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Axial Polytopes

• Monostratic Polytopes(i.e. having 2 vertex layers)
  • Pyramids
  • Prisms
  • Antiprisms
  • Alterprisms –gyrotrigonism etc.
  • Cupolas
  • Cuploids
  • Cupolaic Blends
  • Fastegia
  • Antifastegia
  • Duoantifastegia & Duoantifastegiaprisms
  • Duoalterprisms
  • ...
  • Segmentotopes
  • Lace Prisms
• Bistratic Polytopes(i.e. having 3 vertex layers)
  • Bipyramids
  • Elongated Pyramids
  • Elongated Cupolas
  • Edge-Expanded Biprisms *)
  • Edge-Expanded Bi-Antiprisms *)
  • Ursatopes
  • Tutsatopes
  • Trippescu series
  • Peratopes
  • ...
  • Bistratic Lace Towers
• (Generally) Multistratic Polytopes
  • Elongated Bipyramids
  • Supersemicupola
  • Cuneprisms
  • ...

*)The part about edge-expanded polytopes was derived in a co-operation with J. McNeill (cf. e.g. his pages onEEBs or onn,n,3-acrons).

Also lots of theEKFs provide interesting axial polytopes (although these are not restricted to axial symmetries in general).

Pyramids   (up)

(E.g. in 3D cf.n/d-py for general {n/d} base.)

The pyramids are the outcome of thepyramid product. Take any polytope as base, a single point with a relative orthogonal offset. Then the projective scaling, centered atthat point, will outline the derived pyramid within the interval from the point up to the given base.At this level nothing is said about dimensions, convexity of the base, nor about edge lengths.

As far as the base polytope is anon-snub, has aDynkin diagram descriptionand isorbiform, the whole pyramid will have a Dynkin diagram too. At least in the sense of alace prism with appropriately scaled lacings. Just prefix any node symbol (thus either being anx or ao) by an unringed node (o).And finally postfix the obtained symbol by "&#y", where the relative size ofy is a function of thedesired height of the pyramid, in fact it provides the lacing length. – Although pyramids can be built on a snubbed base, the required symbol cannot be obtainedin the just described way, because the processes of alternation and setting up the pyramid product do not commute.Orbiformity of the base was required, as else those lacings cannot all be of a single length.

The actual height of a pyramid can be calculated by means of theorem of Pythagoras ash = sqrt(|y|² - r²),where|y| will be the absolute length of the lacing edges andr is the circumradius of the base polytope.For 3D pyramids one further hasr(x-n/d-o ) = 1/(2 sin(π d/n)) and therefore, because ofx being unit edges,

h( ox-n/d-oo&#x ) = sqrt[1 - 1/(2 sin(π d/n))2]

Generally this gives as restriction for possible base polytopesh > 0 or|y| > r.For those 3D pyramids therefore one derives

h( ox-n/d-oo&#x ) > 0   or   1 > 1/(2 sin(π d/n))   or   sin(π d/n) > 1/2   or   π d/n > π/6   or   n/d < 6

Because of the equivalencex-n/d-o = x-n/(n-d)-o we likewise getn/(n-d) < 6, or in other words finally:5/6 < n/d < 6.

Pyramids can be seen assegmentotopes, provided they fullfil their required axioms.In this context those boil down to the requirement that the base polytope has to be a polytope with unique circumradiusr, which has to bestrictly smaller than 1, and all edges are of unit length. Only then the lacings can be chosen to be of unit length as well.Furthermore the circumradiusR of the whole pyramid can be provided then in terms of the circumradiusr of the base as:

R2 = 1/[4 (1-r2)]

The lacing facet polytopes all clearly are pyramids in turn, in fact they are pyramids based on the facets of the base polytope.

Forconvex pyramidal segmentotopes we have:

A special subclass here aremulti-pyramids. Again cf. to thepyramid product.Multi-pyramids are not to be miss-understood here in the sense of abipyramid, i.e.not astegum sum of a line and a perp base, butrather are meant in the sense of iteratedly applying the pyramid operation instead, thereby adding a further dimension each time. Thus, these figures would be "pt || (pt || (...(pt || base)...))". J. Bowers here introduced a sequence of according names:

Q-pyramid = Q-py,Q-scalene = Q-py-py, Q-tettene = Q-py-py-py, Q-pennene = Q-py-py-py-py, Q-hixene  = Q-py-py-py-py-py, ...

Esp. the point-pyramid-pyramid is nothing but a (generally) irregular triangle, which commonly is called a scalene (triangle). This is, where this sequence derives from.
Some examples then would be:

{3} = pt-py-py

-

-

-

-

tet = line-py-py

tet = pt-py-py-py

-

-

-

pen    = {3}-py-pysquasc = {4}-py-pypesc   = {5}-py-pystasc  = {5/2}-py-pyshasc  = {7/2}-py-pyogasc  = {8/3}-py-py...

pen = line-py-py-py

pen = pt-py-py-py-py

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-

hix       =tet-py-pyoctasc    =oct-py-pytrippasc  =trip-py-pyquithesc  =quith-py-pysissidisc =sissid-py-pysquete    =squippy-py-pystate     =stapy-py-py...

hix    = {3}-py-py-pysquete = {4}-py-py-pystate  = {5/2}-py-py-py...

