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As a foregoing consideration one might ask whether euclidean space curvature also does allow for non-simplicial reflection groups, such as thehyperbolic one does. In fact, one should state, that this is the case here indeed.But, on the other hand, that extraordinary regime of varities breaks down in here to one rather tame effect: For D=2 we just can have a tetragonal fundamental domain, with all angles being α=π/2, i.e. a rectangle. In Weyl notation that "extraordinary" symmetry is nothing but A1×A1.
For higher dimensions this same effect generalizes to cartesianhoneycomb products of lower dimensional euclidean tesselations, where the fundamental domains derive likewise by orthogonalprism products of those of the factor tesselations. In terms of symmetries this generally is called areducible (afine) symmetry.In Weyl group description those are marked by an ×.
Just as in hyperbolic geometry, non-simplicial reflection groups in here also provide some additional degrees of freedom:a reducible symmetry with n factors, allows for n-1 independent relative scalings of the orthogonal factor domains. (Sure, the absolut size of the first factor in euclidean space is not restricted either.)
Euclidean tesselations are closely related to cell complexes based onlattices. Best known is theVoronoi complex. This is an isohedral (1 single cell class) tesselation of space. Each cell is the closure of all points nearer to anyspecific lattice point than to any other.
The locally dual complex is theDelone complex. The vertices thus are the lattice points again. Two lattice points are joined by an edge iff the correspondingVoronoi cells are adjoined at an facet, etc.There might be several different types of Delone cells, corresponding to the symmetry inequivalent types of vertices ofthe Voronoi cell. Both these complexes by definition use convex cells only.
Beside to the (unmarked) Dynkin symbols of the symmetry groups the corresponding representations asWeyl groups and asCoxeter groups are given.
Finally, besides of (marked) Dynkin symbols and Bowers acronyms for individual tesselationsthe Olshevsky numbers ("O...") are provided as well (up to dimension 4). Those refer onto his paper onUniform Panoploid Tetracombs, assembled since 2003, made availablein 2006.)
---- 1DSequence (up)----
| o∞o = A1 = W(2) | o∞'o |
x∞o -azex∞x -aze | x∞'o -azex∞'x - ∞-covered edge |
A1, taken aslattice, defines the intervals inbetween as Delone cells, while the Voronoi cells are those intervalsshifted by half a unit length. Both complexes are equivalent (relatively shifted) to the apeirogon.
Generally {n/d} and {(n/d)'} = {n/(n-d)} denote pairs of pro- and retrograde polygons. That is, when considered without context,their geometric realization is the same. Even though, as abstract polytopes they have to be distinguished, esp. when usedas link marks within Dynkin diagrams.–It then is only by transition to infinite values of this quotient of n/d, that {n/d} =x-N/D-o and {2n/d} =x-N/D-x become geometrically identical. But this does not hold true for {n/(n-d)} =x-N/(N-D)-o, were the corresponding limit of the mark value becomes 1(the retrograde aze), and {2n/(n-d)} =x-2N/(N-D)-o, were the limit of that mark value becomes 2 (the ∞-covered edge).These rather behave like a folding rule, in its unfolded resp. its folded state. That is,x-∞-x is a well-behaved polytope,whereasx-∞'-x rather would be some highly degenerateGrünbaumian.
---- 2DTilings (up)----
| o3o6o = G2 = V(3) | o4o4o = C2 = R(3) | o3o3o3*a = A2 = P(3) | o∞o o∞o = A1×A1 = W(2)2 |
x3o6o -trat - O2o3x6o -that - O5o3o6x -hexat - O3x3x6o -hexat - O3x3o6x -srothat - O8o3x6x -toxat - O7x3x6x -grothat - O9 | x4o4o -squat - O1o4x4o -squat - O1x4x4o -tosquat - O6x4o4x -squat - O1x4x4x -tosquat - O6 | x3o3o3*a -trat - O2x3x3o3*a -that - O5x3x3x3*a -hexat - O3 | x∞o x∞o -squat - O1x∞x x∞o -squat - O1x∞x x∞x -squat - O1 |
| snubs of o3o6o | snubs of o4o4o | snubs of o3o3o3*a | snubs of o∞o o∞o |
s6o3o -trat - O2s6x3o -that - O5s6o3x -that - O5s6x3x -hexat - O3s3s6o -trat - O2s3s6x -srothat - O8s3s6s -snathat - O11β6β3o -2that+∞{3}β6β3x -2srothatβ3x6o -2thatβ3o6x -shothatβ3x6x -2toxatx3β6x -2srothat | s4o4o -squat - O1s4x4o -squat - O1s4o4x -tosquat - O6s4x4x -tosquat - O6o4s4o -squat - O1x4s4o -squat - O1x4s4x -squat - O1s4s4o -snasquat - O10s4s4x -squat - O1s4o4s -squat - O1s4x4s -squat - O1s4s4s -snasquat - O10s4o4s' -snasquat - O10s4x4s' -squat - O1ss'4o4x -squat - O1 | s3s3s3*a -trat - O2 | x∞s2s∞o -hexat - O3 |
| other (convex) uniforms | |||
elong( x3o6o ) -etrat - O4x∞o x -azipx∞x x -azips∞o2s -azaps∞s2s -azap | |||
Aslattice A2 and G2 are equivalent. The Voronoi cell is the hexagon, the Delone cell is the regular triangle.The Voronoi complex ishexat, the Delonai complex istrat.
The lattice C2 clearly has squares for Voronoi and Delone cells. Both complexes are relatively shifted representants ofsquat.
The lattice A2 (or the vertex set oftrat) can be represented by the Eisenstein integersE =Z[ω] = {e0 + e1ω | e0,e1∈Z}, ω = (-1+sqrt(-3))/2.
Similarily the lattice C2 (or the vertex set ofsquat) is represented by the Gaussian integersG =Z[i] = {g0 + g1i | g0,g1∈Z}, i = sqrt(-1).
(A nice applet for experimental tiling of the 2D euclidean plane (as well as hyperbolic ones) istyler.)
Beyond the 3 regular tesselations and the 8 semi-regular ones there arenon-convex euclidean tilings too. Those will addfor additional possibilities the usage of
But a corresponding research, if desired to be meaningful, would ask for some special care in general. Two reasons are provided in what follows.
Although euclidean tilings, in contrast to spherical space tesselations (polyhedra), allow for an infinitude of vertices,we still restrict to a globaldiscreteness of the vertex set. Dense modules will be rejected. Thus the restriction derived from curvature, e.g. for linear diagrams (P-2)(Q-2) = 4, which for a rational P=n/d would resolve toQ=2n/(n-2d), is not enough for that purpose: for instance the reflection groupso5o10/3o,o5/2o10o,o7o14/5o,o7/2o14/3o,o7/3o14o, etc. all would bow to this curvature condition. But those all would produce dense modules! These therefore are to be omitted from further consideration.
Not even a mere investigation of possible vertex configurations here is sufficient (although the respective tilings will be enlisted by them).Because in general thoselocal configurations either would not allow for an infinitely extending, uniform,non-Grünbaumian tiling at all, or they would result (globally) again in a dense vertex set.
a) Consideration by additionally allowed symmetry groups
| o3/2o3/2o3*a | o4o4/3o | o4/3o4/3o |
|---|---|---|
x3/2o3/2o3*a -trat °o3/2x3/2o3*a -trat °x3/2x3/2o3*a - (∞-covered {3})x3/2o3/2x3*a -that °x3/2x3/2x3*a - [Grünbaumian] | x4o4/3o -squat °o4x4/3o -squat °o4o4/3x -squat °x4x4/3o -tosquat °x4o4/3x - (∞-covered {4})o4x4/3x -quitsquatx4x4/3x -qrasquit | x4/3o4/3o -squat °o4/3x4/3o -squat °x4/3x4/3o -quitsquatx4/3o4/3x -squat °x4/3x4/3x -quitsquats4/3s4/3s -rasisquat |
| o3/2o6o | o3o6/5o | o3/2o6/5o |
x3/2o6o -trat °o3/2x6o -that °o3/2o6x -hexat °x3/2x6o - [Grünbaumian]x3/2o6x -qrothato3/2x6x -toxat °x3/2x6x - [Grünbaumian] | x3o6/5o -trat °o3x6/5o -that °o3o6/5x -hexat °x3x6/5o -hexat °x3o6/5x -qrothato3x6/5x -quothatx3x6/5x -quitothit | x3/2o6/5o -trat °o3/2x6/5o -that °o3/2o6/5x -hexat °x3/2x6/5o - [Grünbaumian]x3/2o6/5x -srothat °o3/2x6/5x -quothatx3/2x6/5x - [Grünbaumian] |
| o3/2o6o6*a | o3o6o6/5*a | o3/2o6/5o6/5*a |
x3/2o6o6*a -chatito3/2o6x6*a -2hexatx3/2x6o6*a - [Grünbaumian]x3/2o6x6*a -shothatx3/2x6x6*a - [Grünbaumian] | x3o6o6/5*a -chatito3x6o6/5*a -chatito3o6x6/5*a -2hexatx3x6o6/5*a - (∞-covered {6})x3o6x6/5*a -ghothato3x6x6/5*a -shothatx3x6x6/5*a -thotithit | x3/2o6/5o6/5*a -chatito3/2o6/5x6/5*a -2hexatx3/2x6/5o6/5*a - [Grünbaumian]x3/2o6/5x6/5*a -ghothatx3/2x6/5x6/5*a - [Grünbaumian] |
| o3o3/2o∞*a | o3o3o∞'*a | o3/2o3/2o∞'*a |
x3o3/2o∞*a -ditathao3o3/2x∞*a -ditathax3x3/2o∞*a -chatax3o3/2x∞*a -thao3x3/2x∞*a - [Grünbaumian]x3x3/2x∞*a - [Grünbaumian] | x3o3o∞'*a -ditathax3x3o∞'*a -chatax3o3x∞'*a - (contains ∞-covered edge)x3x3x∞'*a - (contains ∞-covered edge) | x3/2o3/2o∞'*a -ditathax3/2x3/2o∞'*a - [Grünbaumian]x3/2o3/2x∞'*a - (contains ∞-covered edge)x3/2x3/2x∞'*a - [Grünbaumian] |
| o4o4/3o∞*a | o4o4o∞'*a | o4/3o4/3o∞'*a |
x4o4/3o∞*a -cosao4o4/3x∞*a -cosax4x4/3o∞*a -gossax4o4/3x∞*a -shao4x4/3x∞*a -sossax4x4/3x∞*a -satsas4s4/3s∞*a - snassa | x4o4o∞'*a -cosax4x4o∞'*a -gossax4o4x∞'*a - (contains ∞-covered edge)x4x4x∞'*a - (contains ∞-covered edge) | x4/3o4/3o∞'*a -cosax4/3x4/3o∞'*a -sossax4/3o4/3x∞'*a - (contains ∞-covered edge)x4/3x4/3x∞'*a - (contains ∞-covered edge) |
| o6o6/5o∞*a | o6o6o∞'*a | o6/5o6/5o∞'*a |
x6o6/5o∞*a -chao6o6/5x∞*a -chax6x6/5o∞*a -ghahax6o6/5x∞*a -2hohao6x6/5x∞*a -shahax6x6/5x∞*a -hatha | x6o6o∞'*a -chax6x6o∞'*a -ghahax6o6x∞'*a - (contains ∞-covered edge)x6x6x∞'*a - (contains ∞-covered edge) | x6/5o6/5o∞'*a -chax6/5x6/5o∞'*a -shahax6/5o6/5x∞'*a - (contains ∞-covered edge)x6/5x6/5x∞'*a - (contains ∞-covered edge) |
| o o∞o ¹ | other | |
x x∞o -azip °x x∞x -azip °s s∞o -azap °s s∞s -azap ° | sost (non-orientable member ofsossa regiment)rassersa (own regiment, asossa superregiment)rarsisresa (inrassersa regiment)rosassa (inrassersa subregiment)rorisassa (inrosassa regiment)sraht (non-orientable member ofsrothat-regimentgraht (non-orientable member ofghothat-regimenthuht (non-orientable member ofshaha-regiment)retrat (retro-elongatedtrat) | |
° : Those such marked tilings here come out again to be convex after all.
' (in the context of P') : Refering to the inversion between prograde and retrograde polygons {n/d} ↔ {n/(n-d)}. (Note that as abstract polytopes {∞} and {∞'} are well to be distinguished, and thus esp. their usage as link marks within Dynkin diagrams, but that their geometric realizations within a euclidean space context are exactly the sameaze.)
¹ : Those can be considered slab cases, i.e. prisms or prism-like structures between base 1D tilings. However, when looking at those base objects in the sense ofhoroscycles instead, i.e. including some body "beyond" the base,this 2D tiling all the same fills all space.Note moreover that within these slab cases there is a finite polytopal factor, therefore the according naturalproduct will be the prism product,while for cases, where it is not, one rather considers the comb product as natural instead.
b) Consideration by additionally allowed vertex configurations
| using{∞} * | using{n/d} | using both |
|---|---|---|
[42,∞] - x x∞o,azip (∞-prism, convex)[33,∞] - s2s∞o,azap (∞-antipr., convex)[(3,∞)3]/2 - x3o3/2o∞*a,ditatha (intrat-reg.) | [3/2,12/5,12/5] - o3x6/5x,quothat (intoxat-army, reg.11)[4,8/5,8/5] - o4x4/3x,quitsquat (intosquat-army, reg.10)[4,6/5,12/5] - x3x6/5x,quitothit (ingrothat-army, reg.8)[4,8/3,8/7] - x4x4/3x,qrasquit (intosquat-army, reg.7)[6,12/5,12/11] - x3x6x6/5*a,thotithit (intoxat-subarmy, reg.9)[3/2,4,6/5,4] - x3/2o6x,qrothat (insrothat-army, reg.4)[3,12/5,6/5,12/5] - x3o6x6/5*a,ghothat (insrothat-army, reg.4)[3/2,12,6,12] - x3/2o6x6*a,shothat (insrothat-regiment,1)[4,12,4/3,12/11]/0 -sraht (insrothat-regiment,1)[4,12/5,4/3,12/7]/0 -graht (insrothat-army, reg.4)[8/3,8,8/5,8/7]/0 -sost (intosquat-army, reg.2)[12/5,12,12/7,12/11]/0 -huht (intoxat-army, reg.3)[3,3,3,4/3,4/3] -retrat (inetrat-superreg.,12)[3,3,4/3,3,4/3] - s4/3s4/3s,rasisquat (insnasquat-army, reg.13) | [8,8/3,∞] - x4x4/3x∞*a,satsa (intosquat-subarmy, reg.5)[12,12/5,∞] - x6x6/5x∞*a,hatha (intoxat-subarmy, reg.6)[3,∞,3/2,∞]/0 - x3o3/2x∞*a,tha (inthat-regiment)[4,∞,4/3,∞]/0 - x4o4/3x∞*a,sha (insquat-regiment)[6,∞,6/5,∞]/0 -hoha (or x6o6/5x∞*a,2hoha) (inthat-regiment)[8,4/3,8,∞] - x4x4/3o∞*a,gossa (intosquat-army, reg.2)[8/3,4,8/3,∞] - o4x4/3x∞*a,sossa (intosquat-army, reg.2)[12,6/5,12,∞] - x6x6/5o∞*a,ghaha (intoxat-army, reg.3)[12/5,6,12/5,∞] - o6x6/5x∞*a,shaha (intoxat-army, reg.3)[3,4,3,4/3,3,∞] - snassa (insnasquat-army, reg.14)[4,8,8/3,4/3,∞] -rorisassa (intosquat-army, inrassersa-subreg.†)[4,8/3,8,4/3,∞] -rosassa (intosquat-army, inrassersa-subreg.†)[4,8,4/3,8,4/3,∞] -rarsisresa (insossa-superreg.†)[4,8/3,4,8/3,4/3,∞] -rassersa (insossa-superreg.†) |
* Note, that some authors would count in the "tiling by 2azes",but there the 2 tiles would connect at more than a single edge, and therefore in the enlisting is not shown.
