o3o3o3o3x	x3o3o *b3o3o3o3o3o3x	x3o3o *b3o3o3o3o3o3o3x	x3o3o *b3o...o3x
Acronym	suthesa	spuho	?	?	max-exp. n-demihyp.cube
Vertex Count	448	1024	2304	5120	n 2^n-1
Facet Count simplex	64`hop`	128`oca`	256`ene`	512`day`	2^n-1
Facet Count exp. simpl.	64`staf`	128`suph`	256`soxeb`	512`?`	2^n-1
Facet Count duoprism V			256`soxeb`	15360`tethop`	n! 2⁷/[7!(n-7)!]
Facet Count duoprism IV			5376`tethix`	7680`hexhix`	n! 2⁶/[6!(n-6)!]
Facet Count duoprism III		1792`tetpen`	4032`penhex`	8064`penhin`	n! 2⁵/[5!(n-5)!]
Facet Count duoprism II	560`tetdip`	1120`tethex`	2016`tethin`	3360`tethax`	n! 2⁴/[4!(n-4)!]
Facet Count duoprism I	280`trahex`	448`trahin`	672`trahax`	960`trahesa`	4n(n-1)(n-2)/3 n! 2³/[3!(n-3)!]
Facet Count prism	84`hinnip`	112`haxip`	144`hesape`	180`hoctope`	2n(n-1) n! 2²/[2!(n-2)!]
Facet Count demihyp.cube	14`hax`	16`hesa`	18`hocto`	20`henne`	2n n! 2¹/[1!(n-1)!]
Circumradius	sqrt(15/8) 1.369306	sqrt(2) 1.414214	sqrt(17/8) 1.457738	3/2 1.5	sqrt[(n+8)/8]
Inradius wrt. simplex	9/sqrt(56) 1.202676	5/4 1.25	11/sqrt(72) 1.296362	3/sqrt(5) 1.341641	(n+2)/sqrt(8n)
Inradius wrt. exp. simpl.	sqrt(7/8) 0.935414	1	3/sqrt(8) 1.060660	sqrt(5)/2 1.118034	sqrt(n/8)
Inradius wrt. duoprism V			3/sqrt(8) 1.060660	9/sqrt(56) 1.202676	9/sqrt(56) 1.202676 (3+6)/sqrt[(1+6)8]
Inradius wrt. duoprism IV			2/sqrt(3) 1.154701	2/sqrt(3) 1.154701	2/sqrt(3) 1.154701 (3+5)/sqrt[(1+5)8]
Inradius wrt. duoprism III		7/sqrt(40) 1.106797	7/sqrt(40) 1.106797	7/sqrt(40) 1.106797	7/sqrt(40) 1.106797 (3+4)/sqrt[(1+4)8]
Inradius wrt. duoprism II	3/sqrt(8) 1.060660	3/sqrt(8) 1.060660	3/sqrt(8) 1.060660	3/sqrt(8) 1.060660	3/sqrt(8) 1.060660 (3+3)/sqrt[(1+3)8]
Inradius wrt. duoprism I	5/sqrt(24) 1.020621	5/sqrt(24) 1.020621	5/sqrt(24) 1.020621	5/sqrt(24) 1.020621	5/sqrt(24) 1.020621 (3+2)/sqrt[(1+2)8]
Inradius wrt. prism	1	1	1	1	1 (3+1)/sqrt[(1+1)8]
Inradius wrt. demihyp.cube	3/sqrt(8) 1.060660	3/sqrt(8) 1.060660	3/sqrt(8) 1.060660	3/sqrt(8) 1.060660	3/sqrt(8) 1.060660 (3+0)/sqrt[(1+0)8]
Volume	?	?	?	?	?
Surface	?	?	?	?	?
Dihedral angles simpl. - e.sim.	arccos(-5/7) 135.584691°	arccos[-3/4) 138.590378°	arccos(-7/9) 141.057559°	arccos(-4/5) 143.130102°	arccos[-(n-2)/n]
Dihedral angles e.sim. - e.sim.	arccos(-3/7) 115.376934°	120°	arccos(-5/9) 123.748989°	arccos(-3/5) 126.869898°	arccos[-(n-4)/n]
Dihedral angles e.sim. - d.pr. V			arccos(-5/9) 123.748989°	?	?
Dihedral angles e.sim. - d.pr. IV			?	?	?
Dihedral angles e.sim. - d.pr. III		?	?	?	?
Dihedral angles e.sim. - d.pr. II	?	?	?	?	?
Dihedral angles e.sim. - d.pr. I	?	?	?	?	?
Dihedral angles e.sim. - prism	?	?	?	?	?
Dihedral angles e.sim. - demi.	arccos[-1/sqrt(7)] 112.207654°	arccos[-1/sqrt(8)] 110.704811°	arccos(-1/3) 109.471221°	arccos[-1/sqrt(10)] 108.434949°	arccos[-1/sqrt(n)]
Dihedral angles prism - d.pr. V			arccos(-1/3) 109.471221°	?	?
Dihedral angles prism - d.pr. IV			?	?	?
Dihedral angles prism - d.pr. III		?	?	?	?
Dihedral angles prism - d.pr. II	?	?	?	?	?
Dihedral angles prism - d.pr. I	?	?	?	?	?
Dihedral angles prism - demi.	135°	135°	135°	135°	135°

Symmetry E_n

It is known that those series clearly terminate for n=8, i.e. that for n=9 they would result in aflat tesselations instead. This accordingly reflects itself in the provided dimension formulae:measures like circumradii and inradii all would become infinite for n=9and dihedrals likewise would all become 180° then.

Gossetic n_2,1 (up)

Dimension	4D	5D	6D	7D	8D	nD
Dynkin diagram	o3o3x3o	o3o3o3x *c3o	o3o3o3o3x *c3o	o3o3o3o3o3x *c3o	o3o3o3o3o3o3x *c3o	o3o...o3x *c3o
Acronym	rap	hin	jak	naq	fy	(n-4)_2,1
Vertex Count	10`trip`	16`rap`	27`hin`	56`jak`	240`naq`	?
Facet Count simplex	5`tet`	16`pen`	72`hix`	576`hop`	17280`oca`	?
Facet Count orthoplex	5`oct`	10`hex`	27`tac`	126`gee`	2160`zee`	?
Circumradius	sqrt(3/5) 0.774597	sqrt(5/8) 0.790569	sqrt(2/3) 0.816497	sqrt(3)/2 0.866025	1	sqrt[(10-n)/(18-2n)]
Inradius wrt. simplex	3/sqrt(40) 0.474342	3/sqrt(40) 0.474342	1/2 0.5	3/sqrt(28) 0.566947	3/4 0.75	3/sqrt[n(18-2n)]
Inradius wrt. orthoplex	1/sqrt(10) 0.316228	1/sqrt(8) 0.353553	1/sqrt(6) 0.408248	1/2 0.5	1/sqrt(2) 0.707107	1/sqrt(18-2n)
Volume	11 sqrt(5)/96 0.256216	13 sqrt(2)/120 0.153206	sqrt(3)/16 0.108253	17/140 0.121429	57/112 0.508929	?
Surface	25 sqrt(2)/12 2.946278	[10+sqrt(5)]/6 2.039345	[18 sqrt(2)+3 sqrt(3)]/20 1.532600	[14+sqrt(7)]/10 1.664575	[6+24 sqrt(2)]/7 5.705875	?
Dihedral angles simpl. - ortho.	arccos(-1/4) 104.477512°	arccos[-1/sqrt(5)] 116.565051°	arccos[-sqrt(3/8)] 127.761244°	arccos[-2/sqrt(7)] 139.106605°	arccos[-5/sqrt(32)] 152.114433°	arccos[-(n-3)/sqrt(4n)]
Dihedral angles ortho. - ortho.	arccos(1/4) 75.522488°	90°	arccos(-1/4) 104.477512°	120°	arccos(-3/4) 138.590378°	arccos[-(n-5)/4]

Gossetic 2_n,1 (up)