hix    = line-py-py-py-py

hix    = pt-py-py-py-py-py

-

hop         =pen-py-pyhexasc      =hex-py-pyrapesc      =rap-py-pytepasc      =tepe-py-pytriddipasc  =triddip-py-pysquippypasc =squippyp-py-pyoctete      =octpy-py-pytrippete    =trippy-py-pysquepe      =squasc-py-py...

hop      =tet-py-py-pyoctete   =oct-py-py-pytrippete =trip-py-py-pysquepe   =squippy-py-py-py...

hop    = {3}-py-py-py-pysquepe = {4}-py-py-py-py...

hop    = line-py-py-py-py-py

...

oca      =hix-py-pyrixasc   =rix-py-pytaccasc  =tac-py-pyhinsc    =hin-py-pypenpasc  =penp-py-pyrapete   =rappy-py-pyhexete   =hexpy-py-pyoctepe   =octasc-py-pytrippepe =trippasc-py-pysquix    =squete-py-py...

oca      =pen-py-py-pyrapete   =rap-py-py-pyhexete   =hex-py-py-pyoctepe   =octpy-py-py-pytrippepe =trippy-py-py-pysquix    =squasc-py-py-py...

oca      =tet-py-py-py-pyoctepe   =oct-py-py-py-pytrippepe =trip-py-py-py-pysquix    =squippy-py-py-py-py...

oca   = {3}-py-py-py-py-pysquix = {4}-py-py-py-py-py...

...

geeasc =gee-py-pyhexepe =hexesc-py-pyrapepe =rapesc-py-pytaccete =tacpy-py-py...

hexepe =hexpy-py-py-pyrapepe =rappy-py-py-pytaccete =tac-py-py-py...

hexepe =hex-py-py-py-pyrapepe =rap-py-py-py-py...

...

...

Applying here, the above note, that pyramids onorbiform bases with base radius lesser than 1 generally aresegmentotopes, could be iterated. At the first level we have the alignment point || base. Scalenes with such an orbiform base generally do allow for a further such description as line || perp base-base.Tettenes with such an orbiform base generally do allow for a third such description as triangle || perp base-base-base. And the general item of that sequance then allows for the description as simplex || perp iterated base.

Prisms   (up)

(E.g. in 3D cf.n/d-p for general {n/d} bases.)

Similar to the pyramids, prisms are the outcome of theprism product of any base polytope with a lacing edge. Again nothing is said in general about dimension, convexity nor edge lengths.In fact, they also can be seen as elongated ditopes.

If the base polytope has anyDynkin diagram description, this product will have one too.Just add a further ringed node (inline:x) to the diagram but no further links. Note, this wouldwork forsnubbed base polytopes alike. For non-snubbed ones however, theDynkin diagram can be rewritten as alace prism just by doubling any node symbolof the base polytope, and by postfixing a "&#y", wherey provides the relative edge length of the lacings. – Again snubbing does not commute with the product. In fact, for3D prisms, commutation would lead to theantiprisms.

Prisms can be seen assegmentotopes, provided they fullfil the required axioms.In this context those boil down to the requirement, that the base polytope just has to beorbiform. Further, the lacings will have to be of unit length as well. I.e. for the height of prismatic segmentotopes one generally hash = 1.

The lacing facet polytopes all clearly are prisms in turn, in fact they are prisms based on the facets of the base polytope.

Forconvex prismatic segmentotopes we have:

x o       = oo&#x

x x       = xx&#x

x x-n-o   = xx-n-oo&#xx x4o     = xx4oo&#x

x x-n-x   = xx-n-xx&#x

x x3o3o   = xx3oo3oo&#x

x x3x3o   = xx3xx3oo&#x

x x3o4o   = xx3oo4oo&#xx o3x3o   = oo3xx3oo&#x

x o3x4o   = oo3xx4oo&#x

x o3o4x   = oo3oo4xx&#x

x x3x4o   = xx3xx4oo&#xx x3x3x   = xx3xx3xx&#x

x x3o4x   = xx3oo4xx&#x

x o3x4x   = oo3xx4xx&#x

x x3x4x   = xx3xx4xx&#x

x x3o5o   = xx3oo5oo&#xx s3s3s

x o3x5o   = oo3xx5oo&#x

x o3o5x   = oo3oo5xx&#x

x x3x5o   = xx3xx5oo&#x

x x3o5x   = xx3oo5xx&#x

x o3x5x   = oo3xx5xx&#x

x x3x5x   = xx3xx5xx&#x

x s3s4s

x s3s5s

x x x-n-o = xx  xx-n-oo&#x

x s-2-s-n-s

Of course,multi-prisms do exist as well. And this not only with respect to several perpendicular axes (as in: prism of (prism of (prism of ...)) ),but also in the sense of larger perpendicular objects than a mere product with a line. Cf. here again to theprism product,then within its full generality.

Antiprisms   (up)

(E.g. in 3D cf.n/d-ap for general {n/d} bases.)

As such an antiprism as such is defined only for 3D. Here it might be described as a gyroelongated dihedron.For an according extrapolation into other dimensions it depends on the chosen construction ansatz, which then gets extrapolated.

• Antiprisms as snubs
Antiprisms within 3D can be derived as thesnub (i.e. alternated faceting)of the prisms with even numbered base polygons. Sure, this concept could be extended to higher dimensions as well,then being known asJohnson antiprisms. But because of the decreasing relative amount of degrees of freedom when trying to come back to uniform figures (i.e. equal edge lengths) after the (generally applicable) alternated faceting, this ansatz becomes not too effective. (Rare examples in that sense would besidtidap andgidtidap.)