† Those were found by H. Sakamoto, cf. hiswebsite.What he calls "complex" has nothing to do with the complex numbers, rather it means "complicated". As such it refers to tilings of the euclidean plane, which are non-Wythoffian.
The provided regiment numbers (for regiments, which occur exclusively within this section) correspond to thelisting by J. McNeill.(The corresponding individual number-links would point to his respective page.)
All the above is about at leastuniform tilings, which esp. have just a single vertex type. However there will be numerous further tilings (also with regular faces only), which not become dense, but would use more than just a single vertex type.A nice such non-uniform example might be provided by the right sketch:
---- 3DHoneycombs (up)----
| o4o3o4o = C3 = R(4) | o3o3o *b4o = B3 = S(4) | o3o3o3o3*a = A3 = P(4) | o∞o o3o6o = A1×G2 = W(2)×V(3) |
x4o3o4o -chon - O1o4x3o4o -rich - O15x4x3o4o -tich - O14x4o3x4o -srich - O17x4o3o4x -chon - O1o4x3x4o -batch - O16x4x3x4o -grich - O18x4x3o4x -prich - O19x4x3x4x -gippich - O20s4o3o4o -octet - O21s4x3o4o -rich - O15s4o3x4o -tatoh - O25s4o3o4x -sratoh - O26s4o3x4x -gratoh - O28s4x3x4o -batch - O16s4x3o4x -srich - O17s4x3x4x -grich - O18s4o3o4s -octet - O21s4o3o4s' -cytatoh - O27s4x3o4s -rusch - **s4x3x4s -gabreth - **o4s3s4o -bisch - **x4s3s4o -casch - **x4s3s4x -cabisch - **s4s3s4o -serch - **s4s3s4x -esch - **s4s3s4s -snich - ** | x3o3o *b4o -octet - O21o3x3o *b4o -rich - O15o3o3o *b4x -chon - O1x3x3o *b4o -tatoh - O25x3o3x *b4o -rich - O15x3o3o *b4x -sratoh - O26o3x3o *b4x -tich - O14x3x3x *b4o -batch - O16x3x3o *b4x -gratoh - O28x3o3x *b4x -srich - O17x3x3x *b4x -grich - O18o3o3o *b4s -octet - O21x3o3o *b4s -cytatoh - O27x3x3o *b4s -tatoh - O25s3s3s *b4o -bisch - **s3s3s *b4x -casch - **s3s3s *b4s -serch - ** | x3o3o3o3*a -octet - O21x3x3o3o3*a -cytatoh - O27x3o3x3o3*a -rich - O15x3x3x3o3*a -tatoh - O25x3x3x3x3*a -batch - O16s3s3s3s3*a -bisch - ** | x∞o x3o6o -tiph - O2x∞o o3x6o -thiph - O5x∞o o3o6x -hiph - O3x∞x x3o6o -tiph - O2x∞x o3x6o -thiph - O5x∞x o3o6x -hiph - O3x∞o x3x6o -hiph - O3x∞o x3o6x -srothaph - O8x∞o o3x6x -thaph - O7x∞x x3x6o -hiph - O3x∞x x3o6x -srothaph - O8x∞x o3x6x -thaph - O7x∞o x3x6x -grothaph - O9x∞x x3x6x -grothaph - O9x∞o s3s6s -snathaph - O11x∞x s3s6s -snathaph - O11s∞o2o3o6s -ditoh - *s∞o2x3o6s -gyrich - °s∞o2o3x6s -(?) - *|s∞o2s3s6o -ditoh - *s∞o2s3s6x -(?) - **s∞o2x3x6s -(?) - *|s∞o2s3s6s -(?) - **s∞x2o3o6s -editoh - *s∞x2x3o6s -gyerich - *s∞x2o3x6s -(?) - *|s∞x2s3s6o -editoh - *s∞x2s3s6s -(?) - ** |
| o∞o o4o4o = A1×C2 = W(2)×R(3) | o∞o o3o3o3*c = A1×A2 = W(2)×P(3) | o∞o o∞o o∞o = A1×A1×A1 = W(2)3 | other uniforms |
x∞o x4o4o -chon - O1x∞o o4x4o -chon - O1x∞x x4o4o -chon - O1x∞x o4x4o -chon - O1x∞o x4x4o -tassiph - O6x∞o x4o4x -chon - O1x∞x x4x4o -tassiph - O6x∞x x4o4x -chon - O1x∞o x4x4x -tassiph - O6x∞x x4x4x -tassiph - O6x∞o s4s4s -sassiph - O10x∞x s4s4s -sassiph - O10s∞o2s4o4o -octet - O21s∞o2o4s4o -octet - O21s∞o2s4o4x -pacratoh - *s∞o2x4s4o -(?) - **s∞o2s4s4o -(?) - **s∞o2s4s4x -(?) - **s∞o2s4x4s -(?) - **s∞o2s4s4s -(?) - **s∞x2s4o4o -pextoh - *s∞x2o4s4o -pextoh - * | x∞o x3o3o3*c -tiph - O2x∞x x3o3o3*c -tiph - O2x∞o x3x3o3*c -thiph - O5x∞x x3x3o3*c -thiph - O5x∞o x3x3x3*c -hiph - O3x∞x x3x3x3*c -hiph - O3x∞o s3s3s3*c -tiph - O2x∞x s3s3s3*c -tiph - O2s∞o2s3s3s3*c -ditoh - * | x∞o x∞o x∞o -chon - O1x∞x x∞o x∞o -chon - O1x∞x x∞x x∞o -chon - O1x∞x x∞x x∞x -chon - O1 | gyro( x3o3o3o3*a ) -gytoh - O22elong( x3o3o3o3*a ) -etoh - O23gyroelong( x3o3o3o3*a ) -gyetoh - O24gyro( x3o3o *b4o ) -gytoh - O22elong( x3o3o *b4o ) -etoh - O23gyroelong( x3o3o *b4o ) -gyetoh - O24gyro( x∞o x3o6o ) -gytoph - O12elong( x∞o x3o6o ) -etoph - O4gyroelong( x∞o x3o6o ) -gyetaph - O13x∞o elong( x3o6o ) -etoph - O4x∞x elong( x3o6o ) -etoph - O4gyro( x∞o x3o3o3*c ) -gytoph - O12elong( x∞o x3o3o3*c ) -etoph - O4gyroelong( x∞o x3o3o3*c ) -gyetaph - O13x∞o elong( x3o3o3*c ) -etoph - O4x∞x elong( x3o3o3*c ) -etoph - O4x∞o x-n-o -n-azedipx∞x x-n-o -n-azedipx∞x s-n-s -n-azedip |
* Thesealternated facetings well can be varied to get all equal edge lengths, but still will not become uniform, because they incorporateJohnson solids. They not even will becomescaliform then, because some of the used cells are notorbiform. Hence those count at most asCRFhoneycombs. – This is why those do not have Olshevsky numbers either.
*| Those still can be varied to get all equal edge lengths, but then would yield cells with coplanar adjoining faces.Thence those even disqualify to be CRF.
** Those not even are relaxable to unit edges only. They simply remain to be merealternated facetings,or, when re-scaled, some stillisogonal only variant thereof.
° Those on the other hand would become truescaliforms.
In 3D there are several differentlattices, known as theBravais lattices of crystallography. Only the higher symmetrical ones are related to uniform tesselations. As lattices A1×G2 and A1×A2 again are equivalent, the Voronoi complex of it ishiph, the Delone complex istiph.
In the cubical area there are 3 different lattices. First there is the primitive cubical one, C3. Its Voronoi complex and its Delone complex both are relatively shiftedchon.
Next there is the body-centered cubical (bcc) lattice. Thus it is the union of the primitive cubical lattice plus its Voronoi cell vertices ("holes"). Alternatively it also is described by A3* or equivalently by D3*.The Voronoi complex here isbatch. The Delone complex would be a squashed variant ofoctet.
Finally there is the face-centered cubical (fcc) lattice, as lattice equivalently described as A3 or D3, and likewise described as mod 2 of the sum of the vertex coordinates of a (smaller) primitive cubical lattice. Its Voronoi complex isradh and the Voronoi cell the rhombic dodecahedron (rad). The Delone complex isoctet, with 2 different Delone cells,oct corresponding to the 4-fold vertices ofrad ("deep holes"), whiletet corresponds to the 3-fold vertices ofrad ("shallow holes").
| In 2005 J. McNeill did a research onelementary honeycombs (cf. thiscross-link to his website), that is (non-uniform)scaliform honeycombs usingelementary solids only.Here elementary in turn means: regular faced solids, which are not subdivisable. He then listed 13 such honeycombs (plus some more 4 non-elementary ones). But in fact, 4 of the vertex surroundings, he is displaying, do further split into 3 different modes each (i.e. into a 2-periodic alternating one, resp. into right- or left-handed, larger periodic helical ones). Thus the count increases rather to 21 scaliform elementary honeycombs. The therein being used solids are: Pn - n-gonalprismQn - n-gonalcupola (Q3 isnot elementary)Sn - n-gonalantiprism (S3 isnot elementary)T - tetrahedronT3 - truncated tetrahedron (T3 isnot elementary)Yn - n-gonalpyramid (As he uses in their original namings the attributesortho resp.gyro heavily, but most often in a different sense as those occur in theJohnson solids, I try to avoid those terms then completely.) The non-uniformpartial Stott expansion examples have no elementary counterparts, as thecoes are used in an axial surrounding, which selects their 4-fold symmetry (not their 3-fold one). In 2020 a purelyscaliform (though not elementary) honeycomb was found, which also belongs to an octet subregiment,5Y4-3T-3Q3-T3. |
| Those can be consideredslab cases, i.e. prisms or prism-like structures between base 2D tilings. However, when looking at those base objects in the sense ofhorospheres instead, i.e. including some body "beyond" the base,this 3D honeycomb all the same fills all space.Note moreover that within these slab cases there is a finite polytopal factor, therefore the according naturalproduct will be the prism product,while for cases, where it is not, one rather considers the comb product as natural instead. For sure, instead of considering a single stratos / slab / freeze only, those surely could be stacked infinitely instead,thereby thenblending out the base tilings.However then they would come back to already above described cases. |
:oxxx:3:xxox:3:xoxo:3*a&##xt -rigytoh... | Non-uniform convex honeycombs using axialJohnson solids for building blocks easily can be derived from infinite periodic stacks of 2D tilings, provided those use the same axial symmetry and have the same periodicity scaling in each (independent) around-symmetrical direction. The (infinite) segments then are filled from accordinglace prisms.As these are piled up to linear columns each, those can be considered to become multistraticblends, as long as those still remain convex. |
Beyond the convex honeycombs there arenon-convex euclidean honeycombs too. Those will addfor additional possibilities the usage of
(Several ones can be seen to beGrünbaumian a priori right from having both nodes being markedat links with even denominator. The same holds true forx∞'x, which by itself would be nothing but an infinitely many covered edge. All these then are represented below accordingly. Others might be recognised to be Grünbaumian a posteriory, e.g.when using such polytopes for cells or as vertex figure. These then are mentioned by referenciation to at least one of those elements.)