Dimension	4D	5D	6D	7D	8D	nD
Dynkin diagram	x3o3o3o	x3o3o3o *c3o	x3o3o3o3o *c3o	x3o3o3o3o3o *c3o	x3o3o3o3o3o3o *c3o	x3o...o3o *c3o
Acronym	pen	tac	jak	laq	bay	2_n,1
Vertex Count	5`tet`	10`hex`	27`hin`	126`hax`	2160`hesa`	?
Facet Count simplex	5`tet`	16`pen`	72`hix`	576`hop`	17280`oca`	?
Facet Count Gossetic	5`tet`	16`pen`	27`tac`	56`jak`	240`laq`	?
Circumradius	sqrt(2/5) 0.632456	1/sqrt(2) 0.707107	sqrt(2/3) 0.816497	1	sqrt(2) 1.414214	sqrt[2/(9-n)]
Inradius wrt. simplex	1/sqrt(40) 0.158114	1/sqrt(10) 0.316228	1/2 0.5	2/sqrt(7) 0.755929	5/4 1.25	(n-3)/sqrt[2n(9-n)]
Inradius wrt. Gossetic	1/sqrt(40) 0.158114	1/sqrt(10) 0.316228	1/sqrt(6) 0.408248	1/sqrt(3) 0.577350	1	sqrt[2/((10-n)(9-n))]
Volume	sqrt(5)/96 0.023292	sqrt(2)/30 0.047140	sqrt(3)/16 0.108253	37/70 0.528571	1791/112 15.991071	?
Surface	5 sqrt(2)/12 0.589256	sqrt(5)/3 0.745356	[18 sqrt(2)+3 sqrt(3)]/20 1.532600	[35 sqrt(3)+sqrt(7)]/10 6.326753	894/7 127.714286	?
Dihedral angles simpl. - Goss.	arccos(1/4) 75.522488° simpl. - simpl.	arccos(-3/5) 126.869898°	arccos[-sqrt(3/8)] 127.761244°	arccos[-sqrt(3/7)] 130.893395°	arccos(-3/4) 138.590378°	arccos[-3/sqrt(n(10-n))]
Dihedral angles Goss. - Goss.	arccos(1/4) 75.522488° simpl. - simpl.	arccos(-3/5) 126.869898°	arccos(-1/4) 104.477512°	arccos(-1/3) 109.471221°	120°	arccos[-1/(10-n)]

Gossetic 1_n,2 (up)

Dimension	4D	5D	6D	7D	8D	nD
Dynkin diagram	o3o3o3x	o3o3o3o *c3x	o3o3o3o3o *c3x	o3o3o3o3o3o *c3x	o3o3o3o3o3o3o *c3x	o3o...o3o *c3x
Acronym	pen	hin	mo	lin	bif	1_n,2
Vertex Count	5`tet`	16`rap`	72`dot`	576`bril`	17280`broc`	?
Facet Count demihypercube	5`tet`	10`hex`	27`hin`	126`hax`	2160`hesa`	?
Facet Count Gossetic	5`tet`	16`pen`	27`hin`	56`mo`	240`lin`	?
Circumradius	sqrt(2/5) 0.632456	sqrt(5/8) 0.790569	1	sqrt(7)/2 1.322876	2	sqrt[n/(18-2n)]
Inradius wrt. demihypercube	1/sqrt(40) 0.158114	1/sqrt(8) 0.353553	sqrt(3/8) 0.612372	1	5/sqrt(8) 1.767767	(n-3)/sqrt[8(9-n)]
Inradius wrt. Gossetic	1/sqrt(40) 0.158114	3/sqrt(40) 0.474342	sqrt(3/8) 0.612372	sqrt(3)/2 0.866025	3/2 1.5	3/sqrt[2(10-n)(9-n)]
Volume	sqrt(5)/96 0.023292	13 sqrt(2)/120 0.153206	39 sqrt(3)/80 0.844375	8	44985/112 401.651786	?
Surface	5 sqrt(2)/12 0.589256	[10+sqrt(5)]/6 2.039345	117 sqrt(2)/20 8.273149	[301+546 sqrt(3)]/20 62.334987	[13440+933 sqrt(2)]/7 2108.494465	?
Dihedral angles demi. - demi.	arccos(1/4) 75.522488°	90°	arccos(-1/4) 104.477512°	120°	arccos(-3/4) 138.590378°	arccos[-(n-5)/4]
Dihedral angles demi. - Goss.		arccos[-1/sqrt(5)] 116.565051°	120°	arccos[-1/sqrt(3)] 125.264390°	135°	arccos[-1/sqrt(10-n)]
Dihedral angles Goss. - Goss.		arccos[-1/sqrt(5)] 116.565051°	arccos(-1/4) 104.477512°	arccos(-1/3) 109.471221°	120°	arccos[-1/(10-n)]

RectifiedGossetic r(n_2,1) (up)

Dimension	4D	5D	6D	7D	8D	nD
Dynkin diagram	o3x3o3x	o3o3x3o *c3o	o3o3o3x3o *c3o	o3o3o3o3x3o *c3o	o3o3o3o3o3x3o *c3o	o3o...o3x3o *c3o
Acronym	srip	nit	rojak	ranq	riffy	rect. (n-4)_2,1
Vertex Count	30`xx ox&#q`	80`tisdip`	216`rappip`	756`hinnip`	6720`jakip`	?
Facet Count rect. simpl.	5`oct`	16`rap`	72`rix`	576`ril`	17280`roc`	?
Facet Count Gossetic	10`trip`	16`rap`	27`hin`	56`jak`	240`naq`	?
Facet Count rect. ortho.	5`co`	10`ico`	27`rat`	126`rag`	2160`rez`	?
Circumradius	sqrt(7/5) 1.183216	sqrt(3/2) 1.224745	sqrt(5/3) 1.290994	sqrt(2) 1.414214	sqrt(3) 1.732051	sqrt[(11-n)/(9-n)]
Inradius wrt. rect. simpl.	3/sqrt(10) 0.948683	3/sqrt(10) 0.948683	1	3/sqrt(7) 1.133893	3/2 1.5	sqrt[18/(n(9-n))]
Inradius wrt. Gossetic	7/sqrt(60) 0.903696	3/sqrt(10) 0.948683	5/sqrt(24) 1.020621	2/sqrt(3) 1.154701	3/2 1.5	(11-n)/sqrt[2(10-n)(9-n)]
Inradius wrt. rect. ortho.	sqrt(2/5) 0.632456	1/sqrt(2) 0.707107	sqrt(2/3) 0.816497	1	sqrt(2) 1.414214	sqrt[2/(9-n)]
Volume	73 sqrt(5)/48 3.400687	31 sqrt(2)/10 4.384062	601 sqrt(3)/160 6.506016	1053/70 15.042857	3597/28 128.464286	?
Surface	[20 sqrt(2)+5 sqrt(3)]/2 18.472263	[60+11 sqrt(5)]/3 28.198916	[1089 sqrt(2)+156 sqrt(3)]/40 45.256962	[812+35 sqrt(3)+57 sqrt(7)]/10 102.342960	[924+2904 sqrt(2)]/7 718.696598	?
Dihedral angles r.sim. - Goss.	arccos[-sqrt(3/8)] 127.761244°	arccos(-3/5) 126.869898°	arccos[-sqrt(3/8)] 127.761244°	arccos[-sqrt(3/7)] 130.893395°	arccos(-3/4) 138.590378°	arccos[-3/sqrt(n(10-n))]
Dihedral angles r.sim. - r.orth.	arccos(-1/4) 104.477512°	arccos[-1/sqrt(5)] 116.565051°	arccos[-sqrt(3/8)] 127.761244°	arccos[-2/sqrt(7)] 139.106605°	arccos[-5/sqrt(32)] 152.114433°	arccos[-(n-3)/sqrt(4n)]
Dihedral angles Goss. - r.orth.	arccos[-1/sqrt(6)] 114.094843°	arccos[-1/sqrt(5)] 116.565051°	120°	arccos[-1/sqrt(3)] 125.264390°	135°	arccos[-1/sqrt(10-n)]
Dihedral angles r.orth. - r.orth.	arccos(1/4) 75.522488°	90°	arccos(-1/4) 104.477512°	120°	arccos(-3/4) 138.590378°	arccos[-(n-5)/4]

RectifiedGossetic r(2_n,1) (up)