• Antiprisms as segmentotopes with dual bases
An alternate idea would be to consider the base polygons of 3D antiprisms as a dual pair ofregular polytopes.This ansatz, vialace prisms, clearly extends to any dimension, for 1D it just is point || point and therefore identical to the prism itself, but beyond one uses any linear reflection group graph, assigns for the top layer (left symbol at each node position of the graph) the ringed node "x" at the left-most position,while all others will be marked "o"; for the bottom layer (right symbol at each node position) the ringed node"x" then will be placed at the right-most position, and again all others will be marked "o".Finally this Dynkin diagram gets postfixed by "&#y", wherey gives the relative length of the lacing edges.– As an aside, extending beyond the topic of axials, this ansatz furthermore could be extended onto n-dental reflection group diagramsas well, replacing the lace prisms by (n layered) lace simplices, with a single ringed node at a different end for each layer.

External links

(disambiguation)  (alternated prism / antiprism as a snub)  (antiprism as a segmentotope with dual bases)

It should be emphasized here, by taking dual pairs of regular polytopes, the bases generally will not be the same polytopes,i.e. the top-bottom symmetry generally is lost. It is retained only whenever those are a self-dual pair (as this was the casefor any regular polygon).

Sometimes antiprisms with lateral facet pyramids, which do cross the axis of global symmetry, are also calledretroprisms. E.g. for the5/2-ap the lateral triangles don't, whereas for5/3-ap they do.

Whenever those lacing edges can be chosen to be of the same length as the ones of the base polytopes, we will have a validsegmentotope. Just as for any segmentotope, the lacing facet polytopes all will be segmentotopes in turn.In fact their bases always will be co-dimensional: vertex atop facet (i.e. bottom-up pyramids), edge atop ridge, etc. ..., ridge atop edge, facet atop vertex (i.e. top-down pyramids).

The height of a (uniform)3D antiprism can be calculated using thepolygonal circumradiusr( x-n/d-o ) = 1/(2 sin(π d/n)), the inradiusρ( x-n/d-o ) = sqrt[r² - (1/2)²] = 1/(2 tan(π d/n))and the height of the lacing triangleh( x3o ) = r( x3o ) + ρ( x3o ) = sqrt(3)/2

h( xo-n/d-ox&#x ) = sqrt[(r( x-n/d-o ) - ρ( x-n/d-o ))2 - (h( x3o ))2] = sqrt[(1 + 2 cos(π d/n))/(2 + 2 cos(π d/n))]

Forconvex antiprismatic segmentotopes we have:

Alterprisms   (up)

Alterprisms are just a special case oflace prisms and somehow closely related to a further 3D concept of antiprisms. In fact, a lace prism (in the more specific sense of the term) generally is someA || B, where bothA andB both are given wrt. the same symmetry group. Within this setting then an alterprism would be such an lace prism, whereB=A again, but would benot a mereprism – in fact we can distinguish here several subcases:

• Base polytope allows for adifferent orientation within the same axial symmetry
This type of alterprisms occurs for axial stacks of a symmetry group with an additionalouter symmetry of the (undecorated) Dynkin graph.E.g. for linear Dynkin graphs with additional reflection symmetry. Or for tridental graphs with additional rotation symmetry around the bifurcation nodeor at least with a mirror symmetry between 2 of its legs.(Wrt. the first mentioned group, the linear ones, it becomes clear, thatantiprisms with self-dual bases will be alterprisms as well.)

• Both base polytopes are still aligned in the same orientation, but allow for ashorter height of separation than the mere prism
This would be the case, when the lacing edges won't connect each top vertex with the one directly underneath (as for the prism), but rather a different one instead.Examples here would be all the 3Dpolygrammic antiprisms with even denomiator, but also theJohnson antiprismswith non-chiral bases. Still, there are further such known cases:

• hocucup
• hossdap
• sishia
• ...

External links

For obvious reasonsall these alterprisms (of either type) are at leastscaliform (in its inclusive sense). Therefore lots of examplesare provided on that according page. The most prominent one for sure is the first known scaliform polytope itself,tut || inv tut,which, in the sense of mere lace prisms has been nicknamed "tut-al-tut" (... atop alternated ...), but now can abreviated even shorter to "tut-a" (... alterprism).

Multiple applications of alterprismation will result in analtersquarism, analtercubism, analtertessism, etc. Even so there is a severe restriction to altersquarisms and beyond: the height of the according alterprism itself is bound to be 1/sqrt(2), for else the diagonalsof the according lace city display would not have the right size for additional edges (when all unit edged figures are to be considered).

So for instance the altersquarism oftet is evidently possible: as the alterprism oftet happens to behex and the alterprism ofhex is justhin. Thereforehin is nothing but the altersquarism oftet. The same holds true for all the other hemicubes too: the alter-n-hypercubism of am-hemihypercube is just then+m-hemihypercube!But, nonetheless, this works beyond as well. Consider for instance tutas (aka:pexhin), which is the altersquarism oftut,orritas is the altersquarism ofrit.