_o_ _P- | 3/2 o< 3 >o -P_ | _3- -o- | _o_ _3- | -P_ o< 3/2 >o 3/2 | _P- -o- | ||
| o3o3o3o3/2*a3*c | o3/2o3/2o3o3/2*a3*c | o3o3/2o3o3*a3/2*c | o3o3/2o3/2o3/2*a3/2*c |
|---|---|---|---|
x3o3o3o3/2*a3*c - (contains2tet)o3x3o3o3/2*a3*c -tehtrah (old: datha)o3o3x3o3/2*a3*c - (contains2tet)o3o3o3x3/2*a3*c - (contains2tet)x3x3o3o3/2*a3*c - (contains2tet)x3o3x3o3/2*a3*c -ridathax3o3o3x3/2*a3*c - [Grünbaumian]o3x3x3o3/2*a3*c - (contains2tet)o3x3o3x3/2*a3*c - (contains2tet)o3o3x3x3/2*a3*c - tetatit tritrigonary apeiratitritetratic HCx3x3x3o3/2*a3*c - tutehtrah (old: tedatha) truncated tetrahemitriangular HC (truncated ditetratihemiapeiratic HC)x3x3o3x3/2*a3*c - [Grünbaumian]x3o3x3x3/2*a3*c - [Grünbaumian]o3x3x3x3/2*a3*c - (contains2thah)x3x3x3x3/2*a3*c - [Grünbaumian] | x3/2o3/2o3o3/2*a3*c - (contains2tet)o3/2x3/2o3o3/2*a3*c -tehtraho3/2o3/2x3o3/2*a3*c - (contains2tet)o3/2o3/2o3x3/2*a3*c - (contains2tet)x3/2x3/2o3o3/2*a3*c - [Grünbaumian]x3/2o3/2x3o3/2*a3*c -ridathax3/2o3/2o3x3/2*a3*c - [Grünbaumian]o3/2x3/2x3o3/2*a3*c - [Grünbaumian]o3/2x3/2o3x3/2*a3*c - (contains2tet)o3/2o3/2x3x3/2*a3*c - tetatitx3/2x3/2x3o3/2*a3*c - [Grünbaumian]x3/2x3/2o3x3/2*a3*c - [Grünbaumian]x3/2o3/2x3x3/2*a3*c - (contains2tut)o3/2x3/2x3x3/2*a3*c - [Grünbaumian]x3/2x3/2x3x3/2*a3*c - [Grünbaumian] | x3o3/2o3o3*a3/2*c - (contains2tet)o3x3/2o3o3*a3/2*c - (has2oct-verf)o3o3/2x3o3*a3/2*c - (contains2tet)o3o3/2o3x3*a3/2*c - (contains2tet)x3x3/2o3o3*a3/2*c - (contains2tet)x3o3/2x3o3*a3/2*c - [Grünbaumian]x3o3/2o3x3*a3/2*c - (contains ∞-covered {3})o3x3/2x3o3*a3/2*c - [Grünbaumian]o3x3/2o3x3*a3/2*c - (contains2tet)o3o3/2x3x3*a3/2*c - tetatitx3x3/2x3o3*a3/2*c - [Grünbaumian]x3x3/2o3x3*a3/2*c - (contains2thah)x3o3/2x3x3*a3/2*c - [Grünbaumian]o3x3/2x3x3*a3/2*c - [Grünbaumian]x3x3/2x3x3*a3/2*c - [Grünbaumian] | x3o3/2o3/2o3/2*a3/2*c - (contains2tet)o3x3/2o3/2o3/2*a3/2*c - (has2oct-verf)o3o3/2x3/2o3/2*a3/2*c - (contains2tet)o3o3/2o3/2x3/2*a3/2*c - (contains2tet)x3x3/2o3/2o3/2*a3/2*c - (contains2tet)x3o3/2x3/2o3/2*a3/2*c - [Grünbaumian]x3o3/2o3/2x3/2*a3/2*c - (contains2oct)o3x3/2x3/2o3/2*a3/2*c - [Grünbaumian]o3x3/2o3/2x3/2*a3/2*c - (contains2tet)o3o3/2x3/2x3/2*a3/2*c - [Grünbaumian]x3x3/2x3/2o3/2*a3/2*c - [Grünbaumian]x3x3/2o3/2x3/2*a3/2*c - [Grünbaumian]x3o3/2x3/2x3/2*a3/2*c - [Grünbaumian]o3x3/2x3/2x3/2*a3/2*c - [Grünbaumian]x3x3/2x3/2x3/2*a3/2*c - [Grünbaumian] |
o---3---o | | 3/2 P' | | o---P---o | o--3/2--o | | 3/2 3/2 | | o--3/2--o | _o _3- | o-3-o< ∞ 3/2 | -o | |
| o3o3o3/2o3/2*a | o3o3/2o3o3/2*a | o3/2o3/2o3/2o3/2*a | o3o3o∞o3/2*b |
x3o3o3/2o3/2*a -octet °o3x3o3/2o3/2*a -octet °o3o3o3/2x3/2*a -octet °x3x3o3/2o3/2*a -cytatoh °x3o3x3/2o3/2*a -rich °x3o3o3/2x3/2*a - [Grünbaumian]o3x3o3/2x3/2*a - (contains2thah)x3x3x3/2o3/2*a -tatoh °x3x3o3/2x3/2*a - [Grünbaumian]x3o3x3/2x3/2*a - [Grünbaumian]x3x3x3/2x3/2*a - [Grünbaumian] | x3o3/2o3o3/2*a - x3x3/2o3o3/2*a - x3o3/2x3o3/2*a - x3o3/2o3x3/2*a - [Grünbaumian]x3x3/2x3o3/2*a - [Grünbaumian]x3x3/2x3x3/2*a - [Grünbaumian] | x3/2o3/2o3/2o3/2*a - x3/2x3/2o3/2o3/2*a - [Grünbaumian]x3/2o3/2x3/2o3/2*a - x3/2x3/2x3/2o3/2*a - [Grünbaumian]x3/2x3/2x3/2x3/2*a - [Grünbaumian] | o3o3x∞o3/2*b - o3o3o∞x3/2*b -dattehax3o3x∞o3/2*b - x3o3o∞x3/2*b - (contains2thah)o3x3x∞o3/2*b - o3x3o∞x3/2*b - [Grünbaumian]o3o3x∞x3/2*b - x3x3x∞o3/2*b - x3x3o∞x3/2*b - [Grünbaumian]x3o3x∞x3/2*b - (contains2thah)o3x3x∞x3/2*b - [Grünbaumian]x3x3x∞x3/2*b - [Grünbaumian] |
_o _3- | o-3/2-o< ∞ 3/2 | -o | _o _P- | o-3-o< ∞' -P_ | -o | _o _3- | o-3/2-o< ∞' -3_ | -o | |
| o3/2o3o∞o3/2*b | o3o3o∞'o3*b | o3o3/2o∞'o3/2*b | o3/2o3o∞'o3*b |
o3/2o3x∞o3/2*b - o3/2o3o∞x3/2*b - x3/2o3x∞o3/2*b - (contains2thah)x3/2o3o∞x3/2*b - o3/2x3x∞o3/2*b - o3/2x3o∞x3/2*b - [Grünbaumian]o3/2o3x∞x3/2*b - x3/2x3x∞o3/2*b - [Grünbaumian]x3/2x3o∞x3/2*b - [Grünbaumian]x3/2o3x∞x3/2*b - (contains2thah)o3/2x3x∞x3/2*b - [Grünbaumian]x3/2x3x∞x3/2*b - [Grünbaumian] | o3o3x∞'o3*b - x3o3x∞'o3*b - o3x3x∞'o3*b - o3o3x∞'x3*b - [Grünbaumian]x3x3x∞'o3*b - x3o3x∞'x3*b - [Grünbaumian]o3x3x∞'x3*b - [Grünbaumian]x3x3x∞'x3*b - [Grünbaumian] | o3o3/2x∞'o3/2*b - x3o3/2x∞'o3/2*b - o3x3/2x∞'o3/2*b - [Grünbaumian]o3o3/2x∞'x3/2*b - [Grünbaumian]x3x3/2x∞'o3/2*b - [Grünbaumian]x3o3/2x∞'x3/2*b - [Grünbaumian]o3x3/2x∞'x3/2*b - [Grünbaumian]x3x3/2x∞'x3/2*b - [Grünbaumian] | o3/2o3x∞'o3*b - x3/2o3x∞'o3*b - o3/2x3x∞'o3*b - o3/2o3x∞'x3*b - [Grünbaumian]x3/2x3x∞'o3*b - [Grünbaumian]x3/2o3x∞'x3*b - [Grünbaumian]o3/2x3x∞'x3*b - [Grünbaumian]x3/2x3x∞'x3*b - [Grünbaumian] |
_o 3/2 | o-3/2-o< ∞' 3/2 | -o | |||
| o3/2o3/2o∞'o3/2*b | |||
o3/2o3/2x∞'o3/2*b - x3/2o3/2x∞'o3/2*b - o3/2x3/2x∞'o3/2*b - [Grünbaumian]o3/2o3/2x∞'x3/2*b - [Grünbaumian]x3/2x3/2x∞'o3/2*b - [Grünbaumian]x3/2o3/2x∞'x3/2*b - [Grünbaumian]o3/2x3/2x∞'x3/2*b - [Grünbaumian]x3/2x3/2x∞'x3/2*b - [Grünbaumian] | |||
o-P-o-3-o-4/3-o | o-4-o-3/2-o-P-o | ||
| o4o3o4/3o | o4/3o3o4/3o | o4o3/2o4o | o4o3/2o4/3o |
x4o3o4/3o -chon °o4x3o4/3o -rich °o4o3x4/3o -rich °o4o3o4/3x -chon °x4x3o4/3o -tich °x4o3x4/3o -srich °x4o3o4/3x -chon °o4x3x4/3o -batch °o4x3o4/3x - querch quasirhombated cubic HCo4o3x4/3x -quitchx4x3x4/3o -grich °x4x3o4/3x -paqrichx4o3x4/3x -quipricho4x3x4/3x -gaqrichx4x3x4/3x -gaquapech | x4/3o3o4/3o -chon °o4/3x3o4/3o -rich °x4/3x3o4/3o -quitchx4/3o3x4/3o - querchx4/3o3o4/3x -chon °o4/3x3x4/3o -batch °x4/3x3x4/3o -gaqrichx4/3x3o4/3x -quipqrichx4/3x3x4/3x -quequapech | x4o3/2o4o - o4x3/2o4o - x4x3/2o4o - x4o3/2x4o - x4o3/2o4x - o4x3/2x4o - [Grünbaumian]x4x3/2x4o - [Grünbaumian]x4x3/2o4x - x4x3/2x4x - [Grünbaumian] | x4o3/2o4/3o - o4x3/2o4/3o - o4o3/2x4/3o - o4o3/2o4/3x - x4x3/2o4/3o - x4o3/2x4/3o - x4o3/2o4/3x - o4x3/2x4/3o - [Grünbaumian]o4x3/2o4/3x - o4o3/2x4/3x - x4x3/2x4/3o - [Grünbaumian]x4x3/2o4/3x - x4o3/2x4/3x - o4x3/2x4/3x - [Grünbaumian]x4x3/2x4/3x - [Grünbaumian] |
o-4/3-o-3/2-o-4/3-o | o_ -3_ >o-4/3-o _3- o- | o_ -3_ >o-P-o 3/2 o- | |
| o4/3o3/2o4/3o | o3o3o *b4/3o | o3o3/2o *b4o | o3o3/2o *b4/3o |
x4/3o3/2o4/3o - o4/3x3/2o4/3o - x4/3x3/2o4/3o - x4/3o3/2x4/3o - x4/3o3/2o4/3x - o4/3x3/2x4/3o - [Grünbaumian]x4/3x3/2x4/3o - [Grünbaumian]x4/3x3/2o4/3x - x4/3x3/2x4/3x - [Grünbaumian] | x3o3o *b4/3o -octet °o3x3o *b4/3o -rich °o3o3o *b4/3x -chon °x3x3o *b4/3o -tatoh °x3o3x *b4/3o -rich °x3o3o *b4/3x - qrahch quasirhombic demicubic HCo3x3o *b4/3x -quitchx3x3x *b4/3o -batch °x3x3o *b4/3x - gaqrahch great quasirhombic demicubic HCx3o3x *b4/3x - querchx3x3x *b4/3x -gaqrich | x3o3/2o *b4o - o3x3/2o *b4o - o3o3/2x *b4o - o3o3/2o *b4x - x3x3/2o *b4o - x3o3/2x *b4o - x3o3/2o *b4x - o3x3/2x *b4o - [Grünbaumian]o3x3/2o *b4x - o3o3/2x *b4x - x3x3/2x *b4o - [Grünbaumian]x3x3/2o *b4x - x3o3/2x *b4x - o3x3/2x *b4x - [Grünbaumian]x3x3/2x *b4x - [Grünbaumian] | x3o3/2o *b4/3o - o3x3/2o *b4/3o - o3o3/2x *b4/3o - o3o3/2o *b4/3x - x3x3/2o *b4/3o - x3o3/2x *b4/3o - x3o3/2o *b4/3x - o3x3/2x *b4/3o - [Grünbaumian]o3x3/2o *b4/3x - o3o3/2x *b4/3x - x3x3/2x *b4/3o - [Grünbaumian]x3x3/2o *b4/3x - x3o3/2x *b4/3x - o3x3/2x *b4/3x - [Grünbaumian]x3x3/2x *b4/3x - [Grünbaumian] |
o_ 3/2 >o-P-o 3/2 o- | o---P---o | | P 4/3 | | o---4---o | ||
| o3/2o3/2o *b4o | o3/2o3/2o *b4/3o | o3o3o4o4/3*a | o3/2o3/2o4o4/3*a |
x3/2o3/2o *b4o - o3/2x3/2o *b4o - o3/2o3/2o *b4x - x3/2x3/2o *b4o - [Grünbaumian]x3/2o3/2x *b4o - x3/2o3/2o *b4x - o3/2x3/2o *b4x - x3/2x3/2x *b4o - [Grünbaumian]x3/2x3/2o *b4x - [Grünbaumian]x3/2o3/2x *b4x - x3/2x3/2x *b4x - [Grünbaumian] | x3/2o3/2o *b4/3o - o3/2x3/2o *b4/3o - o3/2o3/2o *b4/3x - x3/2x3/2o *b4/3o - [Grünbaumian]x3/2o3/2x *b4/3o - x3/2o3/2o *b4/3x - o3/2x3/2o *b4/3x - x3/2x3/2x *b4/3o - [Grünbaumian]x3/2x3/2o *b4/3x - [Grünbaumian]x3/2o3/2x *b4/3x - x3/2x3/2x *b4/3x - [Grünbaumian] | x3o3o4o4/3*a - o3x3o4o4/3*a - (has ∞-covered-{4} verf)o3o3x4o4/3*a - o3o3o4x4/3*a -2cuhsquahx3x3o4o4/3*a - x3o3x4o4/3*a - (contains ∞-covered {4})x3o3o4x4/3*a -getdocao3x3x4o4/3*a - o3x3o4x4/3*a - o3o3x4x4/3*a - x3x3x4o4/3*a - (contains ∞-covered {4})x3x3o4x4/3*a - x3o3x4x4/3*a - o3x3x4x4/3*a - x3x3x4x4/3*a - | x3/2o3/2o4o4/3*a - o3/2x3/2o4o4/3*a - o3/2o3/2x4o4/3*a - o3/2o3/2o4x4/3*a -2cuhsquahx3/2x3/2o4o4/3*a - [Grünbaumian]x3/2o3/2x4o4/3*a - x3/2o3/2o4x4/3*a - o3/2x3/2x4o4/3*a - [Grünbaumian]o3/2x3/2o4x4/3*a - o3/2o3/2x4x4/3*a - x3/2x3/2x4o4/3*a - [Grünbaumian]x3/2x3/2o4x4/3*a - [Grünbaumian]x3/2o3/2x4x4/3*a - o3/2x3/2x4x4/3*a - [Grünbaumian]x3/2x3/2x4x4/3*a - [Grünbaumian] |
o---3---o | | 3/2 P | | o---P---o | _o _3- | o-4-o< P' -P_ | -o | ||