Dimension	4D	5D	6D	7D	8D	nD
Dynkin diagram	o3x3o3o	o3x3o3o *c3o	o3x3o3o3o *c3o	o3x3o3o3o3o *c3o	o3x3o3o3o3o3o *c3o	o3x3o...o3o *c3o
Acronym	rap	rat	rojak	rolaq	robay	rectified 2_n,1
Vertex Count	10`trip`	40`ope`	216`rappip`	2016`rixip`	69120`rillip`	?
Facet Count rect. simplex	5`oct`	16`rap`	72`rix`	576`ril`	17280`roc`	?
Facet Count rect. Gossetic	5`oct`	16`rap`	27`rat`	56`rojak`	240`rolaq`	?
Facet Count demihypercube	5`tet`	10`hex`	27`hin`	126`hax`	2160`hesa`	?
Circumradius	sqrt(3/5) 0.774597	1	sqrt(5/3) 1.290994	sqrt(3) 1.732051	sqrt(7) 2.645751	sqrt[(n-1)/(9-n)]
Inradius wrt. rect. simplex	1/sqrt(10) 0.316228	sqrt(2/5) 0.632456	1	4/sqrt(7) 1.511858	5/2 2.5	(n-3) sqrt[2/(n(9-n))]
Inradius wrt. rect. Gossetic	1/sqrt(10) 0.316228	sqrt(2/5) 0.632456	sqrt(2/3) 0.816497	2/sqrt(3) 1.154701	2	sqrt[8/((10-n)(9-n))]
Inradius wrt. demihypercube	3/sqrt(40) 0.474342	1/sqrt(2) 0.707107	5/sqrt(24) 1.020621	3/2 1.5	7/sqrt(8) 2.474874	(n-1)/sqrt[8(9-n)]
Volume	11 sqrt(5)/96 0.256216	9 sqrt(2)/10 1.272792	601 sqrt(3)/160 6.506016	18643/280 66.582143	457563/112 4085.383929	?
Surface	25 sqrt(2)/12 2.946278	(5+11 sqrt(5))/3 9.865583	[1089 sqrt(2)+156 sqrt(3)]/40 45.256962	[301+4207 sqrt(3)+114 sqrt(7)]/20 394.467670	?	?
Dihedral angles r.simp. - r.Goss.	arccos(1/4) 75.522488° r.simp. - r.simp.	arccos(-3/5) 126.869898°	arccos[-sqrt(3/8)] 127.761244°	arccos[-sqrt(3/7)] 130.893395°	arccos(-3/4) 138.590378°	arccos[-3/sqrt(n(10-n))]
Dihedral angles r.Goss. - r.Goss.	arccos(1/4) 75.522488° r.simp. - r.simp.	arccos(-3/5) 126.869898°	arccos(-1/4) 104.477512°	arccos(-1/3) 109.471221°	120°	arccos[-1/(10-n)]
Dihedral angles r.simp. - demi.	arccos(-1/4) 104.477512°	arccos[-1/sqrt(5)] 116.565051°	arccos[-sqrt(3/8)] 127.761244°	arccos[-2/sqrt(7)] 139.106605°	arccos[-5/sqrt(32)] 152.114433°	arccos[-(n-3)/sqrt(4n)]
Dihedral angles r.Goss. - demi.	arccos(-1/4) 104.477512°	arccos[-1/sqrt(5)] 116.565051°	120°	arccos[-1/sqrt(3)] 125.264390°	135°	arccos[-1/sqrt(10-n)]

RectifiedGossetic r(1_n,2) (up)

The rectified Gossetic r(1_n,2) surely can be described likewise as the birectified Gossetic br(2_n,1). In fact it is that polytope, where in itsCoxeter-Dynkin diagram exactly the bifurcation node is marked.

Dimension	4D	5D	6D	7D	8D	nD
Dynkin diagram	o3o3x3o	o3o3x3o *c3o	o3o3x3o3o *c3o	o3o3x3o3o3o *c3o	o3o3x3o3o3o3o *c3o	o3o3x3o...o3o *c3o
Acronym	rap	nit	ram	lanq	buffy	rectified 1_n,2
Vertex Count	10`trip`	80`tisdip`	720`tratrip`	10080`tratepe`	483840`trippen`	?
Facet Count birect. simp.	5`tet`	16`rap`	72`dot`	576`bril`	17280`broc`	?
Facet Count rect. Goss.	5`tet`	16`rap`	27`nit`	56`ram`	240`lanq`	?
Facet Count birect. hyp.c.	5`oct`	10`ico`	27`nit`	126`brox`	2160`bersa`	?
Circumradius	sqrt(3/5) 0.774597	sqrt(3/2) 1.224745	sqrt(3) 1.732051	sqrt(6) 2.449490	sqrt(15) 3.872983	sqrt[3(n-3)/(9-n)]
Inradius wrt. birect. simp.	3/sqrt(40) 0.474342	3/sqrt(10) 0.948683	3/2 1.5	6/sqrt(7) 2.267787	15/4 3.75	3(n-3)/sqrt[2n(9-n)]
Inradius wrt. rect. Gossetic	3/sqrt(40) 0.474342	3/sqrt(10) 0.948683	sqrt(3/2) 1.224745	sqrt(3) 1.732051	3	sqrt[18/((10-n)(9-n))]
Inradius wrt. birect. hyp.c.	1/sqrt(10) 0.316228	1/sqrt(2) 0.707107	sqrt(3/2) 1.224745	2	5/sqrt(2) 3.535534	(n-3)/sqrt(18-2n)
Volume	11 sqrt(5)/96 0.256216	31 sqrt(2)/10 4.384062	243 sqrt(3)/8 52.611043	?	?	?
Surface	25 sqrt(2)/12 2.946278	[60+11 sqrt(5)]/3 28.198916	[1674 sqrt(2)+99 sqrt(3)]/10 253.886653	?	?	?
Dihedral angles bir.s. - r.Goss.	25 sqrt(2)/12 2.946278	arccos(-3/5) 126.869898°	arccos[-sqrt(3/8)] 127.761244°	arccos[-sqrt(3/7)] 130.893395°	arccos(-3/4) 138.590378°	arccos[-3/sqrt(n(10-n))]
Dihedral angles bir.s. - bir.h.c.	arccos(-1/4) 104.477512°	arccos[-1/sqrt(5)] 116.565051°	arccos[-sqrt(3/8)] 127.761244°	arccos[-2/sqrt(7)] 139.106605°	arccos[-5/sqrt(32)] 152.114433°	arccos[-(n-3)/sqrt(4n)]
Dihedral angles r.Goss. - r.Goss.		arccos[-1/sqrt(5)] 116.565051°	arccos(-1/4) 104.477512°	arccos(-1/3) 109.471221°	120°	arccos[-1/(10-n)]
Dihedral angles r.Goss. - bir.h.c.		arccos[-1/sqrt(5)] 116.565051°	120°	arccos[-1/sqrt(3)] 125.264390°	135°	arccos[-1/sqrt(10-n)]
Dihedral angles bir.h.c. - bir.h.c.	arccos(1/4) 75.522488°	90°	arccos(-1/4) 104.477512°	120°	arccos(-3/4) 138.590378°	arccos[-(n-5)/4]

TruncatedGossetic t(n_2,1) (up)

Dimension	4D	5D	6D	7D	8D	nD
Dynkin diagram	o3x3x3x	o3o3x3x *c3o	o3o3o3x3x *c3o	o3o3o3o3x3x *c3o	o3o3o3o3o3x3x *c3o	o3o...o3x3x *c3o
Acronym	grip	thin	tojak	tanq	tiffy	truncated (n-4)_2,1
Vertex Count	60	160	432	1512	13440	?
Facet Count trunc. simplex	5`tut`	16`tip`	72`tix`	576`til`	17280`toc`	?
Facet Count Gossetic	10`trip`	16`rap`	27`hin`	56`jak`	240`naq`	?
Facet Count trunc. ortho.	5`toe`	10`thex`	27`tot`	126`tag`	2160`taz`	?
Circumradius	sqrt(17/5) 1.843909	sqrt(29/8) 1.903943	2	sqrt(19)/2 2.179449	sqrt(7) 2.645751	sqrt[(54-5n)/(18-2n)]
Inradius wrt. trunc. simplex	9/sqrt(40) 1.423025	9/sqrt(40) 1.423025	3/2 1.5	9/sqrt(28) 1.700840	9/4 2.25	9/sqrt[n(18-2n)]
Inradius wrt. Gossetic	13/sqrt(60) 1.678293	11/sqrt(40) 1.739253	sqrt(27/8) 1.837117	7/sqrt(12) 2.020726	5/2 2.5	(21-2n)/sqrt[(18-2n)(10-n)]
Inradius wrt. trunc. ortho.	3/sqrt(10) 0.948683	3/sqrt(8) 1.060660	sqrt(3/2) 1.224745	3/2 1.5	3/sqrt(2) 2.121320	3/sqrt(18-2n)
Volume	287 sqrt(5)/32 20.054735	623 sqrt(2)/24 36.710627	7251 sqrt(3)/160 78.494378	37109/140 265.064286	?	?
Surface	[595 sqrt(2)+30 sqrt(3)]/12 74.451549	[770+87 sqrt(5)]/6 160.756319	[8685 sqrt(2)+1422 sqrt(3)]/40 368.635526	[10122+35 sqrt(3)+722 sqrt(7)]/10 1209.285422	?	?
Dihedral angles tr.simp. - Goss.	arccos[-sqrt(3/8)] 127.761244°	arccos(-3/5) 126.869898°	arccos[-sqrt(3/8)] 127.761244°	arccos[-sqrt(3/7)] 130.893395°	arccos(-3/4) 138.590378°	arccos[-3/sqrt(n(10-n))]
Dihedral angles tr.sim. - tr.orth.	arccos(-1/4) 104.477512°	arccos[-1/sqrt(5)] 116.565051°	arccos[-sqrt(3/8)] 127.761244°	arccos[-2/sqrt(7)] 139.106605°	arccos[-5/sqrt(32)] 152.114433°	arccos[-(n-3)/sqrt(4n)]
Dihedral angles Goss. - tr.orth.	arccos[-1/sqrt(6)] 114.094843°	arccos[-1/sqrt(5)] 116.565051°	120°	arccos[-1/sqrt(3)] 125.264390°	135°	arccos[-1/sqrt(10-n)]
Dihedral angles tr.orth. - tr.orth.	arccos(1/4) 75.522488°	90°	arccos(-1/4) 104.477512°	120°	arccos(-3/4) 138.590378°	arccos[-(n-5)/4]