Finally a further extension should be mentioned within this context, although no longer belonging truely toaxial polytopes.Consider the alterprisms with an axial symmetry group with an additionalouter symmetry of the (undecorated) Dynkin graph, which isnot only a mere reflection.This applies in the realm of finite polytopes e.g. for the 4Dtridental Dynkin diagramso3o3o *b3oor for the 3Dcyclical Dynkin diagramso5/2o5/2o5/2*a.Those also allow for some rotational or gyrational symmetry, this time by 120° within the diagrammal representation space.Accordingly we no longer could stay with the mere stack of 2 mutually gyrated copies, but we well can consider the simplicial (here: trigonal) arrangement ofallgyrated versions at the same time. Here we consider a 4D resp. 3D "base" space of respective layers and a further 2D position space for the lace city. Accordingly those here described polytopes then would live within 6D resp. 5D. – As alterprisms with symmetrical (undecorated) base diagrams in this view well could be consideredgyroprisms, that considered extended class of polypetons resp. polyterons thus generally is called the one ofgyrotrigonisms.

Right as it is true for the all the gyroprisms, these gyrotrigonisms too all happen to bescaliform, becauseobviously all vertices do fall into a single orbit of global symmetry. Moreover, because a trigon clearly is a 2D segmentotope, these gyrotrigonisms could be seen as a monostratic stack of one such 4D resp. 3D base atop ofa 5D resp. 4D gyrostack of the 2 other such bases. Accordingly these gyrotrigonisms all qualify furthermore asconvex segmentopetaresp.non-convex segmentotera.– Despite of this involved description, there happen to be just 4 gyrotrigonisms within 6D in total. These are named according to their respective base polychoraas hexgyt (aka:tedjak), thexgyt (aka:pextedjak),ritgyt,andtahgyt. Because the undecorated 3D base diagram belongs toGrünbaumian figuresonly, the according 5D gyrotrigonisms likewise would become Grünbaumian in the first run. However there are their respectingly according reductions as well.This then provides 2 more (non-Grünbaumian) gyrotrigonisms:hossidgyt andhodidgyt.

As an aside it should be noted that the further 3D finite cyclical symmetryo5/4o5/4o5/4*a does not allow forgyroprisms because according heights then would become imaginary and that thence gyrotrigonisms are not possible either.

However, this concept in 2022 then got continued conceptually with 7Dgyro-octahedronisms (like hexgyo (aka:odinaq),ritgyo,...), where each oriented "base" is "positioned" at the axial ends of anoctof position space,gyro-cuboctahedronisms (like hexgyco (aka:ihdlaq),ritgyco,...),and 8Dgyro-icositetrachoronisms (like hexgyi (aka:codify),...),where each oriented "base" is "positioned" at all vertices of either of the 3ico-inscribed q-hexes. (Although, as their alternate acronyms reveil, the base individuals each had already been known independently before.)In fact those all happen to exist only because of the height of there being used 5D gyroprisms of those tridental bases being 1/sqrt(2), because then the diagonals of according altersquarisms become unity and thus get used as further edges. However the height of the 4D gyroprisms of those cyclical bases differs so that the above structures of position space become not possible in those cases.

Moreover, in retrospective, it just occurs that examples for alterprisms could be obtained by using simply 2 out of 3 "layers" of a gyrotrigonism,examples for altersquarisms could be obtained by using simply 2 out of 3 "layers" of a gyro-octahedronism, and examples for altertessisms could be obtained by using simply 2 out of 3 "layers" of a gyro-icositetrachoronisms. (The seeming lack of the altercubism here is because the former 2 subsettings become subdimensional, while the latter is not.)

A further idea for application turned up in 2024 by using the concept of gyrotrigonisms witheuclidean components, which then, like in hyperbolic space too, simplycount as horocells (or higher dimensional equivalents). Clearly, by construction these would contain both bounded and infinite directions (like a 3D pillar): the former being built by the position subspace, i.e. the triangular cross-section in case of trigonisms, while the latter is due to the here infinite object subspace.The simplest such figures would betratgyt andthatgyt. But also the symmetryE₆⁺ would allow for several further such cases.

Cupolas   (up)

(E.g. in 3D cf.n/d-cu for general {n/d} and {2n/d} bases.)

As such a cupola is defined onlyfor 3D. Although it is ment as monostratic face-parallel top-section of larger (uniform) polyhedra,it is best defined directly as lace prismxx-n/d-ox&#y. As such, the cupola is nothing but a Stott expansion of thepyramid (as the first node position changes from "oo" to "xx"),accordingly fory = x we get the same heights, i.e.h(xx-n/d-ox&#x ) = h( oo-n/d-ox&#x ), and therefrom the same restriction:6/5 < n/d < 6.Further the base polygon, in order to not become a Grünbaumian double cover, requiresd to be odd.

In order to extrapolate cupolas into spaces ofhigher dimensions, there are different valid possibilities, even within the realm ofsegmentotopes:

1. This extrapolation is based on the observation, that thebottom polytope is thekernel of intersection of a dual pair of the top polytope.(Here the basex-n/d-x of a 3D cupola is read as being the kernel of thecompound ofx-n/d-o witho-n/d-x.)Speaking of dual, the top figures here will be restricted toregular polytopes. Dealing with theirDynkin diagrams those kernels of intersection, (as is described in thetruncation series) in case of odd dimensional top facets, just have the singlemiddle node ringed, resp., for even dimensional top facets, just the two central nodes ringed.