| o3o3/2o4o4*a | o3o3/2o4/3o4/3*a | o4o3o4o4/3*b | o4o3o4/3o4*b |
x3o3/2o4o4*a - o3x3/2o4o4*a - o3o3/2x4o4*a - o3o3/2o4x4*a -2cuhsquahx3x3/2o4o4*a - x3o3/2x4o4*a - x3o3/2o4x4*a - o3x3/2x4o4*a - [Grünbaumian]o3x3/2o4x4*a - o3o3/2x4x4*a - x3x3/2x4o4*a - [Grünbaumian]x3x3/2o4x4*a - x3o3/2x4x4*a - o3x3/2x4x4*a - [Grünbaumian]x3x3/2x4x4*a - [Grünbaumian] | x3o3/2o4/3o4/3*a - o3x3/2o4/3o4/3*a - o3o3/2x4/3o4/3*a - o3o3/2o4/3x4/3*a -2cuhsquahx3x3/2o4/3o4/3*a - x3o3/2x4/3o4/3*a - x3o3/2o4/3x4/3*a - o3x3/2x4/3o4/3*a - [Grünbaumian]o3x3/2o4/3x4/3*a - o3o3/2x4/3x4/3*a - x3x3/2x4/3o4/3*a - [Grünbaumian]x3x3/2o4/3x4/3*a - x3o3/2x4/3x4/3*a - o3x3/2x4/3x4/3*a - [Grünbaumian]x3x3/2x4/3x4/3*a - [Grünbaumian] | x4o3o4o4/3*b - (hasoct+6{4}-verf)o4x3o4o4/3*b - (containsoct+6{4})o4o3x4o4/3*b - (containsoct+6{4})o4o3o4x4/3*b - (contains2cube)x4x3o4o4/3*b - (containsoct+6{4})x4o3x4o4/3*b - (containsoct+6{4})x4o3o4x4/3*b - (contains2cube)o4x3x4o4/3*b - (contains2cho)o4x3o4x4/3*b -wavicocao4o3x4x4/3*b - scoca small cubaticubatiapeiratic HCx4x3x4o4/3*b - (contains2cho)x4x3o4x4/3*b - gepdica great prismatodiscubatiapeiratic HCx4o3x4x4/3*b - (contains ∞-covered {4})o4x3x4x4/3*b - caquiteca cubatiquasitruncated cubatiapeiratic HCx4x3x4x4/3*b -caquitpica | x4o3o4/3o4*b - (hasoct+6{4}-verf)o4x3o4/3o4*b - (containsoct+6{4})o4o3x4/3o4*b - (containsoct+6{4})o4o3o4/3x4*b - (contains2cube)x4x3o4/3o4*b - (containsoct+6{4})x4o3x4/3o4*b - (containsoct+6{4})x4o3o4/3x4*b - (contains2cube)o4x3x4/3o4*b - (contains2cho)o4x3o4/3x4*b - rawvicoca retrosphenoverted cubaticubatiapeiratic HCo4o3x4/3x4*b -gacocax4x3x4/3o4*b - (contains2cho)x4x3o4/3x4*b - spadica small prismated discubatiapeiratic HCx4o3x4/3x4*b -skivpacocao4x3x4/3x4*b - cuteca cubatitruncated cubatiapeiratic HCx4x3x4/3x4*b -cutpica |
_o 3/2 | o-4-o< P -P_ | -o | _o _3- | o-4/3-o< P' -P_ | -o | ||
| o4o3/2o4o4*b | o4o3/2o4/3o4/3*b | o4/3o3o4o4/3*b | o4/3o3o4/3o4*b |
x4o3/2o4o4*b - o4x3/2o4o4*b - o4o3/2x4o4*b - o4o3/2o4x4*b - x4x3/2o4o4*b - x4o3/2x4o4*b - x4o3/2o4x4*b - o4x3/2x4o4*b - [Grünbaumian]o4x3/2o4x4*b - o4o3/2x4x4*b - x4x3/2x4o4*b - [Grünbaumian]x4x3/2o4x4*b - x4o3/2x4x4*b - o4x3/2x4x4*b - [Grünbaumian]x4x3/2x4x4*b - [Grünbaumian] | x4o3/2o4/3o4/3*b - o4x3/2o4/3o4/3*b - o4o3/2x4/3o4/3*b - o4o3/2o4/3x4/3*b - x4x3/2o4/3o4/3*b - x4o3/2x4/3o4/3*b - x4o3/2o4/3x4/3*b - o4x3/2x4/3o4/3*b - [Grünbaumian]o4x3/2o4/3x4/3*b - o4o3/2x4/3x4/3*b - x4x3/2x4/3o4/3*b - [Grünbaumian]x4x3/2o4/3x4/3*b - x4o3/2x4/3x4/3*b - o4x3/2x4/3x4/3*b - [Grünbaumian]x4x3/2x4/3x4/3*b - [Grünbaumian] | x4/3o3o4o4/3*b - (hasoct+6{4}-verf)o4/3x3o4o4/3*b - (containsoct+6{4})o4/3o3x4o4/3*b - (containsoct+6{4})o4/3o3o4x4/3*b - (contains2cube)x4/3x3o4o4/3*b - (containsoct+6{4})x4/3o3x4o4/3*b - (containsoct+6{4})x4/3o3o4x4/3*b - (contains2cube)o4/3x3x4o4/3*b - (contains2cho)o4/3x3o4x4/3*b -wavicocao4/3o3x4x4/3*b - scocax4/3x3x4o4/3*b - (contains2cho)x4/3x3o4x4/3*b - gapdica grand prismated discubatiapeiratic HCx4/3o3x4x4/3*b -gikkiv pacocao4/3x3x4x4/3*b - caquitecax4/3x3x4x4/3*b -gacquitpica | x4/3o3o4/3o4*b - (hasoct+6{4}-verf)o4/3x3o4/3o4*b - (containsoct+6{4})o4/3o3x4/3o4*b - (containsoct+6{4})o4/3o3o4/3x4*b - (contains2cube)x4/3x3o4/3o4*b - (containsoct+6{4})x4/3o3x4/3o4*b - (containsoct+6{4})x4/3o3o4/3x4*b - (contains2cube)o4/3x3x4/3o4*b - (contains2cho)o4/3x3o4/3x4*b - rawvicocao4/3o3x4/3x4*b -gacocax4/3x3x4/3o4*b - (contains2cho)x4/3x3o4/3x4*b - mapdica medial prismated discubatiapeiratic HCx4/3o3x4/3x4*b - (contains ∞-covered {4})o4/3x3x4/3x4*b - cutecax4/3x3x4/3x4*b -gactipeca |
_o 3/2 | o-4/3-o< P -P_ | -o | _o _4- | o-P-o< ∞ 4/3 | -o | ||
| o4/3o3/2o4o4*b | o4/3o3/2o4/3o4/3*b | o3o4o∞o4/3*b | o3/2o4o∞o4/3*b |
x4/3o3/2o4o4*b - o4/3x3/2o4o4*b - o4/3o3/2x4o4*b - o4/3o3/2o4x4*b - x4/3x3/2o4o4*b - x4/3o3/2x4o4*b - x4/3o3/2o4x4*b - o4/3x3/2x4o4*b - [Grünbaumian]o4/3x3/2o4x4*b - o4/3o3/2x4x4*b - x4/3x3/2x4o4*b - [Grünbaumian]x4/3x3/2o4x4*b - x4/3o3/2x4x4*b - o4/3x3/2x4x4*b - [Grünbaumian]x4/3x3/2x4x4*b - [Grünbaumian] | x4/3o3/2o4/3o4/3*b - o4/3x3/2o4/3o4/3*b - o4/3o3/2x4/3o4/3*b - o4/3o3/2o4/3x4/3*b - x4/3x3/2o4/3o4/3*b - x4/3o3/2x4/3o4/3*b - x4/3o3/2o4/3x4/3*b - o4/3x3/2x4/3o4/3*b - [Grünbaumian]o4/3x3/2o4/3x4/3*b - o4/3o3/2x4/3x4/3*b - x4/3x3/2x4/3o4/3*b - [Grünbaumian]x4/3x3/2o4/3x4/3*b - x4/3o3/2x4/3x4/3*b - o4/3x3/2x4/3x4/3*b - [Grünbaumian]x4/3x3/2x4/3x4/3*b - [Grünbaumian] | o3o4x∞o4/3*b - o3o4o∞x4/3*b - x3o4x∞o4/3*b - x3o4o∞x4/3*b - o3x4x∞o4/3*b - o3x4o∞x4/3*b - wavicac sphenoverted cubitiapeiraticubatic HCo3o4x∞x4/3*b - x3x4x∞o4/3*b - x3x4o∞x4/3*b - sacpaca small cubatiprismatocubatiapeiratic HCx3o4x∞x4/3*b - o3x4x∞x4/3*b - dacta dicubatitruncated apeiratic HCx3x4x∞x4/3*b - dactapa dicubiatitruncated prismato-apeiratic HC | o3/2o4x∞o4/3*b - o3/2o4o∞x4/3*b - x3/2o4x∞o4/3*b - x3/2o4o∞x4/3*b - o3/2x4x∞o4/3*b - o3/2x4o∞x4/3*b - o3/2o4x∞x4/3*b - x3/2x4x∞o4/3*b - [Grünbaumian]x3/2x4o∞x4/3*b - [Grünbaumian]x3/2o4x∞x4/3*b - o3/2x4x∞x4/3*b - x3/2x4x∞x4/3*b - [Grünbaumian] |
_o _3- | o-P-o< ∞ 3/2 | -o | |||
| o4o3o∞o3/2*b | o4/3o3o∞o3/2*b | ||
x4o3o∞o3/2*b - o4x3o∞o3/2*b - o4o3x∞o3/2*b - o4o3o∞x3/2*b - x4x3o∞o3/2*b - x4o3x∞o3/2*b - x4o3o∞x3/2*b - o4x3x∞o3/2*b - o4x3o∞x3/2*b - [Grünbaumian]o4o3x∞x3/2*b - doha disoctahemiapeiratic HCx4x3x∞o3/2*b - x4x3o∞x3/2*b - [Grünbaumian]x4o3x∞x3/2*b - o4x3x∞x3/2*b - [Grünbaumian]x4x3x∞x3/2*b - [Grünbaumian] | x4/3o3o∞o3/2*b - o4/3x3o∞o3/2*b - o4/3o3x∞o3/2*b - o4/3o3o∞x3/2*b - x4/3x3o∞o3/2*b - x4/3o3x∞o3/2*b - x4/3o3o∞x3/2*b - o4/3x3x∞o3/2*b - o4/3x3o∞x3/2*b - [Grünbaumian]o4/3o3x∞x3/2*b - doha disoctahemiapeiratic HCx4/3x3x∞o3/2*b - x4/3x3o∞x3/2*b - [Grünbaumian]x4/3o3x∞x3/2*b - o4/3x3x∞x3/2*b - [Grünbaumian]x4/3x3x∞x3/2*b - [Grünbaumian] | ||
_o _P- | o-3-o< ∞' -P_ | -o | _o _P- | o-3/2-o< ∞' -P_ | -o | ||
| o3o4o∞'o4*b | o3o4/3o∞'o4/3*b | o3/2o4o∞'o4*b | o3/2o4/3o∞'o4/3*b |
o3o4x∞'o4*b - x3o4x∞'o4*b - o3x4x∞'o4*b - o3o4x∞'x4*b - [Grünbaumian]x3x4x∞'o4*b - x3o4x∞'x4*b - [Grünbaumian]o3x4x∞'x4*b - [Grünbaumian]x3x4x∞'x4*b - [Grünbaumian] | o3o4/3x∞'o4/3*b - x3o4/3x∞'o4/3*b - o3x4/3x∞'o4/3*b - o3o4/3x∞'x4/3*b - [Grünbaumian]x3x4/3x∞'o4/3*b - x3o4/3x∞'x4/3*b - [Grünbaumian]o3x4/3x∞'x4/3*b - [Grünbaumian]x3x4/3x∞'x4/3*b - [Grünbaumian] | o3/2o4x∞'o4*b - x3/2o4x∞'o4*b - o3/2x4x∞'o4*b - o3/2o4x∞'x4*b - [Grünbaumian]x3/2x4x∞'o4*b - [Grünbaumian]x3/2o4x∞'x4*b - [Grünbaumian]o3/2x4x∞'x4*b - [Grünbaumian]x3/2x4x∞'x4*b - [Grünbaumian] | o3/2o4/3x∞'o4/3*b - x3/2o4/3x∞'o4/3*b - o3/2x4/3x∞'o4/3*b - o3/2o4/3x∞'x4/3*b - [Grünbaumian]x3/2x4/3x∞'o4/3*b - [Grünbaumian]x3/2o4/3x∞'x4/3*b - [Grünbaumian]o3/2x4/3x∞'x4/3*b - [Grünbaumian]x3/2x4/3x∞'x4/3*b - [Grünbaumian] |
_o _P- | o-4-o< ∞' -P_ | -o | _o _P- | o-4/3-o< ∞' -P_ | -o | ||
| o4o3o∞'o3*b | o4o3/2o∞'o3/2*b | o4/3o3o∞'o3*b | o4/3o3/2o∞'o3/2*b |
o4o3x∞'o3*b - x4o3x∞'o3*b - o4x3x∞'o3*b - o4o3x∞'x3*b - [Grünbaumian]x4x3x∞'o3*b - x4o3x∞'x3*b - [Grünbaumian]o4x3x∞'x3*b - [Grünbaumian]x4x3x∞'x3*b - [Grünbaumian] | o4o3/2x∞'o3/2*b - x4o3/2x∞'o3/2*b - o4x3/2x∞'o3/2*b - [Grünbaumian]o4o3/2x∞'x3/2*b - [Grünbaumian]x4x3/2x∞'o3/2*b - [Grünbaumian]x4o3/2x∞'x3/2*b - [Grünbaumian]o4x3/2x∞'x3/2*b - [Grünbaumian]x4x3/2x∞'x3/2*b - [Grünbaumian] | o4/3o3x∞'o3*b - x4/3o3x∞'o3*b - o4/3x3x∞'o3*b - o4/3o3x∞'x3*b - [Grünbaumian]x4/3x3x∞'o3*b - [Grünbaumian]x4/3o3x∞'x3*b - [Grünbaumian]o4/3x3x∞'x3*b - [Grünbaumian]x4/3x3x∞'x3*b - [Grünbaumian] | o4/3o3/2x∞'o3/2*b - x4/3o3/2x∞'o3/2*b - o4/3x3/2x∞'o3/2*b - [Grünbaumian]o4/3o3/2x∞'x3/2*b - [Grünbaumian]x4/3x3/2x∞'o3/2*b - [Grünbaumian]x4/3o3/2x∞'x3/2*b - [Grünbaumian]o4/3x3/2x∞'x3/2*b - [Grünbaumian]x4/3x3/2x∞'x3/2*b - [Grünbaumian] |
_o_ _P- | -4_ o< 3 >o -P_ | 4/3 -o- | _o_ _3- | -3_ o< 4/3 >o -4_ | _4- -o- | _o_ 3/2 | 3/2 o< 4/3 >o 4/3 | 4/3 -o- | |
| o3o3o4/3o4*a3*c | o3/2o3/2o4/3o4*a3*c | o3o4o4o3*a4/3*c | o3/2o4/3o4/3o3/2*a4*c |