TruncatedGossetic t(2_n,1) (up)

Dimension	4D	5D	6D	7D	8D	nD
Dynkin diagram	x3x3o3o	x3x3o3o *c3o	x3x3o3o3o *c3o	x3x3o3o3o3o *c3o	x3x3o3o3o3o3o *c3o	x3x3o...o3o *c3o
Acronym	tip	tot	tojak	talq	toby	truncated 2_n,1
Vertex Count	20	80	432	4032	138240	?
Facet Count trunc. simplex	5`tut`	16`tip`	72`tix`	576`til`	17280`toc`	?
Facet Count trunc. Gossetic	5`tut`	16`tip`	27`tot`	56`tojak`	240`talq`	?
Facet Count demihypercube	5`tet`	10`hex`	27`hin`	126`hax`	2160`hesa`	?
Circumradius	sqrt(8/5) 1.264911	sqrt(5/2) 1.581139	2	sqrt(7) 2.645751	4	sqrt[2n/(9-n)]
Inradius wrt. trunc. simplex	3/sqrt(40) 0.474342	3/sqrt(10) 0.948683	3/2 1.5	6/sqrt(7) 2.267787	15/4 3.75	3(n-3)/sqrt[2n(9-n)]
Inradius wrt. trunc. Gossetic	3/sqrt(40) 0.474342	3/sqrt(10) 0.948683	sqrt(3/2) 1.224745	sqrt(3) 1.732051	3	sqrt[18/((9-n)(10-n))]
Inradius wrt. demihypercube	7/sqrt(40) 1.106797	sqrt(2) 1.414214	sqrt(27/8) 1.837117	5/2 2.5	11/sqrt(8) 3.889087	(n+3)/sqrt[8(9-n)]
Volume	19 sqrt(5)/24 1.770220	119 sqrt(2)/15 11.219428	7251 sqrt(3)/160 78.494378	?	?	?
Surface	10 sqrt(2) 14.142136	(5+76 sqrt(5))/3 58.313722	[8685 sqrt(2)+1422 sqrt(3)]/40 368.635526	?	?	?
Dihedral angles tr.sim. - tr.Goss.	arccos(1/4) 75.522488° tr.sim. - tr.sim.	arccos(-3/5) 126.869898°	arccos[-sqrt(3/8)] 127.761244°	arccos[-sqrt(3/7)] 130.893395°	arccos(-3/4) 138.590378°	arccos[-3/sqrt(n(10-n))]
Dihedral angles tr.Goss. - tr.Goss.	arccos(1/4) 75.522488° tr.sim. - tr.sim.	arccos(-3/5) 126.869898°	arccos(-1/4) 104.477512°	arccos(-1/3) 109.471221°	120°	arccos[-1/(10-n)]
Dihedral angles tr.sim. - demi.	arccos(-1/4) 104.477512°	arccos[-1/sqrt(5)] 116.565051°	arccos[-sqrt(3/8)] 127.761244°	arccos[-2/sqrt(7)] 139.106605°	arccos[-5/sqrt(32)] 152.114433°	arccos[-(n-3)/sqrt(4n)]
Dihedral angles tr.Goss. - demi.	arccos(-1/4) 104.477512°	arccos[-1/sqrt(5)] 116.565051°	120°	arccos[-1/sqrt(3)] 125.264390°	135°	arccos[-1/sqrt(10-n)]

TruncatedGossetic t(1_n,2) (up)

Dimension	4D	5D	6D	7D	8D	nD
Dynkin diagram	o3o3x3x	o3o3x3o *c3x	o3o3x3o3o *c3x	o3o3x3o3o3o *c3x	o3o3x3o3o3o3o *c3x	o3o3x3o...o3o *c3x
Acronym	tip	thin	tim	tolin	tabif	truncated 1_n,2
Vertex Count	20	160	1440	20160	967680	?
Facet Count birect. simp.	5`tet`	16`rap`	72`dot`	576`bril`	17280`broc`	?
Facet Count trunc. Goss.	5`tet`	16`tip`	27`thin`	56`tim`	240`tolin`	?
Facet Count trunc. demih.c.	5`tut`	10`thex`	27`thin`	126`thax`	2160`thesa`	?
Circumradius	sqrt(8/5) 1.264911	sqrt(29/8) 1.903943	sqrt(7) 2.645751	sqrt(55)/2 3.708099	sqrt(34) 5.830952	sqrt[(13n-36)/(18-2n)]
Inradius wrt. birect. simp.	7/sqrt(40) 1.106797	11/sqrt(40) 1.739253	5/2 2.5	19/sqrt(28) 3.590662	23/4 5.75	(4n-9)/sqrt[2n(9-n)]
Inradius wrt. trunc. Goss.	7/sqrt(40) 1.106797	9/sqrt(40) 1.423025	sqrt(27/8) 1.837117	sqrt(27)/2 2.598076	9/2 4.5	9/sqrt[2(9-n)(10-n)]
Inradius wrt. trunc. demih.c.	3/sqrt(40) 0.474342	3/sqrt(8) 1.060660	sqrt(27/8) 1.837117	3	15/sqrt(8) 5.303301	3(n-3)/sqrt[8(9-n)]
Volume	19 sqrt(5)/24 1.770220	623 sqrt(2)/24 36.710627	5673 sqrt(3)/16 614.120264	?	?	?
Surface	10 sqrt(2) 14.142136	[770+87 sqrt(5)]/6 160.756319	[28035 sqrt(2)+198 sqrt(3)]/20 1999.521164	?	?	?
Dihedral angles bir.s. - tr.Goss.	10 sqrt(2) 14.142136	arccos(-3/5) 126.869898°	arccos[-sqrt(3/8)] 127.761244°	arccos[-sqrt(3/7)] 130.893395°	arccos(-3/4) 138.590378°	arccos[-3/sqrt(n(10-n))]
Dihedral angles bir.s. - tr.demi.	arccos(-1/4) 104.477512°	arccos[-1/sqrt(5)] 116.565051°	arccos[-sqrt(3/8)] 127.761244°	arccos[-2/sqrt(7)] 139.106605°	arccos[-5/sqrt(32)] 152.114433°	arccos[-(n-3)/sqrt(4n)]
Dihedral angles tr.Goss. - tr.Goss.		arccos[-1/sqrt(5)] 116.565051°	arccos(-1/4) 104.477512°	arccos(-1/3) 109.471221°	120°	arccos[-1/(10-n)]
Dihedral angles tr.Goss. - tr.demi.		arccos[-1/sqrt(5)] 116.565051°	120°	arccos[-1/sqrt(3)] 125.264390°	135°	arccos[-1/sqrt(10-n)]
Dihedral angles tr.demi. - tr.demi.	arccos(1/4) 75.522488°	90°	arccos(-1/4) 104.477512°	120°	arccos(-3/4) 138.590378°	arccos[-(n-5)/4]

All-Ends ExpandedGossetic eE_n (up)