   For according segmentochoraxPoQo ||oPxQo, i.e. the lace prismsxoPoxQoo&#x,the lacings thus would beantiprisms (as subdiagrams:xoPox ..&#x) andpyramids (as:.. oxQoo&#x) only.

2. Thisdifferent extrapolation sticks to the idea of being acap of a larger uniform polytope. It also starts withregular polytopesfor top facets, but asking thebottom facet being the correspondingStott expanded version, i.e. its Dynkin diagram hasboth end nodes ringed.The Dynkin diagram of that larger uniform polytope (of which the cupola would be a cap of) furthermore could be derived by adding "...3x" to the diagram of the top facet.

   The accordingly extrapolated segmentochoraxPoQo ||xPoQx, i.e. the lace prismsxxPooQox&#x have for lacing facetsprisms (as subdiagrams:xxPoo ..&#x),trips (as:xx .. ox&#x, i.e. used as digonal 3D cupola in here), andpyramids (as:.. ooQox&#x). Those then would be thexPoQo-cap of the polychoronxPoQo3x.

3. Other authors like tostick to the 3D axial segment to be an according cupola, while the higher-dimensional tail of the Dynkin diagram remains unringed.I.e. here we would have esp.xPoQo ||xPxQo to be "the" 4D cupolae, then having as lacing facets these saidcupolaexxPox ..&#x in turn andpyramids.. oxQoo&#x.

4. It should be noted, that in amuch looser sense, sometimesany possible monostratic stacking of Dynkin symbols, i.e. anylace prism, with non-degenerate bases (esp. neither pyramid nor wedge), which not qualifies as prism or other more specific terms (e.g. not an antiprism), might be termed "cupola".

Case A) is the reading of the term "cupola", which the authorprefers. For case of B) the author rather prefers the termcap instead.Noteworthy type C) clearly can start for 3D only (the 3D cupolae themselves) and additionally would have even less entries for spherical space.Finally D) is mentioned here for awareness only, and not too much endorsed by the author.

Forconvex cupolaic segmentotopes we have:

Cuploids   (up)

Cuploids are a completely3D specific concept. They are somehow related toseveral of the uniform polyhedra, which do not emanate directly byWythoff's construction, but in fact are reduced forms ofGrünbaumian polyhedra.The same holds true here: as pointed outabove, the denominatord of the top polygonx-n/d-ohas to be odd, else the cupola gets a Grünbaumian double-cover polygon for bottom base. Exactly in those prohibited cases, i.e. ford beingeven, that offending face will just be withdrawn, and the openbut pairwise coincident edges will be reconnected in the obvious way.

This picture, showing a {7/4}-cuploid, reflected in a mirror, was rendered by C. Tuveson in 2001 in reply to a post of mine: "But your structure reminds me to a true polyhedron with a{7/2}-heptagrammic edge circuit at the bottom and a {7/3}-face at the topside, joined to one another by squares plus trigons (the latter pointingtowards the vertices of the top-face). It is the retrograd {7/3}-cuploid,or might also be called the {7/4}-cuploid. As it is well known, thebottom-face of a {n/d}-cupola is a {(2n)/d}; but for 7/4 this becomes thereducible number 14/4, which is nothing but the (reduced) {7/2} with adouble circuit. Thereby the latteral sides (squares and trigons) join atthe bottom edges, and the bottom face is obsolete (or would have to becounted twice, giving rise to pairwise coincident edges)."

It should be noted additionally that, as long as the top face is not retrograde, i.e. as long asn/d > 2, central partsof the top baseoffers both sides to the outside because of the bottom base reduction. Faces having this property generally are calledmembranes.

Within the bounds, also provided above, cuploids exist as segmentohedra. Their bottom face just will have to be marked "pseudo".As this adjective is not transferable into Dynkin diagrams, a lace prism description does not exist.Further, asd has to be even and thusn/d can no longer be integral, clearly there isno convex segmentotope.– As (non-convex) examplesthah = {3/2}|| pseudo {6/2} andstiscu = {5/2}|| pseudo {10/2} might serve.Within 4D there isfirp = tet|| pseudo 2thah and theco retro-cuploid = co|| pseudo 2oct+6{4}.

Cupolaic Blends   (up)

©  

Also being ment for3D in the first run, those are miming thecuploids in the complemental cases, i.e. ford being odd again.In those cases nomal cupolas do exist. Sure, in a locally similar manner, 2 copies each can beblended in an axially gyrated way.This blending operation thereby withdraws the doubled up bottom face, while the top face becomes a regularcompound.(Although those clearly aresegmentotopes, those top face compounds willnot be convex.)The easiest ones here are:

• "2{2}|| pseudo {4}" (tutrip), the blend of 2trip, which looks kind a Phillips head.
• "2{3}|| pseudo {6}", which again has amembrane.