x3o3o4/3o4*a3*c - [Grünbaumian]o3x3o4/3o4*a3*c - [Grünbaumian]o3o3x4/3o4*a3*c - [Grünbaumian]o3o3o4/3x4*a3*c - [Grünbaumian]x3x3o4/3o4*a3*c - [Grünbaumian]x3o3x4/3o4*a3*c - [Grünbaumian]x3o3o4/3x4*a3*c -getit cadocao3x3x4/3o4*a3*c - [Grünbaumian]o3x3o4/3x4*a3*c - [Grünbaumian]o3o3x4/3x4*a3*c -stut cadocax3x3x4/3o4*a3*c - [Grünbaumian]x3x3o4/3x4*a3*c -gikkivcadachx3o3x4/3x4*a3*c - dichac dicubatihexacubatic HCo3x3x4/3x4*a3*c -skivcadachx3x3x4/3x4*a3*c - dicroch dicubatirhombated cubihexagonal HC | x3/2o3/2o4/3o4*a3*c - [Grünbaumian]o3/2x3/2o4/3o4*a3*c - [Grünbaumian]o3/2o3/2x4/3o4*a3*c - [Grünbaumian]o3/2o3/2o4/3x4*a3*c - [Grünbaumian]x3/2x3/2o4/3o4*a3*c - [Grünbaumian]x3/2o3/2x4/3o4*a3*c - [Grünbaumian]x3/2o3/2o4/3x4*a3*c -getit cadocao3/2x3/2x4/3o4*a3*c - [Grünbaumian]o3/2x3/2o4/3x4*a3*c - [Grünbaumian]o3/2o3/2x4/3x4*a3*c -stut cadocax3/2x3/2x4/3o4*a3*c - [Grünbaumian]x3/2x3/2o4/3x4*a3*c - [Grünbaumian]x3/2o3/2x4/3x4*a3*c - dichac dicubatihexacubatic HCo3/2x3/2x4/3x4*a3*c - [Grünbaumian]x3/2x3/2x4/3x4*a3*c - [Grünbaumian] | x3o4o4o3*a4/3*c - (containsoct+6{4})o3x4o4o3*a4/3*c - (containsoct+6{4})o3o4x4o3*a4/3*c - (contains2cube)x3x4o4o3*a4/3*c - (contains2cho)x3o4x4o3*a4/3*c -gacocao3x4x4o3*a4/3*c - (contains2cube)o3x4o4x3*a4/3*c - (containsoct+6{4})x3x4x4o3*a4/3*c - x3x4o4x3*a4/3*c - (contains2cho)o3x4x4x3*a4/3*c - x3x4x4x3*a4/3*c - | x3/2o4/3o4/3o3/2*a4/3*c - (containsoct+6{4})o3/2x4/3o4/3o3/2*a4/3*c - (containsoct+6{4})o3/2o4/3x4/3o3/2*a4/3*c - (contains2cube)x3/2x4/3o4/3o3/2*a4/3*c - [Grünbaumian]x3/2o4/3x4/3o3/2*a4/3*c -gacocao3/2x4/3x4/3o3/2*a4/3*c - (contains2cube)o3/2x4/3o4/3x3/2*a4/3*c - (containsoct+6{4})x3/2x4/3x4/3o3/2*a4/3*c - x3/2x4/3o4/3x3/2*a4/3*c - [Grünbaumian]o3/2x4/3x4/3x3/2*a4/3*c - x3/2x4/3x4/3x3/2*a4/3*c - [Grünbaumian] |
| Some non-Wythoffians | |||
cuhsquah - withinchon regimenthatiph - withintiph regimentetratip blendretratip blendhexatip blend | |||
° : Those such marked honeycombs come out again to be convex after all.
' (in the context of P') : Refering to the inversion between prograde and retrograde polygons {n/d} ↔ {n/(n-d)}. (Note, that as abstract polytopes {∞} and {∞'} are well to be distinguished, and thus esp. their usage as link marks within Dynkin diagrams, but that their geometric realizations within a euclidean space context are exactly the sameaze.However this changes when having both nodes thereof being ringed:x∞x still isaze, as both alternating edges head into the same direction, whilex∞'x on the other hand becomes highly Grünbaumian,because the alternating edges head into opposite directions.)
---- 4DTetracombs (up)----
| o4o3o3o4o = C4 = R(5) | o3o3o *b3o4o = B4 = S(5) | o3o3o *b3o *b3o = D4 = Q(5) | o3o3o3o3o3*a = A4 = P(5) |
x4o3o3o4o -test - O1o4x3o3o4o -rittit - O87o4o3x3o4o -icot - O88x4x3o3o4o -tattit - O89x4o3x3o4o -srittit - O90x4o3o3x4o -spittit - O91x4o3o3o4x -test - O1o4x3x3o4o -batitit - O92o4x3o3x4o -ricot - O93x4x3x3o4o -grittit - O94x4x3o3x4o -potatit - O95x4x3o3o4x -capotat - O96x4o3x3x4o -prittit - O97x4o3x3o4x -cartit - O98o4x3x3x4o -ticot - O99x4x3x3x4o -gippittit - O100x4x3x3o4x -cagratit - O101x4x3o3x4x -captatit - O102x4x3x3x4x -gacotat - O103s4o3o3o4o -hext - O104s4o3x3o4o -thext - O105s4o3o3x4o -bricot - O106s4o3o3o4x -siphatit - O108s4o3x3x4o -bithit - O107s4o3x3o4x -pithatit - O109s4o3o3x4x -pirhatit - O110s4o3x3x4x -giphatit - O111s4o3x3o4s -cesratit - **)s4o3o3o4s -hext - O104o4s3s3s4o -sadit - O133s4o3o3o4s' -rittit - O87 | x3o3o *b3o4o -hext - O104o3x3o *b3o4o -icot - O88o3o3o *b3x4o -rittit - O87o3o3o *b3o4x -test - O1x3x3o *b3o4o -thext - O105x3o3x *b3o4o -rittit - O87x3o3o *b3x4o -bricot - O106x3o3o *b3o4x -siphatit - O108o3x3o *b3x4o -batitit - O92o3x3o *b3o4x -srittit - O90o3o3o *b3x4x -tattit - O89x3x3x *b3o4o -batitit - O92x3x3o *b3x4o -bithit - O107x3x3o *b3o4x -pithatit - O109x3o3x *b3x4o -ricot - O93x3o3x *b3o4x -spittit - O91x3o3o *b3x4x -pirhatit - O110o3x3o *b3x4x -grittit - O94x3x3x *b3x4o -ticot - O99x3x3x *b3o4x -prittit - O97x3x3o *b3x4x -giphatit - O111x3o3x *b3x4x -potatit - O95x3x3x *b3x4x -gippittit - O100x3o3o *b3o4s -rittit - O87s3s3s *b3s4o -sadit - O133 | x3o3o *b3o *b3o -hext - O104o3x3o *b3o *b3o -icot - O88x3x3o *b3o *b3o -thext - O105x3o3x *b3o *b3o -rittit - O87x3x3x *b3o *b3o -batitit - O92x3o3x *b3x *b3o -bricot - O106x3x3x *b3x *b3o -bithit - O107x3o3x *b3x *b3x -ricot - O93x3x3x *b3x *b3x -ticot - O99s3s3s *b3s *b3s -sadit - O133 | x3o3o3o3o3*a -cypit - O134x3x3o3o3o3*a -cytopit - O135x3o3x3o3o3*a -scyropot - O136x3x3x3o3o3*a -gocyropit - O137x3x3o3x3o3*a -cypropit - O138x3x3x3x3o3*a -gocypapit - O139x3x3x3x3x3*a -otcypit - O140 |
| o3o3o4o3o = F4 = U(5) | o∞o o4o3o4o = A1×C3 = W(2)×R(4) | o∞o o3o3o *d4o = A1×B3 = W(2)×S(4) | o∞o o3o3o3o3*c = A1×A3 = W(2)×P(4) |
x3o3o4o3o -hext - O104o3x3o4o3o -icot - O88o3o3x4o3o -bricot - O106o3o3o4x3o -ricot - O93o3o3o4o3x -icot - O88x3x3o4o3o -thext - O105x3o3x4o3o -ricot - O93x3o3o4x3o -spaht - O122x3o3o4o3x -scicot - O121o3x3x4o3o -bithit - O107o3x3o4x3o -sibricot - O116o3x3o4o3x -spict - O115o3o3x4x3o -baticot - O113o3o3x4o3x -sricot - O112o3o3o4x3x -ticot - O99x3x3x4o3o -ticot - O99x3x3o4x3o -pataht - O128x3x3o4o3x -capicot - O127x3o3x4x3o -praht - O125x3o3x4o3x -caricot - O124x3o3o4x3x -capoht - O123o3x3x4x3o -gibricot - O119o3x3x4o3x -pricot - O118o3x3o4x3x -paticot - O117o3o3x4x3x -gricot - O114x3x3x4x3o -gipaht - O131x3x3x4o3x -cagoraht - O130x3x3o4x3x -capticot - O129x3o3x4x3x -cagricot - O126o3x3x4x3x -gippict - O120x3x3x4x3x -gacicot - O132 o3o3o4s3s -sadit - O133s3s3s4o3o -sadit - O133x3o3o4s3s -capshot - **)o3x3o4s3s -paltite - **)o3o3x4s3s -sricot - O112s3s3s4o3x -capirsit - **)x3x3o4s3s -capsthat - **)x3o3x4s3s -caricot - O124o3x3x4s3s -pricot - O118x3x3x4s3s -cagricot - O130 | x∞o x4o3o4o -test - O1x∞o o4x3o4o - ricpit - O15 rectified-cubic prismatic TCx∞x x4o3o4o -test - O1x∞x o4x3o4o - ricpit - O15x∞o x4x3o4o - ticpit - O14 truncated-cubic prismatic TCx∞o x4o3x4o - sricpit (old: cacpit) - O17 small-rhombated-cubic-HC prismatic TC cantellated-cubic prismatic TCx∞o x4o3o4x -test - O1x∞o o4x3x4o - bitticpit - O16 bitruncated-cubic prismatic TCx∞x x4x3o4o - ticpit - O14x∞x x4o3x4o - sricpit (old: cacpit) - O17x∞x x4o3o4x -test - O1x∞x o4x3x4o - bitticpit - O16x∞o x4x3x4o - gricpit (old: catcupit) - O18 great-rhombated-cubic-HC prismated TC cantitruncated-cubic prismatic TCx∞o x4x3o4x - pricpit (old: rutcupit) - O19 prismatorhombated-cubic-HC prismated TC runcitruncated-cubic prismatic TCx∞x x4x3x4o - gricpit (old: catcupit) - O18x∞x x4x3o4x - pricpit (old: rutcupit) - O19x∞o x4x3x4x - gippicpit (old: otacpit) - O20 great-prismated-cubic-HC prismated TC omnitruncated-cubic prismatic TCx∞x x4x3x4x - gippicpit (old: otacpit) - O20 | x∞o x3o3o *d4o - tepit (old: acpit) - O21 tetrahedral-octahedral-HC prismated TC alternated-cubic prismatic TCx∞o o3x3o *d4o - ricpit - O15x∞o o3o3o *d4x -test - O1x∞x x3o3o *d4o - tepit (old: acpit) - O21x∞x o3x3o *d4o - ricpit - O15x∞x o3o3o *d4x -test - O1x∞o x3x3o *d4o - tatopit (old: tacpit) - O25 truncated-tetrahedral-octahedral-HC prismated TC truncated-alternated-cubic prismatic TCx∞o x3o3x *d4o - ricpit - O15x∞o x3o3o *d4x - sratopit (old: racpit) - O26 small-rhombated-tetrahedral-octahedral-HC prismated TC runcinated-alternated-cubic prismatic TCx∞o o3x3o *d4x - ticpit - O14x∞x x3x3o *d4o - tatopit (old: tacpit) - O25x∞x x3o3x *d4o - ricpit - O15x∞x x3o3o *d4x - sratopit (old: racpit) - O26x∞x o3x3o *d4x - ticpit - O14x∞o x3x3x *d4o - bitticpit - O16x∞o x3x3o *d4x - gratopit (old: rucacpit) - O28 great-rhombated-tetrahedral-octahedral-HC prismated TC runcicantic-cubic prismatic TCx∞o x3o3x *d4x - sricpit (old: cacpit) - O17x∞x x3x3x *d4o - bitticpit - O16x∞x x3x3o *d4x - gratopit (old: rucacpit) - O28x∞x x3o3x *d4x - sricpit (old: cacpit) - O17x∞o x3x3x *d4x - gricpit (old: catcupit) - O18x∞x x3x3x *d4x - gricpit (old: catcupit) - O18 | x∞o x3o3o3o3*c - tepit (old: acpit) - O21x∞x x3o3o3o3*c - tepit (old: acpit) - O21x∞o x3x3o3o3*c - cytatopit (old: quacpit) - O27 cyclotruncated-tetrahedral-octahedral-HC prismated TC quarter-cubic prismatic TCx∞o x3o3x3o3*c - ricpit - O15x∞x x3x3o3o3*c - cytatopit (old: quacpit) - O27x∞x x3o3x3o3*c - ricpit - O15x∞o x3x3x3o3*c - tatopit (old: tacpit) - O25x∞x x3x3x3o3*c - tatopit (old: tacpit) - O25x∞o x3x3x3x3*c - bitticpit - O16x∞x x3x3x3x3*c - bitticpit - O16 |
| o3o6o o3o6o = G2×G2 = V(3)2 | o3o6o o4o4o = G2×C2 = V(3)×R(3) | o3o6o o3o3o3*d = G2×A2 = V(3)×P(3) | o4o4o o4o4o = C2×C2 = R(3)2 |