Dimension	4D	5D	6D	7D	8D	nD
Dynkin diagram	x3o3x3x	x3o3o3x *c3x	x3o3o3o3x *c3x	x3o3o3o3o3x *c3x	x3o3o3o3o3o3x *c3x	x3o...o3x *c3x
Acronym	prip	spat	spam	sethalq	spuffy	all-ends exp. E_n
Vertex Count	60	320	2160	24192	?	?
Facet Count exp. simpl.	5`co`	16`spid`	72`scad`	576`staf`	?`suph`	?
Facet Count exp. Goss.	10`trip`	16`spid`	27`siphin`	56`hejak`	?`shilq`	?
Facet Count exp. Goss. pr.			216`spiddip`	756`siphinnip`	?`hejakip`	?
Facet Count duoprism I				4032`traspid`	?`trasiphin`	?
Facet Count duoprism II				4032`traspid`	?`tetspid`	?
Facet Count dippip		80`tisdip`	720`tratrip`	10080`tratepe`	?`trippen`	?
Facet Count exp. simpl. pr.	10`hip`	40`cope`	216`spiddip`	2016`scadip`	?`staffip`	?
Facet Count exp. demicube	5`tut`	10`rit`	27`siphin`	126`sochax`	?`suthesa`	?
Circumradius	sqrt(13/5) 1.612452	sqrt(7/2) 1.870829	sqrt(5) 2.236068	sqrt(8) 2.828427	sqrt(17) 4.123106	sqrt[(9+n)/(9-n)]
Inradius wrt. exp. simpl.	sqrt(8/5) 1.264911	sqrt(5/2) 1.581139	2	sqrt(7) 2.645751	4	sqrt[2n/(9-n)]
Inradius wrt. exp. Goss.	11/sqrt(60) 1.420094	sqrt(5/2) 1.581139	sqrt(27/8) 1.837117	4/sqrt(3) 2.309401	7/2 3.5	(15-n)/sqrt[2(10-n)(9-n)]
Inradius wrt. exp. Goss. pr.			sqrt(15)/2 1.936492	7/sqrt(8) 2.474874	13/sqrt(12) 3.752777	(21-n)/sqrt[4(11-n)(9-n)]
Inradius wrt. duoprism I				sqrt(20/3) 2.581989	?	?
Inradius wrt. duoprism II				sqrt(20/3) 2.581989	?	?
Inradius wrt. dippip		4/sqrt(6) 1.632993	7/sqrt(12) 2.020726	13/sqrt(24) 2.653614	31/sqrt(60) 4.002083	(5n-9)/sqrt[12(n-3)(9-n)]
Inradius wrt. exp. simpl. pr.	sqrt(27/20) 1.161895	3/2 1.5	sqrt(15)/2 1.936492	sqrt(27)/2 2.598076	sqrt(63)/2 3.968627	sqrt[(9n-9)/(36-4n)]
Inradius wrt. exp. demicube	7/sqrt(40) 1.106797	sqrt(2) 1.414214	sqrt(27/8) 1.837117	5/2 2.5	11/sqrt(8) 3.889087	(n+3)/sqrt[8(9-n)]
Volume	237 sqrt(5)/32 16.560878	142 sqrt(2)/3 66.939442	17811 sqrt(3)/80 385.619462	34715/8 4339.375	?	?
Surface	[215 sqrt(2)+210 sqrt(3)]/12 55.648882	5 [23+40 sqrt(2)+ +12 sqrt(3)+14 sqrt(5)]/3 219.430173	[2700+4203 sqrt(2)+ +756 sqrt(3)+6300 sqrt(5)]/20 1202.029914	[19159+58842 sqrt(2)+ +38962 sqrt(3)+4200 sqrt(6)+ +1848 sqrt(7)+14700 sqrt(15)]/20 12098.418927	?	?
Dihedral angles e.sim. - e.Goss.	arccos[-sqrt(3/8)] 127.761244°	arccos(-3/5) 126.869898°	arccos[-sqrt(3/8)] 127.761244°	arccos[-sqrt(3/7)] 130.893395°	arccos(-3/4) 138.590378°	arccos[-3/sqrt(n(10-n))]
Dihedral angles e.sim. - e.Goss.p.			?	?	?	?
Dihedral angles e.Goss. - e.Goss.p.			?	?	?	?
Dihedral angles e.sim. - duop.I				?	?	?
Dihedral angles e.Goss.p. - duop.I				?	?	?
Dihedral angles e.sim. - duop.II					?	?
Dihedral angles duop.I - duop.II					?	?
Dihedral angles e.sim. - dippip		?	?	?	?	?
Dihedral angles duop.II - dippip		?	?	?	?	?
Dihedral angles e.sim. - e.sim.p.	arccos[-sqrt(1/6)] 114.094843°	arccos[-sqrt(2/5)] 129.231520°	?	?	?	?
Dihedral angles e.Goss. - e.sim.p.	arccos(-2/3) 131.810315°	arccos[-sqrt(2/5)] 129.231520°	?	?	?	?
Dihedral angles e.Goss.p. - e.sim.p.			?	?	?	?
Dihedral angles duop.I - e.sim.p.				?	?	?
Dihedral angles duop.II - e.sim.p.				?	?	?
Dihedral angles dippip - e.sim.p.		?	?	?	?	?
Dihedral angles e.sim. - e.demic.	arccos(-1/4) 104.477512°	arccos[-1/sqrt(5)] 116.565051°	arccos[-sqrt(3/8)] 127.761244°	arccos[-2/sqrt(7)] 139.106605°	arccos[-5/sqrt(32)] 152.114433°	arccos[-(n-3)/sqrt(4n)]
Dihedral angles e.Goss. - e.demic.		arccos[-1/sqrt(5)] 116.565051°	120°	arccos[-1/sqrt(3)] 125.264390°	135°	arccos[-1/sqrt(10-n)]
Dihedral angles e.Goss.p. - e.demic.			?	?	?	?
Dihedral angles duop.I - e.demic.				?	?	?
Dihedral angles duop.II - e.demic.				?	?	?
Dihedral angles e.sim.p. - e.demic.	arccos[-sqrt(3/8)] 127.761244°	135°	?	?	?	?

OmnitruncatedGossetic otE_n (up)