Nonetheless, this type of operation surely might look to apply also to pairs of (either way)higher dimensionally extrapolated cupolas withdual top bases.But because then those dual top bases generally no longer have the same circumradius, the height of the to be blended cupolas toomust no longer be the same. Thus the top bases then generally would not result in a compound but are arranged in parallel layers, i.e. the blend would not be monostratic anymore. Therefore the research for cupolaic blend segmentotopes (i.e. being monostratic)would have to restrict to top bases which areselfdual only (or otherwise would assure to have the same circumradius). Examples here are

• "so|| pseudooct" (turap) – as type A cupola extrapolation –, cell count is: 1so, 8tet, 8oct
• "so|| pseudoco" (hocoaso) – as type B cupola extrapolation –,cell count is: 1so, 8+12trip, 8tet (here the 12 trips themselves might be blended further into 6tutrip, thereby omitting even the bottom square each. Its cell count thus becomes instead:1so, 8trip, 8tet, 6tutrip)

Clearly there is a way to extrapolate at least the Philips head (tutrip) into higher dimensions. However the bottom base dimension then corresponds to the number of orthogonally beingblendedhypercubes, so that for odd dimensions that very bottom base no longer would result in a hollow figure. Moreover those polytopes then would become wild, because e.g. within 4D the square and the triangle of a latteraltutripand a square of the bottomcube all are incident to a common edge and thereby still are corealmic. Within 4D we have 2 such extrapolations, the blend of 3squippyps ("coord-axes edge star"|| cube) and the somewhat tallerblend of 3tisdips ("coord-planes square star"|| cube).As a third possibility the then non-wild blend of 3squafs ("thah-squares star"|| cube) is to be added here.

But then there is a further application of this very concept of cupolaic blends into higher dimensions as well.Note that the main 3D examples shown within the picture above all have in common that one base of the to beblendedidentical components has some symmetry which the other base does not have. Within the examples of the picture those are an halved angle of axial rotation. In April 2023 a similar applications of this concept got found for4D:

• "siddo|| pseudocube" (hocubasiddo) – wrt. a pair ofcubaikes,thereby doubling up theike base into the accordingcompound of 2 (siddo), while the oppositebase will be blended out – containing 6tutrip in turn,thence the hollowness of the bottom base creeps up at those lacing facets as well
• "so|| pseudoco" (hocoaso) – wrt. a pair oftetacoes,thereby doubling up thetet base into the accordingcompound of 2 (so), while the oppositebase will be blended out – containing 6tutrip in turn,thence the hollowness of the bottom base creeps up at those lacing facets as well

Shortly thereafter it even was driven into5D as well:

• "disti|| pseudorico" (horico adisti) – wrt. a pair ofricoasadis,thereby doubling up thesadi base into the accordingcompound of 2 (disti), while the oppositebase will be blended out – containing 24hocubasiddo and 24hocoaso in turn,thence the hollowness of the bottom base creeps up at those lacing facets as well

Fastegia   (up)

Fastegium here derives from latinfastigium, the pediment. Accordingly, asarbitrary dimensional analogue just

• take 3 copies of any polytope,
• use 2 of them to build a prism,
• use that prism as lower base,
• and place the 3rd copy atop that.

Thus, written as a lace prism, it is justoy ...xx...&#z, wherex, y, z all define edges of possibly different lengths.The lacing facets accordingly are 2 prisms, similar the bottom one, but now with lacing edgesz (while the bottom one hasy-lacings),plus, as far as the starting figure had any facets itself, the subdimensional fastegiums derived by those.(Sometimes even the top layer is allowed to differ in size, yieldingoy ...wx...&#z.)So, fastegia are special cases ofwedges.

Ifz = y (andw = x) this figure can be rewritten as lace simplex or even as duoprism.oy ...xx...&#y = ...xxx...&#y = y3o ...x....If furtherall edges have equal size and the starting polytope was someorbiform, sayQ, then the fastegium will be a validsegmentotope (in factQ|| Q-p).Accordingly the segmentotopal heightthen generally will beh = sqrt(3)/2.

Moreover, wheneverQ would be additionally uniform, then the fastegium clearly will be uniform itself, being nothing but the3,Q-dip.

Forconvex fastegmal segmentotopes we have:

ox oo&#x      = ooo&#x       = x3o o

ox xx&#x      = xxx&#x       = x3o x

ox xx-n-oo&#x = xxx-n-ooo&#x = x3o x-n-o

This small table already shows, provided P would be any convex, unit-edged, andorbiform polytope of any dimension, that the set of convex fastegmal segmentotopes could equivalently be described as the set of according 3,P-duoprisms.

Antifastegia / Alterfastegia   (up)

The antifastegium is essentially built the same way as a (normal)fastegium, just that its subdimensional top layer polytope is replaced by its dual. Speaking of duals, this already asks the starting figure itself to be a regular polytope.Those generally areWythoffian and therefore moreoverorbiform.

Any antifastegium can be written as lace prism, which isoy xo...ox&#z in general, where againx, y, z all define edges of possibly different lengths.The facets of an antifastegium are its bottom prism(.y) (.o)...(.x), the 2 antiprism connecting either bottom base to the top base.. xo...ox&#z, and finally for any other sub-element of the bottom prism there will by an according co-dimensional sub-element of the top base, which has to be adjoined.

Finally, antifastegia which have unit edges only, clearly aresegmentotopes.

Forconvex antifastegmal segmentotopes we have:

ox oo&#x             = ooo&#x

ox xx&#x             = xxx&#x

ox xo-n-ox&#x        = xxo-n-oox&#x

ox xo3oo3ox&#x       = xxo3ooo3oox&#x

ox xo3oo3oo3ox&#x    = xxo3ooo3ooo3oox&#x

ox xo3oo4oo3ox&#x    = xxo3ooo4ooo3oox&#x

ox xo3oo3oo3oo3ox&#x = xxo3ooo3ooo3ooo3oox&#x

(The more general, not necessarily convex case of 4D then is then/d-af. Various individual examples then are provided in those general group with links each.)