x3o6o x3o6o -tribbit - O29x3o6o o3x6o - tathibbit - O32 triangular-trihexagonal duoprismatic TCx3o6o o3o6x - thibbit - O30 triangular-hexagonal duoprismatic TCo3x6o o3x6o - thabbit - O56 trihexagonal duoprismatic TCo3x6o o3o6x - hithibbit - O41 hexagonal-trihexagonal duoprismatic TCo3o6x o3o6x -hibbit - O39x3x6o x3o6o - thibbit - O30x3x6o o3x6o - hithibbit - O41x3x6o o3o6x -hibbit - O39x3o6x x3o6o - trithit - O35 triangular-rhombitrihexagonal TCx3o6x o3x6o -thrathibbit - O59x3o6x o3o6x - harhibit - O44 hexagonal-rhombihexagonal duoprismatic TCo3x6x x3o6o - tathobit - O34 triangular-tomohexagonal duoprismatic TCo3x6x o3x6o - thathobit - O58 trihexagonal-tomohexagonal duoprismatic TCo3x6x o3o6x - hithobit - O43 hexagonal-tomohexagonal duoprismatic TCx3x6x x3o6o - totuthit - O36 triangular-omnitruncated-trihexagonal TCx3x6x o3x6o -thot thibbit - O60x3x6x o3o6x -hot thibbit - O45x3x6o x3x6o -hibbit - O39x3x6o x3o6x - harhibit - O44x3x6o o3x6x - hithobit - O43x3o6x x3o6x - rithbit - O74 rhombitrihexagonal duoprismatic TCx3o6x o3x6x -thorahbit - O70o3x6x o3x6x - thobit - O69 tomohexagonal duoprismatic TCx3x6x x3x6o -hot thibbit - O45x3x6x x3o6x -rathotathibit - O75x3x6x o3x6x -thoot thibbit - O71x3x6x x3x6x - otathibbit - O78 omnitruncated-trihexagonal duoprismatic TCx3o6o s3s6s -tisthit - O38o3x6o s3s6s - thisthibbit - O62 trihexagonal-simotrihexagonal duoprismatic TCo3o6x s3s6s - hasithbit - O47 hexagonal-simotrihexagonal duoprismatic TCx3x6o s3s6s - hasithbit - O47x3o6x s3s6s -rithsithbit - O77o3x6x s3s6s - thosithbit - O73 tomohexagonal-simotrihexagonal duoprismatic TCx3x6x s3s6s - otsithbit - O80 omnitruncated-simotrihexagonal duoprismatic TCs3s6s s'3s'6s' - sithbit - O83 simotrihexagonal duoprismatic TC | x3o6o x4o4o - tisbat - O2 triangular-square duoprismatic TCx3o6o o4x4o - tisbat - O2o3x6o x4o4o - thisbit - O5 trihexagonal-square duoprismatic TCo3x6o o4x4o - thisbit - O5o3o6x x4o4o - shibbit - O3 square-hexagonal duoprismatic TCo3o6x o4x4o - shibbit - O3x3x6o x4o4o - shibbit - O3x3x6o o4x4o - shibbit - O3x3o6x x4o4o - rithsibbit - O8 rhombitrihexagonal-square duoprismatic TCx3o6x o4x4o - rithsibbit - O8o3x6x x4o4o - thosbit - O7 tomohexagonal-square duoprismatic TCo3x6x o4x4o - thosbit - O7x3o6o x4x4o - tatosbit - O33 triangular-tomosquare duoprismatic TCx3o6o x4o4x - tisbat - O2o3x6o x4x4o - thatosbit - O57 trihexagonal-tomosquare duoprismatic TCo3x6o x4o4x - thisbit - O5o3o6x x4x4o - hitosbit - O42 hexagonal-tomosquare duoprismatic TCo3o6x x4o4x - shibbit - O3x3x6x x4o4o -otathisbit - O9x3x6x o4x4o -otathisbit - O9x3x6o x4x4o - hitosbit - O42x3x6o x4o4x - shibbit - O3x3o6x x4x4o - tosrithbit - O65 tomosquare-rhombitrihexagonal duoprismatic TCx3o6x x4o4x - rithsibbit - O8o3x6x x4x4o - tosthobit - O64 tomosquare-tomohexagonal duoprismatic TCo3x6x x4o4x - thosbit - O7x3o6o x4x4x - tatosbit - O33o3x6o x4x4x - thatosbit - O57o3o6x x4x4x - hitosbit - O42x3x6x x4x4o -tosot thibbit - O66x3x6x x4o4x -otathisbit - O9x3x6o x4x4x - hitosbit - O42x3o6x x4x4x - tosrithbit - O65o3x6x x4x4x - tosthobit - O64x3x6x x4x4x - tosot thibbit - O66x3o6o s4s4s - tasist - O37 triangular-simosquare TCo3x6o s4s4s - thisosbit - O61 trihexagonal-simosquare duoprismatic TCo3o6x s4s4s - hisosbit - O46 hexagonal-simosquare duoprismatic TCx3x6o s4s4s - hisosbit - O46x3o6x s4s4s - rithsisbit - O76 rhombitrihexagonal-simosquare duoprismatic TCo3x6x s4s4s - thosisbit - O72 tomohexagonal-simosquare duoprismatic TCx3x6x s4s4s - otsisbit - O79 omnitruncated-simosquare duoprismatic TC s3s6s x4o4o - sithsobit - O11 simotrihexagonal-square duoprismatic TCs3s6s o4x4o - sithsobit - O11s3s6s x4x4o - tosasithbit - O68 tomosquare-simotrihexagonal duoprismatic TCs3s6s x4o4x - sithsobit - O11s3s6s x4x4x - tosasithbit - O68s3s6s s'4s'4s' - sissithbit - O82 simosquare-simotrihexagonal duoprismatic TC | x3o6o x3o3o3*d -tribbit - O29o3x6o x3o3o3*d - tathibbit - O32o3o6x x3o3o3*d - thibbit - O30x3x6o x3o3o3*d - thibbit - O30x3o6x x3o3o3*d - trithit - O35o3x6x x3o3o3*d - tathobit - O34x3o6o x3x3o3*d - tathibbit - O32o3x6o x3x3o3*d - thabbit - O56o3o6x x3x3o3*d - hithibbit - O41x3x6x x3o3o3*d - totuthit - O36x3x6o x3x3o3*d - hithibbit - O41x3o6x x3x3o3*d -thrathibbit - O59o3x6x x3x3o3*d - thathobit - O58x3o6o x3x3x3*d - thibbit - O30o3x6o x3x3x3*d - hithibbit - O41o3o6x x3x3x3*d -hibbit - O39x3x6x x3x3o3*d -thot thibbit - O60x3x6o x3x3x3*d -hibbit - O39x3o6x x3x3x3*d - harhibit - O44o3x6x x3x3x3*d - hithobit - O43x3x6x x3x3x3*d -hot thibbit - O45s3s6s x3o3o3*d -tisthit - O38s3s6s x3x3o3*d - thisthibbit - O62s3s6s x3x3x3*d - hasithbit - O47 | x4o4o x4o4o -test - O1x4o4o o4x4o -test - O1o4x4o o4x4o -test - O1x4x4o x4o4o - tososbit - O6 tomosquare-square duoprismatic TCx4x4o o4x4o - tososbit - O6x4o4x x4o4o -test - O1x4o4x o4x4o -test - O1x4x4x x4o4o - tososbit - O6x4x4x o4x4o - tososbit - O6x4x4o x4x4o - tosbit - O63 tomosquare duoprismatic TCx4x4o x4o4x - tososbit - O6x4o4x x4o4x -test - O1x4x4x x4x4o - tosbit - O63x4x4x x4o4x - tososbit - O6x4x4x x4x4x - tosbit - O63s4s4s x4o4o - sisosbit - O10 simosquare-square duoprismatic TCs4s4s o4x4o - sisosbit - O10s4s4s x4x4o - tosisasbit - O67 tomosquare-simosquare duoprismatic TCs4s4s x4o4x - sisosbit - O10s4s4s x4x4x - tosisasbit - O67s4s4s s4s4s - sisbit - O81 simosquare duoprismatic TC |
| o4o4o o3o3o3*d = C2×A2 = R(3)×P(3) | o3o3o3*a o3o3o3*d = A2×A2 = P(3)2 | o∞o o∞o o3o6o = A1×A1×G2 = W(2)2×V(3) | o∞o o∞o o4o4o = A1×A1×C2 = W(2)2×R(3) |
x4o4o x3o3o3*d - tisbat - O2o4x4o x3o3o3*d - tisbat - O2x4x4o x3o3o3*d - tatosbit - O33x4o4x x3o3o3*d - tisbat - O2x4o4o x3x3o3*d - thisbit - O5o4x4o x3x3o3*d - thisbit - O5x4x4x x3o3o3*d - tatosbit - O33x4x4o x3x3o3*d - thatosbit - O57x4o4x x3x3o3*d - thisbit - O5x4o4o x3x3x3*d - shibbit - O3o4x4o x3x3x3*d - shibbit - O3x4x4x x3x3o3*d - thatosbit - O57x4x4o x3x3x3*d - hitosbit - O42x4o4x x3x3x3*d - shibbit - O3x4x4x x3x3x3*d - hitosbit - O42s4s4s x3o3o3*d - tasist - O37s4s4s x3x3o3*d - thisosbit - O61s4s4s x3x3x3*d - hisosbit - O46 | x3o3o3*a x3o3o3*d -tribbit - O29x3o3o3*a x3x3o3*d - tathibbit - O32x3x3o3*a x3x3o3*d - thabbit - O56x3o3o3*a x3x3x3*d - thibbit - O30x3x3o3*a x3x3x3*d - hithibbit - O41x3x3x3*a x3x3x3*d -hibbit - O39 | x∞o x∞o x3o6o - tisbat - O2x∞o x∞o o3x6o - thisbit - O5x∞o x∞o o3o6x - shibbit - O3x∞x x∞o x3o6o - tisbat - O2x∞x x∞o o3x6o - thisbit - O5x∞x x∞o o3o6x - shibbit - O3x∞o x∞o x3x6o - shibbit - O3x∞o x∞o x3o6x - rithsibbit - O8x∞o x∞o o3x6x - thosbit - O7x∞x x∞x x3o6o - tisbat - O2x∞x x∞x o3x6o - thisbit - O5x∞x x∞x o3o6x - shibbit - O3x∞x x∞o x3x6o - shibbit - O3x∞x x∞o x3o6x - rithsibbit - O8x∞x x∞o o3x6x - thosbit - O7x∞o x∞o x3x6x -otathisbit - O9x∞x x∞x x3x6o - shibbit - O3x∞x x∞x x3o6x - rithsibbit - O8x∞x x∞x o3x6x - thosbit - O7x∞x x∞o x3x6x -otathisbit - O9x∞x x∞x x3x6x -otathisbit - O9x∞o x∞o s3s6s - sithsobit - O11 | x∞o x∞o x4o4o -test - O1x∞o x∞o o4x4o -test - O1x∞x x∞o x4o4o - test - O1x∞x x∞o o4x4o - test - O1x∞o x∞o x4x4o - tososbit - O6x∞o x∞o x4o4x -test - O1x∞x x∞x x4o4o - test - O1x∞x x∞x o4x4o - test - O1x∞x x∞o x4x4o - tososbit - O6x∞x x∞o x4o4x - test - O1x∞o x∞o x4x4x - tososbit - O6x∞x x∞x x4x4o - tososbit - O6x∞x x∞x x4o4x - test - O1x∞x x∞o x4x4x - tososbit - O6x∞x x∞x x4x4x - tososbit - O6x∞o x∞o s4s4s - sisosbit - O10 |
| o∞o o∞o o3o3o3*e = A1×A1×A2 = W(2)2×P(3) | o∞o o∞o o∞o o∞o = A1×A1×A1×A1 = W(2)4 | other uniforms | |
x∞o x∞o x3o3o3*e - tisbat - O2x∞x x∞o x3o3o3*e - tisbat - O2x∞o x∞o x3x3o3*e - thisbit - O5x∞x x∞x x3o3o3*e - tisbat - O2x∞x x∞o x3x3o3*e - thisbit - O5x∞o x∞o x3x3x3*e - shibbit - O3x∞x x∞x x3x3o3*e - thisbit - O5x∞x x∞o x3x3x3*e - shibbit - O3x∞x x∞x x3x3x3*e - shibbit - O3 | x∞o x∞o x∞o x∞o - test - O1x∞x x∞o x∞o x∞o - test - O1x∞x x∞x x∞o x∞o - test - O1x∞x x∞x x∞x x∞o - test - O1x∞x x∞x x∞x x∞x - test - O1 | elong( x3o3o3o3o3*a ) -ecypit - O141 elongated cyclopentachoric TC, elongated pentachoric-dispentachoric TCschmo( x3o3o3o3o3*a ) -zucypit - O142 schmoozed cyclopentachoric TC, schmoozed pentachoric-dispentachoric TCelongschmo( x3o3o3o3o3*a ) - ezucypit - O143 elongated schmoozed cyclopentachoric TC, elongated schmoozed pentachoric-dispentachoric TCelong( x3o6o x3o6o ) - etbit - O31 elongated triangular duoprismatic TCelong( x3o6o o3x6o ) - etothbit - O49 elongated triangular-trihexagonal duoprismatic TCelong( x3o6o o3o6x ) - ethibit - O40 elongated triangular-hexagonal duoprismatic TCelong( x3o6o x3x6o ) - ethibit - O40elong( x3o6o x3o6x ) - etrithit - O52 elongated triangular-rhombitrihexagonal TCelong( x3o6o o3x6x ) - etathobit - O51 elongated triangular-tomohexagonal duoprismatic TCelong( x3o6o x3x6x ) - etotithat - O53 elongated triangular-omnitruncated-trihexagonal TCelong( x3o6o s3s6s ) - etasithit - O55 elongated triangular-simotrihexagonal TCelong( elong( x3o6o