Dimension	4D	5D	6D	7D	8D	nD
Dynkin diagram	x3x3x3x	x3x3x3x *c3x	x3x3x3x3x *c3x	x3x3x3x3x3x *c3x	x3x3x3x3x3x3x *c3x	x3x...x3x3x *c3x
Acronym	gippid	gippit	gopam	gotanq	gupofy	omnitr. (n-4)_2,1
Vertex Count	120	1920	51840	2903040	696729600	?
Facet Count wrt. type 1	5`toe`	16`gippid`	72`gocad`	576`gotaf`	17280`guph`	?
Facet Count wrt. type 2	10`hip`	16`gippid`	27`gippit`	56`gopam`	240`gotanq`	?
Facet Count wrt. type 3	10`hip`	80`shiddip`	216`gippiddip`	756`gippitip`	6720`gopamp`	?
Facet Count wrt. type 4	5`toe`	40`tope`	720`hahip`	4032`hagippid`	60480`hagippit`	?
Facet Count wrt. type 5		10`tico`	216`gippiddip`	2016`gocadip`	241920`toegippid`	?
Facet Count wrt. type 6			27`gippit`	10080`hatope`	483840`hagippiddip`	?
Facet Count wrt. type 7				126`gocog`	69120`gotafip`	?
Facet Count wrt. type 8				126`gocog`	2160`gotaz`	?
Circumradius	sqrt(5) 2.236068	sqrt(15) 3.872983	sqrt(39) 6.244998	sqrt(399)/2 9.987492	sqrt(310) 17.606817	?
Inradius wrt. facet type 1	sqrt(5/2) 1.581139	sqrt(10) 3.162278	11/2 5.5	sqrt(343)/2 9.260130	17	?
Inradius wrt. facet type 2	sqrt(15)/2 1.936492	sqrt(10) 3.162278	sqrt(24) 4.898980	sqrt(243)/2 7.794229	29/2 14.5	?
Inradius wrt. facet type 3	sqrt(15)/2 1.936492	sqrt(27/2) 3.674235	sqrt(135)/2 5.809475	13/sqrt(2) 9.192388	sqrt(1083)/2 16.454483	?
Inradius wrt. facet type 4	sqrt(5/2) 1.581139	7/2 3.5	sqrt(147)/2 6.062178	sqrt(375)/2 9.682458	sqrt(294) 17.146428	?
Inradius wrt. facet type 5		sqrt(8) 2.828427	sqrt(135)/2 5.809475	sqrt(363)/2 9.526279	sqrt(605/2) 17.392527	?
Inradius wrt. facet type 6			sqrt(24) 4.898980	sqrt(96) 9.797959	sqrt(1215)/2 17.428425	?
Inradius wrt. facet type 7				17/2 8.5	sqrt(1183)/2 17.197384	?
Inradius wrt. facet type 8				17/2 8.5	sqrt(529/2) 16.263456	?
Volume	125 sqrt(5)/4 69.877124	?	?	?	?	?
Surface	?	?	?	?	?	?
Dihedral angles types 1 - 2	arccos[-sqrt(3/8)] 127.761244°	arccos(-3/5) 126.869898°	arccos[-sqrt(3/8)] 127.761244°	arccos[-sqrt(3/7)] 130.893395°	arccos(-3/4) 138.590378°	arccos[-3/sqrt(n(10-n))]
Dihedral angles types 1 - 3	arccos[-sqrt(1/6)] 114.094843°	?	?	?	?	?
Dihedral angles types 1 - 4	arccos(-1/4) 104.477512°	arccos[-sqrt(2/5)] 129.231520°	?	?	?	?
Dihedral angles types 1 - 5		arccos[-1/sqrt(5)] 116.565051°	?	?	?	?
Dihedral angles types 1 - 6			arccos[-sqrt(3/8)] 127.761244°	?	?	?
Dihedral angles types 1 - 7				arccos[-2/sqrt(7)] 139.106605°	?	?
Dihedral angles types 1 - 8				arccos[-2/sqrt(7)] 139.106605°	arccos[-5/sqrt(32)] 152.114433°	?
Dihedral angles types 2 - 3	arccos(-2/3) 131.810315°	?	?	?	?	?
Dihedral angles types 2 - 4	arccos(-1/4) 104.477512°	arccos[-sqrt(2/5)] 129.231520°	?	?	?	?
Dihedral angles types 2 - 5		arccos[-1/sqrt(5)] 116.565051°	?	?	?	?
Dihedral angles types 2 - 6			120°	?	?	?
Dihedral angles types 2 - 7				arccos[-1/sqrt(3)] 125.264390°	?	?
Dihedral angles types 2 - 8				arccos[-1/sqrt(3)] 125.264390°	135°	?
Dihedral angles types 3 - 4	arccos[-sqrt(3/8)] 127.761244°	?	?	?	?	?
Dihedral angles types 3 - 5		?	?	?	?	?
Dihedral angles types 3 - 6			?	?	?	?
Dihedral angles types 3 - 7				?	?	?
Dihedral angles types 3 - 8				?	?	?
Dihedral angles types 4 - 5		135°	?	?	?	?
Dihedral angles types 4 - 6			?	?	?	?
Dihedral angles types 4 - 7				?	?	?
Dihedral angles types 4 - 8				?	?	?
Dihedral angles types 5 - 6			?	?	?	?
Dihedral angles types 5 - 7				?	?	?
Dihedral angles types 5 - 8				?	?	?
Dihedral angles types 6 - 7				?	?	?
Dihedral angles types 6 - 8					?	?
Dihedral angles types 7 - 8					?	?

Some Axial Cases

Simplexial Ursatope U_n (up)

The name of theUrsatopes derives from the acronym of the 3D sequence member,teddi (J63), being homonym to the toy-bear, or Latinized "urs". The simplexial ones are defined generally as thebistratic lace towersofx3xoo3ooo...ooo3ooo&#xt, i.e. the n-dimensional simplexial ursatope U_n can be described astherectified simplex rS_n-1 atop thef-scaledregular simplexf·S_n-1 atop the (unit) regular simplex S_n-1. All those ursatopes happen to beorbiform CRFs, i.e. are circumscribable, convex, and regular faced.

It could be mentioned here additionally that the simplexial ursatope U_n generally is nothing but thevertex figure ofs3s4o3o...o3o, which for low dimensions is spherical, at rank 5 (i.e. 5 nodes) becomes aneuclidean tetracomb, and thereafter will belong tohyperbolic geometry. This then gets reflected too in the table below by the values of the circumradii of U_n, which traverse unity at n=4.

Further thevertex figures of these polytopes could be described uniformely.At the lower 2 of its vertex types one hasox3oo...oo3oo&#f spike-like tall simplex pyramides,the top vertices however arexf xo...oo3oo&#x, i.e. simplex prism wedges, where the additionalwedge-edge has sizef and runs axis parallel to the base (simplex prism).

Dimension	2D	3D	4D	5D	nD
Dynkin diagram	ofx&#xt	ofx3xoo&#xt	ofx3xoo3ooo&#xt	ofx3xoo3ooo3ooo&#xt	ofx3xoo3ooo...ooo3ooo&#xt
Acronym	peg	teddi	tetu	penu	simpl. n-ursatope
Vertex Count top layer	1	3	6	10	n(n-1)/2
Vertex Count medial layer	2	3	4	5	n
Vertex Count bottom layer	2	3	4	5	n
Facet Count top	2	1`trig`	1`oct`	1`rap`	1
Facet Count upper lacing	2`line`	3`trig`	4`tet`	5`pen`	n
Facet Count lower lacing	2`line`	3`peg`	4`teddi`	5`tetu`	n
Facet Count bottom	1`line`	1`trig`	1`tet`	1`pen`	1
Circumradius	sqrt[(5+sqrt(5))/10] 0.850651	sqrt[(5+sqrt(5))/8] 0.951057	1	sqrt[2+sqrt(5)]/2 1.029086	sqrt[((29n-22)+(13n-10) sqrt(5)) / ((22n+6)+(10n+2) sqrt(5))]
Inradius wrt. top facet	sqrt[(5+sqrt(5))/10] 0.850651	sqrt[(7+3 sqrt(5))/24] 0.755761	1/sqrt(2) 0.707107	sqrt[(5 sqrt(5)-2)/20] 0.677508	?
Inradius wrt. upper lacing	sqrt[(5+2 sqrt(5))/20] 0.688191	sqrt[(7+3 sqrt(5))/24] 0.755761	sqrt(5/8) 0.790569	sqrt[(1+3 sqrt(5))/12] 0.801468	?
Inradius wrt. lower lacing	sqrt[(5+2 sqrt(5))/20] 0.688191	sqrt[(5+sqrt(5))/40] 0.425325	[sqrt(5)-1]/4 0.309017	sqrt[sqrt(5)-2]/2 0.242934	?
Inradius wrt. bottom facet	sqrt[(5+2 sqrt(5))/20] 0.688191	sqrt[(7+3 sqrt(5))/24] 0.755761	sqrt(5/8) 0.790569	sqrt[(1+3 sqrt(5))/12] 0.801468	?
Volume	sqrt[25+10 sqrt(5)]/4 1.720477	[15+7 sqrt(5)]/24 1.277186	[28+13 sqrt(5)]/96 0.594468	[11 sqrt(5 sqrt(5)-2)+2 sqrt(15+45 sqrt(5))+ +(140+65 sqrt(5)) sqrt(sqrt(5)-2)]/960 0.201536	?
Surface	5	[5 sqrt(3)+3 sqrt(25+10 sqrt(5))]/4 7.326496	[30+9 sqrt(2)+14 sqrt(5)]/12 6.169406	[70+41 sqrt(5)]/48 3.368308	?
Dihedral angles top - upper	108° upper - upper	arccos(-sqrt(5)/3) 138.189685°	?	?	?
Dihedral angles top - lower	108° upper - upper	arccos(-sqrt(5)/3) 138.189685°	?	?	?
Dihedral angles lower - upper	108°	arccos(-sqrt[(5-2 sqrt(5))/15]) 100.812317°	?	?	?
Dihedral angles lower - lower	108°	arccos(1/sqrt(5)) 63.434949°	?	?	?
Dihedral angles lower - bottom	108°	arccos(-sqrt[(5-2 sqrt(5))/15]) 100.812317°	?	?	?
Dimension	6D	7D	8D	9D	nD
Dynkin diagram	ofx3xoo3ooo3ooo3ooo&#xt	ofx3xoo3ooo3ooo3ooo3ooo&#xt	ofx3xoo3ooo3ooo3ooo3ooo3ooo&#xt	ofx3xoo3ooo3ooo3ooo3ooo3ooo3ooo&#xt	ofx3xoo3ooo...ooo3ooo&#xt
Acronym	hixu	hopu	ocu	enu	simpl. n-ursatope
Vertex Count top layer	15	21	28	36	n(n-1)/2
Vertex Count medial layer	6	7	8	9	n
Vertex Count bottom layer	6	7	8	9	n
Facet Count top	1`rix`	1`ril`	1`roc`	1`rene`	1
Facet Count upper lacing	6`hix`	7`hop`	8`oca`	9`ene`	n
Facet Count lower lacing	6`penu`	7`hixu`	8`hopu`	9`ocu`	n
Facet Count bottom	1`hix`	1`hop`	1`oca`	1`ene`	1
Circumradius	sqrt[(13+5 sqrt(5))/22] 1.048383	sqrt[(25+9 sqrt(5))/40] 1.062128	sqrt[(20+7 sqrt(5))/31] 1.072418	sqrt[(29+10 sqrt(5))/44] 1.080411	sqrt[((29n-22)+(13n-10) sqrt(5)) / ((22n+6)+(10n+2) sqrt(5))]
Inradius wrt. top facet	sqrt[(15 sqrt(5)-5)/66] 0.657601	sqrt[(63 sqrt(5)-25)/280] 0.643296	sqrt[(28 sqrt(5)-13)/124] 0.632519	sqrt[(90 sqrt(5)-47)/396] 0.624108	?
Inradius wrt. upper lacing	sqrt[(23+30 sqrt(5))/132] 0.826099	sqrt[(55+63 sqrt(5))/280] 0.836387	sqrt[(103+112 sqrt(5))/496] 0.844144	sqrt[(85+90 sqrt(5))/396] 0.850202	?
Inradius wrt. lower lacing	sqrt[(4-sqrt(5))/44] 0.200223	sqrt[(15-sqrt(5))/440] 0.170320	sqrt[(25+sqrt(5))/1240] 0.148204	sqrt[(19+2 sqrt(5))/1364] 0.131180	?
Inradius wrt. bottom facet	sqrt[(23+30 sqrt(5))/132] 0.826099	sqrt[(55+63 sqrt(5))/280] 0.836387	sqrt[(103+112 sqrt(5))/496] 0.844144	sqrt[(85+90 sqrt(5))/396] 0.850202	?
Volume	?	?	?	?	?
Surface	?	?	?	?	?
Dihedral angles top - upper	?	?	?	?	?
Dihedral angles top - lower	?	?	?	?	?
Dihedral angles lower - upper	?	?	?	?	?
Dihedral angles lower - lower	?	?	?	?	?
Dihedral angles lower - bottom	?	?	?	?	?