Quite similar as for theantiprisms there has been an extension towards non-regular bases by means of the term ofalterprisms. Here too we could definealterfastegia to be thelace simplex of somepolytope (the symmetry group of which has some additional outer symmetry) atop the prism of an alternately oriented version of the same polytope. Accordingly e.g.tutaltuttip could be shortened to "tutaf".Or there even is something like arappaf, ahexaf, arixaf, ahinaf, or ajakaf.

Duoantifastegia & Duoantifastegiaprisms   (up)

Aduoantifastegiaprism always can be written as a lace prism in the formxo...ox yo...oy&#z, where againx, y, z all define edges of possibly different lengths. Here the subelementsxo...ox&#z andyo...oy&#z would berequired to describe antiprisms each. Accordingly the bases here would be required to be (bidually aligned copies of) duoprisms oftwo regulars each. (In fact, the term duoantifastegiaprism was chosen as a contraction from duoprism duoantifastegium.)

The generalconvex duoantifastegiaprismal segmentoteron here would ben,m-dafup =xo-n-ox xo-m-ox&#x.In higher dimensions however, the 2 duoprism factors no longer would be bound to equating dimensions.Examples would be the 6Dtratetdafup =xo3ox xo3oo3ox&#x or the 7Dtrapendafup =xo3ox xo3oo3oo3ox&#x andtrahex dafup =xo3ox xo3oo3ox *d3oo&#x.

Similarily to the generalization of alterprisms above antiprisms here too the duoantifastegiaprism could be extrapolated a bit, when not being restricted tobases which are duoprisms of regular components only, but allowing for duoprisms of not necessarily regular components as well.Thus then that more general duoantifastegiaprism would require the other layer to be its bi-alternate version (instead of just bi-dual).An example here would be the 7Dtrarapdafup =xo3ox oo3xo3ox3oo&#x.

Aduoantifastegium then can be given as the specific cases, where one of the required subelemental antiprisms becomes a (stretched)tet (xo ox&#z). This clearly makes the duoprisms of the 2 bases degenerate, i.e. the bases would become subdimensional. (This is what reduces one prismaticpart from the defining duoprism, and therefore too from the contracted name.)

The generalduoantifastegial segmentoteron here would ben/d-daf =xo ox xo-n/d-ox&#x (which moreover is convex for d=1).

As an aside it should be pointed out, that the sequence of used operations (i.e.prism product andstacking) heredoes not commute: e.g.the number of vertices of(x3o x3o) || (o3x o3x) clearly is (3·3) + (3·3) = 18, whereas(x3o ||o3x) (x3o ||o3x) would result in (3+3) · (3+3) = 36.

Duoalterprisms   (up)

The termduoalterprism was made up as a kind of a false hybrid of the terms of analterprism and the aboveduoantifastegiaprism. In fact it is considered to be ategum sum (instead of the lace prism) of duoprismatic layers as in the latter case, however those needn't be taken as factor-wiseantiprismsbut rather are allowed to be alterprisms instead. Thus, in short, those could simply be considered to betegum sums of 2 bi-alternated duoprismatic "layers".Esp. those always happen to bescaliform.

Thence as examples there are e.g.

• pexhax -xo3xx3ox xo3oo3ox&#zx, the tetrahedral-truncated-tetrahedral duoalterprism
• pabex hax -xo3xx3ox xo3xx3ox&#zx, the truncated-tetrahedral duoalterprism
• idinaq -xo3ox xo3oo3oo3oo3ox&#zx, the triangular-hexateric duoalterprism
• thidlaq -xo3ox oo3xo3oo3ox3oo&#zx, the triangular-rectified-hexateric duoalterprism
• tijakdap -xo3ox xo3oo3oo3oo3ox *e3oo&#zx, the triangular-icosiheptapetic duoalterprism

Bipyramids   (up)

The bipyramids are the outcome of thetegum product. Take any polytope as a base and a single line segment in orthogonal space, either one being centered at the origin. Then the projective scaling, centered atthe ends of the segment, will outline the derived pyramid within the interval from the point up to the given base.At this level nothing is said about dimensions, convexity of the base, nor about edge lengths.

Clearly, bipyramids are closely related topyramids. In fact those are just externalblendsof 2 pyramids, being adjoined at their base polytopes. Therefore that defining polytope itself willnot be contributeas a true facet of the outcome, as it thereby would be blended out. Yet it can be considered a pseudo face.Likewise it might be possible to dissect the bipyramid into 2 pyramids while squeezing inbetween an equatorial prism. This then is what is meant byelongated bipyramids.

As far as the base polytope is anon-snub, has aDynkin diagram descriptionand isorbiform, the whole pyramid, bipyramid and even the elongated bipyramids will have a Dynkin diagram too. At least in the sense of alace tower with appropriately scaled lacings. For bipyramids just pre- and postfix any node symbol (thus either being anx or ao) by an unringed node (o).And finally add to the obtained symbol a final "&#y", where the relative size ofy is again a function of thedesired height of the pyramid on either side, because literally it provides the lacing length. – Although pyramids can be built on a snubbed base, the required symbol cannot be obtainedin the just described way, because the processes of alternation and setting up the tegum product do not commute.Orbiformity of the base was required, as else those lacings cannot all be of a single length.