x3o6o )) - betobit - O48 bielongated triangular duoprismatic TCelong( x3o6o elong( x3o6o )) - betobit - O48x3o6o elong( x3o6o ) - etbit - O31o3x6o elong( x3o6o ) - etothbit - O49o3o6x elong( x3o6o ) - ethibit - O40x3x6o elong( x3o6o ) - ethibit - O40x3o6x elong( x3o6o ) - etrithit - O52o3x6x elong( x3o6o ) - etathobit - O51x3x6x elong( x3o6o ) - etotithat - O53s3s6s elong( x3o6o ) - etasithit - O55elong( x3o6o ) elong( x3o6o ) - betobit - O48elong( x3o6o x4o4o ) - etsobit - O4 elongated triangular-square duoprismatic TCelong( x3o6o x4x4o ) - etatosbit - O50 elongated triangular-tomosquare duoprismatic TCelong( x3o6o x4x4x ) - etatosbit - O50elong( x3o6o s4s4s ) - etasist - O54 elongated triangular-simosquare TCgyro( x3o6o x4o4o ) - gytosbit - O12 gyrated triangular-square duoprismatic TCgyroelong( x3o6o x4o4o ) - egytsobit - O13 elongated gyrated triangular-square duoprismatic TCbigyro( x3o6o x4o4o ) - bigytsbit - O84 bigyrated triangular-square duoprismatic TCbigyroelong( x3o6o x4o4o ) - ebiytsbit - O85 elongated bigyrated triangular-square duoprismatic TCprismatogyro( x3o6o x4o4o ) - pegytsbit - O86 prismatoelongated gyrated triangular-square duoprismatic TCgyro( elong( x3o6o ) x4o4o ) - egytsobit - O13bigyro( elong( x3o6o ) x4o4o ) - ebiytsbit - O85elong( x3o6o ) x4o4o - etsobit - O4elong( x3o6o ) x4x4o - etatosbit - O50elong( x3o6o ) x4x4x - etatosbit - O50elong( x3o6o ) s4s4s - etasist - O54elong( x3o6o x3o3o3*d ) - etbit - O31elong( x3o6o x3x3o3*d ) - etothbit - O49elong( x3o6o x3x3x3*d ) - ethibit - O40elong( o3x6o x3o3o3*d ) - etothbit - O49elong( o3o6x x3o3o3*d ) - ethibit - O40elong( x3x6o x3o3o3*d ) - ethibit - O40elong( x3o6x x3o3o3*d ) - etrithit - O52elong( o3x6x x3o3o3*d ) - etathobit - O51elong( x3x6x x3o3o3*d ) - etotithat - O53elong (s3s6s x3o3o3*d ) - etasithit - O55elong( elong( x3o6o x3o3o3*d )) - betobit - O48elong( x3o6o elong( x3o3o3*d )) - betobit - O48elong( elong( x3o6o ) x3o3o3*d ) - betobit - O48x3o6o elong( x3o3o3*d ) - etbit - O31o3x6o elong( x3o3o3*d ) - etothbit - O49o3o6x elong( x3o3o3*d ) - ethibit - O40x3x6o elong( x3o3o3*d ) - ethibit - O40x3o6x elong( x3o3o3*d ) - etrithit - O52o3x6x elong( x3o3o3*d ) - etathobit - O51x3x6x elong( x3o3o3*d ) - etotithat - O53s3s6s elong( x3o3o3*d ) - etasithit - O55elong( x3o6o ) x3o3o3*d - etbit - O31elong( x3o6o ) x3x3o3*d - etothbit - O49elong( x3o6o ) x3x3x3*d - ethibit - O40elong( x3o6o ) elong( x3o3o3*d ) - betobit - O48elong( x3o3o3*a x4x4o ) - etatosbit - O50elong( x3o3o3*a x4x4x ) - etatosbit - O50elong( x3o3o3*a s4s4s ) - etasist - O54gyro( x3o3o3*a x4o4o ) - gytosbit - O12gyroelong( x3o3o3*a x4o4o ) - egytsobit - O13bigyro( x3o3o3*a x4o4o ) - bigytsbit - O84bigyroelong( x3o3o3*a x4o4o ) - ebiytsbit - O85prismatogyro( x3o3o3*a x4o4o ) - pegytsbit - O86gyro( elong( x3o3o3*a ) x4o4o ) - egytsobit - O13bigyro( elong( x3o3o3*a ) x4o4o ) - ebiytsbit - O85elong( x3o3o3*a ) x4x4o - etatosbit - O50elong( x3o3o3*a ) x4x4x - etatosbit - O50elong( x3o3o3*a ) s4s4s - etasist - O54elong( x3o3o3*a x3o3o3*d ) - etbit - O31elong( x3o3o3*a x3x3o3*d ) - etothbit - O49elong( x3o3o3*a x3x3x3*d ) - ethibit - O40elong( elong( x3o3o3*a x3o3o3*d )) - betobit - O48elong( x3o3o3*a elong( x3o3o3*d )) - betobit - O48x3o3o3*a elong( x3o3o3*d ) - etbit - O31x3x3o3*a elong( x3o3o3*d ) - etothbit - O49x3x3x3*a elong( x3o3o3*d ) - ethibit - O40elong( x3o3o3*a ) elong( x3o3o3*d ) - betobit - O48elong( x∞o x3o3o *d4o ) - eacpit - O23 elongated-alternated-cubic prismatic TCgyro( x∞o x3o3o *d4o ) - gyacpit - O22 gyrated-alternated-cubic prismatic TCgyroelong( x∞o x3o3o *d4o ) - gyeacpit - O24 gyrated-elongated-alternated-cubic prismatic TCx∞o elong( x3o3o *d4o ) - eacpit - O23x∞o gyro( x3o3o *d4o ) - gyacpit - O22x∞o gyroelong( x3o3o *d4o ) - gyeacpit - O24elong( x∞o x3o3o3o3*c ) - eacpit - O23gyro( x∞o x3o3o3o3*c ) - gyacpit - O22gyroelong( x∞o x3o3o3o3*c ) - gyeacpit - O24x∞o elong( x3o3o3o3*c ) - eacpit - O23x∞o gyro( x3o3o3o3*c ) - gyacpit - O22x∞o gyroelong( x3o3o3o3*c ) - gyeacpit - O24gyro( x∞o x∞o x3o6o ) - gytosbit - O12gyroelong( x∞o x∞o x3o6o ) - egytsobit - O13bigyro( x∞o x∞o x3o6o ) - bigytsbit - O84bigyroelong( x∞o x∞o x3o6o ) - ebiytsbit - O85prismatogyro( x∞o x∞o x3o6o ) - pegytsbit - O86elong( x∞o gyro( x∞o x3o6o )) - egytsobit - O13gyro( x∞o gyro( x∞o o3x6o )) - bigytsbit - O84gyroelong( x∞o gyro( x∞o o3x6o )) - ebiytsbit - O85gyro( x∞o x∞o elong( x3o6o )) - egytsobit - O13bigyro( x∞o x∞o elong( x3o6o )) - ebiytsbit - O85x∞o elong( x∞o x3o6o ) - etsobit - O4x∞o gyro( x∞o x3o6o ) - gytosbit - O12x∞o gyroelong( x∞o x3o6o ) - egytsobit - O13gyro( x∞o x∞o x3o3o3*e ) - gytosbit - O12gyroelong( x∞o x∞o x3o3o3*e ) - egytsobit - O13bigyro( x∞o x∞o x3o3o3*e ) - bigytsbit - O84bigyroelong( x∞o x∞o x3o3o3*e ) - ebiytsbit - O85prismatogyro( x∞o x∞o x3o3o3*e ) - pegytsbit - O86elong( x∞o gyro( x∞o x3o3o3*e )) - egytsobit - O13gyro( x∞o gyro( x∞o x3x3o3*e )) - bigytsbit - O84gyroelong( x∞o gyro( x∞o x3x3o3*e )) - ebiytsbit - O85gyro( x∞o x∞o elong( x3o3o3*e )) - egytsobit - O13bigyro( x∞o x∞o elong( x3o3o3*e )) - ebiytsbit - O85x∞o gyro( x∞o x3o3o3*e ) - gytosbit - O12x∞o gyroelong( x∞o x3o3o3*e ) - egytsobit - O13...-schmo- = ...-0-0-... -gyro- = ...-0-1-0-1-...-bigyro- = ...-0-1-2-0-1-2-...-prismatogyro- = ...-0-1-0-2-0-1-0-2-... | |
** Thesealternated facetings well can be varied to get all equal edge lengths, but still will not become uniform.In fact those relaxations would qualify asscaliform tetracombs only. – This is why they do not have an Olshevsky number either.
In 4D too there are several differentlattices.
In the pentachoric area there are 2 lattices, A4 and A4*.The latter of which can be considered as a superposition of 5 of the former, each having a 1/5 rotatedDynkin diagram. The Delone complex of A4 iscypit.The Voronoi complex of A4* isotcypit.
In the tesseractic area there is the primitive cubical one, C4. Its Voronoi complex and its Delone complex both are relatively shiftedtest.
Next there is the body-centered tesseractic (bct) lattice. Thus it is the union of the primitive cubical tesseractic plus its Voronoi cell vertices ("holes"). Alternatively this one can be described either as lattice B4, as lattice D4, or as lattice F4.The Voronoi complex here isicot. The Delone complex ishext.
The lattice C4 = C2×C2 (or the vertex set oftest) can be represented by the Hamiltonian integersHam =Z[i,j] = {a0 + a1i + a2j + a3k | a0,a1,a2,a3∈Z}, i2 = j2 = k2 = ijk = -1.
The lattice A2×A2 (or the vertex set oftribbit) can be represented by the hybrid integersHyb =Z[ω,j] = {b0 + b1ω + b2j + b3ωj | b0,b1,b2,b3∈Z}, ω = (-1+sqrt(-3))/2, j as above.
The lattice F4 (or the vertex set ofhext) can be represented by Hurwitz integersHur =Z[u,v] = {c0 + c1u + c2v + c3w | c0,c1,c2,c3∈Z}, u = (1-i-j+k)/2, v = (1+i-j-k)/2, w = (1-i+j-k)/2, uvw = 1.
:xoqo:4:xxox:3:oooo:4:oxxx:&##x:xoqo:4:xxox:3:xxxx:4:oxxx:&##x... | Non-uniform convex tetracombs using axialCRF polychora for building blocks easily can be derived from infinite periodic stacks of 3D honeycombs, provided those use the same axial symmetry and have the same periodicity scaling in each (independent) around-symmetrical direction. The (infinite) segments then are filled from accordinglace prisms.As these are piled up to linear columns each, those can be considered to become multistraticblends, as long as those still remain convex. |
A further non-uniform tetracomb with all unit-sized edges andCRF polychoral elements can be obtained from the non-Wythoffiansadit viaambification,then resulting inrisadit.
Beyond the convex tetracombs there are uniformnon-convex euclidean tetracombs too. Those will addfor additional possibilities the usage of
(Several ones can be seen to beGrünbaumian a priori right from having both nodes being markedat links with even denominator. The same holds true forx∞'x, which by itself would be nothing but an infinitely many covered edge. All these then are represented below accordingly. Others might be recognised to be Grünbaumian a posteriory, e.g.when using such polytopes for cells or as vertex figure. These then are mentioned by referenciation to at least one of those elements.)
o--3--o_ | | -3_ 4 4/3 _>o | | _4 o--3--o- | |
| o3o4o3o4o3*a4/3*c | |
|---|---|
x3o4x3o4o3*a4/3*c -sazdideta | |
---- 5DPentacombs (up)----
So far Wythoffian elementary ones only.