Orthoplexial Ursatope oU_n (up)

The orthoplexialUrsatopes are defined generally as thebistratic lace towersofx3xoo3ooo...ooo3ooo4ooo&#xt, i.e. the n-dimensional orthoplexial ursatope oU_n can be described astherectified orthoplex rO_n-1 atop thef-scaledregular orthoplexf·O_n-1 atop the (unit) regular orthoplex O_n-1. Nearly all those ursatopes happen to beorbiform CRFs, i.e. are circumscribable, convex, and regular faced. Only the 3D representant shows up external q-edges;none the less it still remains circumscribable.

Dimension	3D	4D	5D	6D	7D	nD
Dynkin diagram	ofx4qoo&#xt (non-orbiform)	ofx3xoo4ooo&#xt	ofx3xoo3ooo4ooo&#xt	ofx3xoo3ooo3ooo4ooo&#xt	ofx3xoo3ooo3ooo3ooo4ooo&#xt	ofx3xoo3ooo...ooo4ooo&#xt
Acronym	--	octu	hexu	tacu	gu	orthopl. n-ursatope
Vertex Count top layer	4	12	24	40	60	2(n-1)(n-2)
Vertex Count medial layer	4	6	8	10	12	2(n-1)
Vertex Count bottom layer	4	6	8	10	12	2(n-1)
Facet Count top	1`q-square`	1`co`	1`ico`	1`rat`	1`rag`	1
Facet Count upper lacing	4`oq&#x`	6`squippy`	8`octpy`	10`hexpy`	12`tacpy`	2(n-1)
Facet Count lower lacing	4`peg`	8`teddi`	16`tetu`	32`penu`	64`hixu`	2^n-1
Facet Count bottom	1`square`	1`oct`	1`hex`	1`tac`	1`gee`	1
Circumradius	sqrt[3+sqrt(5)]/2 1.144123	sqrt[3+sqrt(5)]/2 1.144123	sqrt[3+sqrt(5)]/2 1.144123	sqrt[3+sqrt(5)]/2 1.144123	sqrt[3+sqrt(5)]/2 1.144123	sqrt[3+sqrt(5)]/2 1.144123
Inradius wrt. top facet	sqrt[sqrt(5)-1]/2 0.555893	sqrt[sqrt(5)-1]/2 0.555893	sqrt[sqrt(5)-1]/2 0.555893	sqrt[sqrt(5)-1]/2 0.555893	sqrt[sqrt(5)-1]/2 0.555893	sqrt[sqrt(5)-1]/2 0.555893
Inradius wrt. upper lacing	sqrt[1+sqrt(5)]/2 0.899454	sqrt[1+sqrt(5)]/2 0.899454	sqrt[1+sqrt(5)]/2 0.899454	sqrt[1+sqrt(5)]/2 0.899454	sqrt[1+sqrt(5)]/2 0.899454	sqrt[1+sqrt(5)]/2 0.899454
Inradius wrt. lower lacing	sqrt[(5+3 sqrt(5))/20] 0.765121	sqrt[(1+sqrt(5))/8] 0.636010	sqrt[sqrt(5)-1]/2 0.555893	1/2 0.5	sqrt[(7+sqrt(5)/44] 0.458160	sqrt[(29+13 sqrt(5)) / ((22n-16)+(10n-8)sqrt(5))]
Inradius wrt. bottom facet	sqrt[1+sqrt(5)]/2 0.899454	sqrt[1+sqrt(5)]/2 0.899454	sqrt[1+sqrt(5)]/2 0.899454	sqrt[1+sqrt(5)]/2 0.899454	sqrt[1+sqrt(5)]/2 0.899454	sqrt[1+sqrt(5)]/2 0.899454
Volume	?	?	?	?	?	?
Surface	?	?	?	?	?	?
Dihedral angles top - upper	?	?	?	?	?	?
Dihedral angles top - lower	?	?	?	?	?	?
Dihedral angles lower - upper	?	?	?	?	?	?
Dihedral angles lower - lower	?	?	?	?	?	?
Dihedral angles lower - bottom	?	?	?	?	?	?

Rectified Simplex Pyramid rS_n-py (up)

As such these polytopesoo3ox3oo...oo3oo&#x look just to be a mere similar concept to the pyramids on simplex base, which, for sure, as such are nothing but simplices of the next dimension themselves. However it happens that the demihypercube D_n, when seen as lace towerwith vertex first orientation, becomes generallyooo..-3-oxo..-3-ooo..-3-oox..-...-ooo..&#xt (n-1 node positions, n/2 or (n+1)/2 layers). Thence this very pyramid of consideration is nothing but the vertex pyramid thereof.

Below it is shown that the dihedral angle at the base decreases to zero with increasing dimension. This is what makes the possibilities to augment other polytopes with this component ever more likely, esp. the possibilities for higher dimensionalCRF would explode.

Dimension	3D	4D	5D	6D	nD
Dynkin diagram	oo3ox&#x	oo3ox3oo&#x	oo3ox3oo3oo&#x	oo3ox3oo3oo&#x	oo3ox3oo...oo3oo&#x
Acronym	tet	octpy	rappy	rixpy	rect. n-simplex pyr.
Vertex Count	1+3	1+6	1+10	1+15	1+n(n-1)/2
Facet Count simpl. lacing	3`trig`	4`tet`	5`pen`	6`hix`	n
Facet Count other lacing	3`trig`	4`tet`	5`octpy`	6`rappy`	n
Facet Count base	1`trig`	1`oct`	1`rap`	1`rix`	1
Circumradius	sqrt(3/8) 0.612372	1/sqrt(2) 0.707107	sqrt(5/8) 0.790569	sqrt(3)/2 0.866025	sqrt(n/8)
Inradius wrt. simpl. lacing	1/sqrt(24) 0.204124	1/sqrt(8) 0.353553	3/sqrt(40) 0.474342	1/sqrt(3) 0.577350	(n-2)/sqrt(8n)
Inradius wrt. other lacing	1/sqrt(24) 0.204124	1/sqrt(8) 0.353553	1/sqrt(8) 0.353553	1/sqrt(8) 0.353553	1/sqrt(8) 0.353553
Inradius wrt. base	1/sqrt(24) 0.204124	0	-1/sqrt(40) -0.158114	-1/sqrt(12) -0.288675	-(n-4)/sqrt(8n)
Volume	sqrt(2)/12 0.117851	1/12 0.833333	11 sqrt(2)/480 0.032409	13/1440 0.0090278	(2^n-1-n)/(n! sqrt(2^n-2))
Surface	sqrt(3) 1.732051	sqrt(2) 1.414214	?	?	?
Dihedral angles simp. - other	arccos(1/3) 70.528779° (simp. - simp.)	120°	arccos[-1/sqrt(5)] 116.565051°	arccos[-1/sqrt(6)] 114.094843°	arccos[-1/sqrt(n)]
Dihedral angles other - other	arccos(1/3) 70.528779° (simp. - simp.)	120°	90°	90°	90°
Dihedral angles simp. - base	arccos(1/3) 70.528779°	60°	arccos(3/5) 53.130102°	arccos(2/3) 48.189685°	arccos[(n-2)/n]
Dihedral angles other - base	arccos(1/3) 70.528779°	60°	arccos[1/sqrt(5)] 63.434949°	arccos[1/sqrt(6)] 65.905157°	arccos[1/sqrt(n)]
Height	sqrt(2/3) 0.816497	1/sqrt(2) 0.707107	sqrt(2/5) 0.632456	1/sqrt(3) 0.577350	sqrt(2/n)
Dimension	7D	8D	9D	10D	nD
Dynkin diagram	oo3ox3oo3oo3oo3oo&#x	oo3ox3oo3oo3oo3oo3oo&#x	oo3ox3oo3oo3oo3oo3oo3oo&#x	oo3ox3oo3oo3oo3oo3oo3oo3oo&#x	oo3ox3oo...oo3oo&#x
Acronym	rilpy	rocpy	renepy	?	rect. n-simplex pyr.
Vertex Count	1+21	1+28	1+36	1+45	1+n(n-1)/2
Facet Count simpl. lacing	7`hop`	8`oca`	9`ene`	10`day`	n
Facet Count other lacing	7`rixpy`	8`rilpy`	9`rocpy`	10`renepy`	n
Facet Count base	1`ril`	1`roc`	1`rene`	1`reday`	1
Circumradius	sqrt(7/8) 0.935414	1	3/sqrt(8) 1.060660	sqrt(5)/2 1.118034	sqrt(n/8)
Inradius wrt. simpl. lacing	5/sqrt(56) 0.668153	3/4 0.75	7/sqrt(72) 0.824958	2/sqrt(5) 0.894427	(n-2)/sqrt(8n)
Inradius wrt. other lacing	1/sqrt(8) 0.353553	1/sqrt(8) 0.353553	1/sqrt(8) 0.353553	1/sqrt(8) 0.353553	1/sqrt(8) 0.353553
Inradius wrt. base	-3/sqrt(56) -0.400892	-1/2 -0.5	-5/sqrt(72) -0.589256	-6/sqrt(80) -0.670820	-(n-4)/sqrt(8n)
Volume	19 sqrt(2)/13440 0.0019993	1/2688 0.00037202	247 sqrt(2)/5806080 0.000060163	251/29030400 0.0000086461	(2^n-1-n)/(n! sqrt(2^n-2))
Surface	?	?	?	?	?
Dihedral angles simp. - other	arccos[-1/sqrt(7)] 112.207654°	arccos[-1/sqrt(8)] 110.704811°	arccos(-1/3) 109.471221°	arccos[-1/sqrt(10)] 108.434949°	arccos[-1/sqrt(n)]
Dihedral angles other - other	90°	90°	90°	90°	90°
Dihedral angles simp. - base	arccos(5/7) 44.415309°	arccos(3/4) 41.409622°	arccos(7/9) 38.942441°	arccos(4/5) 36.869898°	arccos[(n-2)/n]
Dihedral angles other - base	arccos[1/sqrt(7)] 67.792346°	arccos[1/sqrt(8)] 69.295189°	arccos(1/3) 70.528779°	arccos[1/sqrt(10)] 71.565051°	arccos[1/sqrt(n)]
Height	sqrt(2/7) 0.534522	1/2 0.5	sqrt(2)/3 0.471405	1/sqrt(5) 0.447214	sqrt(2/n)

Some Duoprismatic Cases

Volumes of duoprisms A×B are easily calculated as the product of the subdimensional volumes of A resp. of B.Thus plain prisms (of unit height, for sure) have the same numeric volume value, as the subdimensional volume of its base.

Vertex counts of duoprisms A×B likewise are given as the product of the vertex counts of A resp. of B.

Simplex Duoprism S_n×S_n (up)

This case results ineven dimensions only.

From the axial representation of one of the factors, i.e. of S_n,it becomes clear that S_n×S_n can well be representedas thesegmentotope of theregular simplex S_natop the simplex duoprism S_n×S_n-1.Thence, by means of thelace prism notation, S_n×S_n
x3o3o...o3o x3o3o...o3o (2n nodes) can be described as well asxx3oo3oo...oo3oo ox3oo...oo3oo&#x (2n-1 nodes).

It could be mentioned here additionally that the simplex duoprism S_n×S_n generally is nothing but thevertex figure of themid-rectified simplex mrS_2n+1.

Dimension	2D	4D	6D	8D	10D	(2n)D
Dynkin diagram	x x	x3o x3o	x3o3o x3o3o	x3o3o3o x3o3o3o	x3o3o3o3o x3o3o3o3o	x3o...o3o x3o...o3o
Acronym	square	triddip	tetdip	pendip	hixdip	n-simplex duoprism
Vertex Count	4	9	16	25	36	(n+1)²
Facet Count	4`line`	6`trip`	8`tratet`	10`tetpen`	12`penhix`	2(n+1)
Circumradius	1/sqrt(2) 0.707107	sqrt(2/3) 0.816497	sqrt(3)/2 0.866025	2/sqrt(5) 0.894427	sqrt(5/6) 0.912871	sqrt[n/(n+1)]
Inradius	1/2 0.5	1/sqrt(12) 0.288675	1/sqrt(24) 0.204124	1/sqrt(40) 0.158114	1/sqrt(60) 0.129099	1/sqrt[2n(n+1)]
Volume	1	3/16 0.1875	1/72 0.013889	5/9216 0.00054253	1/76800 0.000013021	(n+1)/[2ⁿ (n!)²]
Surface	4	sqrt(27)/2 2.598076	1/sqrt(6) 0.408248	5 sqrt(10)/576 0.027450	1/[256 sqrt(15)] 0.0010086	sqrt[(n+1)³/(n 2^2n-3 ((n-1)!)⁴)]
Dihedral angles at S_n-1×S_n-1	90°	90°	90°	90°	90°	90°
Dihedral angles at S_n×S_n-2	90°	60°	arccos(1/3) 70.528779	arccos(1/4) 75.522488	arccos(1/5) 78.463041	arccos(1/n)

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Movatterモバイル変換

Analogs

Symmetry An

Regular Simplex Sn (up)

Rectified Simplex rSn (up)

Facetorectified Simplex frSn (up)

Birectified Simplex brSn (up)

Truncated Simplex tSn (up)

Bitruncated Simplex btSn (up)

Mid-rectified Simplex mrSn (up)

Mid-truncated Simplex mtSn (up)

Maximal Expanded Simplex eSn (up)

Retroexpanded Simplex reSn (up)

Omnitruncated Simplex otSn (up)

Symmetry BCn

Regular Orthoplex On (up)

Regular Hypercube Cn (up)

Rectified Orthoplex rOn (up)

Rectified Hypercube rCn (up)

Facetorectified Hypercube frCn (up)

Birectified Orthoplex brOn (up)

Birectified Hypercube brCn (up)

Truncated Orthoplex tOn (up)

Truncated Hypercube tCn (up)

Quasitruncated Hypercube qtCn (up)

Bitruncated Hypercube btCn (up)

Rhombated Hypercube rbCn (up)

Quasihombateded Hypercube qrbCn (up)

Maximal Expanded Hypercube eCn (up)

Quasiexpanded Hypercube qeCn (up)

Retroexpanded Hypercube reCn (up)

Quasiretroexpanded Hypercube qreCn (up)

Omnitruncated Hypercube otCn (up)

Symmetry Dn

Demihypercube Dn (up)

Demicross hOn (up)

Truncated Demihypercube tDn (up)

Maximal Expanded Demihypercube eDn (up)

Symmetry En

Gossetic n2,1 (up)

Gossetic 2n,1 (up)

Gossetic 1n,2 (up)

RectifiedGossetic r(n2,1) (up)

RectifiedGossetic r(2n,1) (up)

RectifiedGossetic r(1n,2) (up)

TruncatedGossetic t(n2,1) (up)

TruncatedGossetic t(2n,1) (up)

TruncatedGossetic t(1n,2) (up)

All-Ends ExpandedGossetic eEn (up)

OmnitruncatedGossetic otEn (up)