| o4o3o3o3o4o = C5 = R(6) | o3o3o *b3o3o4o = B5 = S(6) | o3o3o o3o3o *b3*e = D5 = Q(6) | o3o3o3o3o3o3*a = A5 = P(6) |
x4o3o3o3o4o -pentho4x3o3o3o4o -rinoho4o3x3o3o4o -brinohx4x3o3o3o4o -tanohx4o3x3o3o4o -sirnohx4o3o3x3o4o - spanohx4o3o3o3x4o - scanohx4o3o3o3o4x -penth (stenoh)o4x3x3o3o4o - bittinoho4x3o3x3o4o - sibranoho4x3o3o3x4o - sibpanoho4o3x3x3o4o -titanohx4x3x3o3o4o - girnohx4x3o3x3o4o - pattinohx4x3o3o3x4o - catanohx4x3o3o3o4x - tetanohx4o3x3x3o4o - prinohx4o3x3o3x4o - carnohx4o3x3o3o4x - tepanohx4o3o3x3x4o - cappinoho4x3x3x3o4o - gibranoho4x3x3o3x4o - biprinohx4x3x3x3o4o - gippinohx4x3x3o3x4o - cogrinohx4x3x3o3o4x - tegranohx4x3o3x3x4o - captinohx4x3o3x3o4x - teptanohx4x3o3o3x4x - tectanohx4o3x3x3x4o - capranohx4o3x3x3o4x - tepranoho4x3x3x3x4o - gibpanohx4x3x3x3x4o - gacnohx4x3x3x3o4x - tegpenohx4x3x3o3x4x - tecgranohx4x3x3x3x4x - gatenoh | x3o3o *b3o3o4o -hinoho3x3o *b3o3o4o -brinoho3o3o *b3x3o4o -brinoho3o3o *b3o3x4o -rinoho3o3o *b3o3o4x -penthx3x3o *b3o3o4o -thinohx3o3x *b3o3o4o -rinohx3o3o *b3x3o4o - sirhinohx3o3o *b3o3x4o - siphinohx3o3o *b3o3o4x - sachinoho3x3o *b3x3o4o -titanoho3x3o *b3o3x4o - sibranoho3x3o *b3o3o4x - spanoho3o3o *b3x3x4o - bittinoho3o3o *b3x3o4x -sirnoho3o3o *b3o3x4x -tanohx3x3x *b3o3o4o - girnohx3x3o *b3x3o4o - girhinohx3x3o *b3o3x4o - pithinohx3x3o *b3o3o4x - cathinohx3o3x *b3x3o4o - sibranohx3o3x *b3o3x4o - sibpanohx3o3x *b3o3o4x - scanohx3o3o *b3x3x4o - pirhinohx3o3o *b3x3o4x - crahinohx3o3o *b3o3x4x - caphinoho3x3o *b3x3x4o - gibranoho3x3o *b3x3o4x - prinoho3x3o *b3o3x4x - pattinoho3o3o *b3x3x4x - girnohx3x3x *b3x3o4o - gibranohx3x3x *b3o3x4o - biprinohx3x3x *b3o3o4x - cappinohx3x3o *b3x3x4o - giphinohx3x3o *b3x3o4x - cograhnohx3x3o *b3o3x4x - copthinohx3o3x *b3x3x4o - biprinohx3o3x *b3x3o4x - carnohx3o3x *b3o3x4x - catanoho3x3o *b3x3x4x - gippinohx3x3x *b3x3x4o - gibpanohx3x3x *b3x3o4x - capranohx3x3x *b3o3x4x - captinohx3x3o *b3x3x4x - gachinohx3o3x *b3x3x4x - cogrinohx3x3x *b3x3x4x - gacnoh | x3o3o o3o3o *b3*e -hinoho3x3o o3o3o *b3*e -brinohx3x3o o3o3o *b3*e -thinohx3o3x o3o3o *b3*e -rinohx3o3o x3o3o *b3*e -spaquinohx3o3o o3x3o *b3*e - sirhinoho3x3o o3x3o *b3*e -titanohx3x3x o3o3o *b3*e - bittinohx3x3o x3o3o *b3*e -praquinohx3x3o o3x3o *b3*e - girhinohx3o3x x3o3o *b3*e - siphinohx3o3x o3x3o *b3*e - sibranohx3x3x x3o3o *b3*e - pirhinohx3x3x o3x3o *b3*e - gibranohx3x3o x3x3o *b3*e -gapquinohx3x3o x3o3x *b3*e - pithinohx3o3x x3o3x *b3*e - sibpanohx3x3x x3x3o *b3*e - giphinohx3x3x x3o3x *b3*e - biprinohx3x3x x3x3x *b3*e - gibpanoh | x3o3o3o3o3o3*a -cyxhx3x3o3o3o3o3*a -cytaxhx3o3x3o3o3o3*a -racyxhx3o3o3x3o3o3*a -spacyxhx3x3x3o3o3o3*a - tacyxhx3x3o3x3o3o3*a - cyprexhx3o3x3o3x3o3*a - sarcyxhx3x3x3x3o3o3*a - cygpoxhx3x3x3o3x3o3*a - parcyxhx3x3o3x3x3o3*a - bitcyxhx3x3x3x3x3o3*a - garcyxhx3x3x3x3x3x3*a -gapcyxh |
Thelattice D5 is the vertex set ofhinoh, or taken the other way round, that one is the Delone complex of this lattice. The vertex count of its vertex figure (rat)displays the highest kissing number of this dimension (40).
The lattice D5* can be constructed either as union of the vertex sets of 4hinoh(x3o3o o3o3o *b3*e + o3o3x o3o3o *b3*e + o3o3o x3o3o *b3*e + o3o3o o3o3x *b3*e), or as the union of the vertex sets of 2penth (x4o3o3o3o4o + o4o3o3o3o4x). The latterdescription shows that this is the body-centered penteractic lattice. Its kissing number is 10. Its Voronoi complex istitanoh.
The lattice A5 is the vertex set ofcyxh, or taken the other way round, that one is the Delone complex of this lattice.
There are different overlays of that vertex set possible, which correspond to overlays by rotations of the respective diagrams: 2 such intervoven latticeswould result from 2 diagrams with opposite nodes ringed (x3o3o3o3o3o3*a + o3o3o3x3o3o3*a), generating in the lattice A52; 3 such intervoven lattices result from 3 diagrams with the ringed node in different triangular positions(x3o3o3o3o3o3*a + o3o3x3o3o3o3*a + o3o3o3o3x3o3*a), generating A53; and finally, using 6 intervoven lattices, corresponding to either orientation of the diagram(x3o3o3o3o3o3*a + o3x3o3o3o3o3*a + o3o3x3o3o3o3*a + o3o3o3x3o3o3*a + o3o3o3o3x3o3*a + o3o3o3o3o3x3*a), generates A56 = A5*. And the Voronoi complex of thatlast one would begapcyxh.
---- 6DHexacombs (up)----
So far just the quasiregular ones of the irreducible groups and some further monotoxal ones.
| o4o3o3o3o3o4o = C6 = R(7) | o3o3o *b3o3o3o4o = B6 = S(7) | o3o3o o3o3o *b3o3*e = D6 = Q(7) | o3o3o3o3o3o3o3*a = A6 = P(7) | o3o3o3o3o *c3o3o = E6 = T(7) |
x4o3o3o3o3o4o -axho4x3o3o3o3o4o -raxho4o3x3o3o3o4o -braxho4o3o3x3o3o4o -traxhx4o3o3o3o3o4x -axho4x3o3o3o3x4o -sibcaxho4o3x3o3x3o4o - straxh | x3o3o *b3o3o3o4o -haxho3x3o *b3o3o3o4o -braxho3o3o *b3x3o3o4o -traxho3o3o *b3o3x3o4o -braxho3o3o *b3o3o3x4o -raxho3o3o *b3o3o3o4x -axhx3o3x *b3o3o3o4o -raxho3x3o *b3o3x3o4o - straxhx3o3x *b3o3o3x4o - sibcaxh | x3o3o o3o3o *b3o3*e -haxho3x3o o3o3o *b3o3*e -braxho3o3o o3o3o *b3x3*e -traxhx3o3x o3o3o *b3o3*e -raxho3x3o o3x3o *b3o3*e - straxhx3o3x x3o3x *b3o3*e - sibcaxh | x3o3o3o3o3o3o3*a -cylohx3x3o3o3o3o3o3*a -cytlohx3o3x3o3o3o3o3*a - scyrlohx3o3o3x3o3o3o3*a - spacyloh...x3x3x3x3x3x3x3*a -otcyloh | x3o3o3o3o *c3o3o -jakoho3x3o3o3o *c3o3o -rojkoho3o3x3o3o *c3o3o -ramohx3o3o3o3x *c3o3o -terjakho3x3o3x3o *c3o3o -trojkohx3o3o3o3x *c3o3x -scajakho3x3o3x3o *c3x3o -saberjakh |
Thelattice D6 is the vertex set ofhaxh, or taken the other way round, that one is the Delone complex of this lattice.
The lattice A6 is the vertex set ofcyloh, or taken the other way round, that one is the Delone complex of this lattice.
The lattice E6 is the vertex set ofjakoh, or taken the other way round, that one is the Delone complex of this lattice. The vertex count of its vertex figure (mo)displays the highest kissing number of this dimension (72).
The lattice D6* can be constructed either as union of the vertex sets of 4haxh(x3o3o o3o3o *b3o3*e + o3o3x o3o3o *b3o3*e + o3o3o x3o3o *b3o3*e + o3o3o o3o3x *b3o3*e), or as the union of the vertex sets of 2axh (x4o3o3o3o3o4o + o4o3o3o3o3o4x). The latterdescription shows that this is the body-centered hexeractic lattice. Its kissing number is 12. Its Voronoi complex istraxh.
The lattice A6* can be constructed as union of the vertex sets of 7cyloh(x3o3o3o3o3o3o3*a + o3x3o3o3o3o3o3*a + o3o3x3o3o3o3o3*a + o3o3o3x3o3o3o3*a + o3o3o3o3x3o3o3*a + o3o3o3o3o3x3o3*a + o3o3o3o3o3o3x3*a).Its Voronoi complex isotcyloh (the omnitruncate).
The lattice E6* can be constructed as union of the vertex sets of 3jakoh(x3o3o3o3o *c3o3o + o3o3o3o3x *c3o3o + o3o3o3o3o *c3o3x).Its Voronoi complex isramoh.
---- 7DHeptacombs (up)----
Just the quasiregular ones of the irreducible groups.
| o4o3o3o3o3o3o4o = C7 = R(8) | o3o3o *b3o3o3o3o4o = B7 = S(8) | o3o3o o3o3o *b3o3o3*e = D7 = Q(8) | o3o3o3o3o3o3o3o3*a = A7 = P(8) | o3o3o3o3o3o3o *d3o = E7 = T(8) |
x4o3o3o3o3o3o4o -heptho4x3o3o3o3o3o4o - rasho4o3x3o3o3o3o4o - brasho4o3o3x3o3o3o4o - trashx4o3o3o3o3o3o4x -hepth | x3o3o *b3o3o3o3o4o - hasho3x3o *b3o3o3o3o4o - brasho3o3o *b3x3o3o3o4o - trasho3o3o *b3o3x3o3o4o - trasho3o3o *b3o3o3x3o4o - brasho3o3o *b3o3o3o3x4o - rasho3o3o *b3o3o3o3o4x -hepth | x3o3o o3o3o *b3o3o3*e - hasho3x3o o3o3o *b3o3o3*e - brasho3o3o o3o3o *b3x3o3*e - trash | x3o3o3o3o3o3o3o3*a -cyooh | x3o3o3o3o3o3o *d3o -naquoho3x3o3o3o3o3o *d3o -rinquoho3o3x3o3o3o3o *d3o - brinquoho3o3o3x3o3o3o *d3o -lanquoho3o3o3o3o3o3o *d3x -linoh |
A slab heptacomb would be eg.jakoha.
---- 8DOctacombs (up)----
Just the quasiregular ones of the irreducible groups.
| o4o3o3o3o3o3o3o4o = C8 = R(9) | o3o3o *b3o3o3o3o3o4o = B8 = S(9) | o3o3o o3o3o *b3o3o3o3*e = D8 = Q(9) | o3o3o3o3o3o3o3o3o3*a = A8 = P(9) | o3o3o3o3o3o3o3o *c3o = E8 = T(9) |
x4o3o3o3o3o3o3o4o - och (old: octh)o4x3o3o3o3o3o3o4o - roch (old: rocth)o4o3x3o3o3o3o3o4o - brocho4o3o3x3o3o3o3o4o - trocho4o3o3o3x3o3o3o4o - teroch | x3o3o *b3o3o3o3o3o4o - hoch (old: hocth)o3x3o *b3o3o3o3o3o4o - brocho3o3o *b3x3o3o3o3o4o - trocho3o3o *b3o3x3o3o3o4o - terocho3o3o *b3o3o3x3o3o4o - trocho3o3o *b3o3o3o3x3o4o - brocho3o3o *b3o3o3o3o3x4o - roch (old: rocth)o3o3o *b3o3o3o3o3o4x - och (old: octh) | x3o3o o3o3o *b3o3o3o3*e - hoch (old: hocth)o3x3o o3o3o *b3o3o3o3*e - brocho3o3o o3o3o *b3x3o3o3*e - trocho3o3o o3o3o *b3o3x3o3*e - teroch | x3o3o3o3o3o3o3o3o3*a - cyenoh | x3o3o3o3o3o3o3o *c3o -bayoho3x3o3o3o3o3o3o *c3o -robyaho3o3x3o3o3o3o3o *c3o -ribfoho3o3o3x3o3o3o3o *c3o - tergoho3o3o3o3x3o3o3o *c3o - trigoho3o3o3o3o3x3o3o *c3o - bargoho3o3o3o3o3o3x3o *c3o -rigoho3o3o3o3o3o3o3x *c3o -goho3o3o3o3o3o3o3o *c3x -bifoh |
Thelattice C8 (or the vertex set of of octh) can be represented by the Caylay-Graves integersOcg =Z[i,j,e] = {a0 + a1i + a2j+ a3k + a4e + a5ie + a6je + a7ke | a0,...,a7∈Z}.
The lattice E8 (or the vertex set of ofgoh) can be represented by (each of the 7 different types of) Coxeter-Dickson integersOcd =Z[i,j,h] = {b0 + b1i + b2j + b3k + b4h + b5ih + b6jh + b7kh | b0,...,b7∈Z}, h = (i+j+k+e)/2.
The lattice F4×F4 (or the vertex set of of {3,3,4,3}2) can be represented by (each of the 7 different types of) coupled Hurwitz integersOch=Z[u,v,e] = {c0 + c1u + c2v + c3w + c4e + c5ue + c6ve + c7we | c0,...,c7∈Z}, u = (1-i-j+k)/2, v = (1+i-j-k)/2, w = (1-i+j-k)/2, uvw = 1.
The lattice A2×A2×A2×A2 (or the vertex set of of {3,6}4) can be represented by the compound Eisenstein integersOce =Z[ω,j,e] = {d0 + d1ω + d2j + d3ωj + d4e + d5ωe + d6je + d7ωje | d0,...,d7∈Z}, ω = (-1+sqrt(-3))/2.
---- 9DEnneacombs (up)----
Just the quasiregular ones of the irreducible groups.
| o4o3o3o3o3o3o3o3o4o = C9 = R(10) | o3o3o *b3o3o3o3o3o3o4o = B9 = S(10) | o3o3o o3o3o *b3o3o3o3o3*e = D9 = Q(10) | o3o3o3o3o3o3o3o3o3o3*a = A9 = P(10) |
x4o3o3o3o3o3o3o3o4o - enneho4x3o3o3o3o3o3o3o4o - reneho4o3x3o3o3o3o3o3o4o - breneho4o3o3x3o3o3o3o3o4o - treneho4o3o3o3x3o3o3o3o4o - terneh | x3o3o *b3o3o3o3o3o3o4o - heneho3x3o *b3o3o3o3o3o3o4o - breneho3o3o *b3x3o3o3o3o3o4o - treneho3o3o *b3o3x3o3o3o3o4o - terneho3o3o *b3o3o3x3o3o3o4o - terneho3o3o *b3o3o3o3x3o3o4o - treneho3o3o *b3o3o3o3o3x3o4o - breneho3o3o *b3o3o3o3o3o3x4o - reneho3o3o *b3o3o3o3o3o3o4x - enneh | x3o3o o3o3o *b3o3o3o3o3*e - heneho3x3o o3o3o *b3o3o3o3o3*e - breneho3o3o o3o3o *b3x3o3o3o3*e - treneho3o3o o3o3o *b3o3x3o3o3*e - terneh | x3o3o3o3o3o3o3o3o3o3*a - cydoh |
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