Forconvex bipyramids, which generally are external blends of segmentotopes, we have:

(Lately Bowers prefers the acronym suffix X-t, here representing X(,line)-tegum, rather then the older X-dpy.)

Elongated Pyramids   (up)

As the base angle of apyramid always is smaller than 90 degrees, an according externalblend with a matchingprism thus is convex whenever thepyramid itself was.

Forconvex elongated pyramids, which generally are external blends of segmentotopes, we have e.g.:

Elongated Cupolas   (up)

Obviously the same construction as forelongated pyramids applies for theirStott expansionswrt. their across symmetry. This is what is understood within 3D as the well-known elongated cupolas in the set ofJohnson solids. But, as has been outlined above, the extension of the meaning of the termcupola is quite ambiguous beyond 3D. In fact, those expansions only would build a subset of thetype C cupolas, or would be a superset of the type B cupolas. The type A cupolas on the other hand have to beinvestigated individually whether the 90 degrees condition is being met forall bottom base angles.

Forconvex elongated cupolas (in the sense of across symmetrically Stott expanded elongated pyramids), which then generally are external blends of segmentotopes, we have e.g.:

Edge-Expanded Biprisms   (up)

This concept is meant for 3D and was invented as an infinite series of polyhedra in summer 1999 by the author. It starts with the (exterior)blend of 2prisms,i.e. thelace towerxxx-n/d-ooo&#xt, withn ≥ 2d (i.e. progrades). This one would be the {n/d}-gonal (0,0)-EEB.

Now unconnect the lacing edges, and insert triangles inbetween the lacing squares.Here generally there are 2 possibilities: either by bending the squares inward, thus looking like an exteriorblend at the bottom face of 2 retrogradecupolas (orcuploids),the {n/d}-gonal (exo-) (1,1)-EEB; or by bending the squares outward, thus looking like the corresponding blend of 2 prograde cupolas (or cuploids), {n/d}-gonal (endo-) (1,0)-EEB.

Instead of inserting a single triangle into that lacing gap at either segment, one equally could insertktriangles each. I.e. the base-vertices are [n/d,4,3^k,4] (up to windings, see below). While at the central layer there are vertex types [3²,4²] (fork>0) and[3⁴] (fork>1). Again there are exo- and endo-types, relating to emanating triangles to the relative outside resp. to the inside (best seen in the equatorial section); equivalently exo means that the lacing squareswill bend inward, endo describes the cases where those squares bend outward.Here one also speaks of k-extended (exo-/endo-) EEBs.

Fork>1 there even are several possibilities when the sequence of the equatorial edges of asquare pair, of those inserted triangle pairs, and of the next square pair winds less or more than once around their orthogonal, vertical axis. This winding numberw will be the second parameter of the general {n/d}-gonal (k,w)-EEB. Those are restricted to0 ≤ w ≤ k-1.(In order to get a planar equatorial layer, the bending of the squares will have to be adapted accordingly.)– Here some examples are in place. The following pictures show parts of the base polygons inred,the equatorial edges between the squares inblue, and those between triangles inblack.

exo (3,0)-EEB, the winding KTSL around O is < 2π	endo (3,0)-EEB, the winding KTSL around O is < 2π	endo (3,1)-EEB, the winding KSTL around O is > 2π, but < 2 · 2π

In order tocalculate the height consider the internal vertex angle of the base polygonx-n/d-o. That one is

∠AOB = π(1-2d/n)

For endo-EEBs (withk>0) one adds 2 right angles (∠AOK,∠LOB) plus2πw. That total angle then will be devided intokequal parts, each being the angle sustained by any equatorial triangle edge:

∠TOS = 2π(1+w-d/n)/k

On the other hand, for exo-EEBs (withk>0) one has to subtract from∠AOB those 2 right angles. But one will have to add2π(w+1) here. Thus the total angle a posteriori will be the same, and so too eachangle sustained by any equatorial triangle edge gets the same number as for the endo case.

Using unit edges and the radius of those equatorial vertex circlesr = |OK| = |OL| = |OT| = |OS| within the right triangle, which is the half ofTOS, one getsr sin(∠TOS/2) = 1/2.On the other hand the total height of the EEB ish = 2 sqrt(1 - r²). Thus(independing of exo- or endo-)

h( {n/d}-gonal (k,w)-EEB ) = sqrt[4 - 1/sin2(π(1+w-d/n)/k)]

This height formula also shows the range ofn/d such that for any given value of(k,w)both the exo- or endo-forms of an {n/d}-gonal (k,w)-EEB do exist, i.e. provide a height withh>0.

endo {7} (k,w)-EEB

clicking the drawings provides the corresponding preview in the canvas right
k = 2 k = 3 k = 4 k = 5 w = 0 endo {7} (2,0)-EEB endo {7} (3,0)-EEB endo {7} (4,0)-EEB endo {7} (5,0)-EEB w = 1 endo {7} (3,1)-EEB endo {7} (4,1)-EEB endo {7} (5,1)-EEB w = 2 endo {7} (4,2)-EEB endo {7} (5,2)-EEB w = 3 endo {7} (5,3)-EEB	⭳©

exo {7} (k,w)-EEB

clicking the drawings provides the corresponding preview in the canvas right
k = 2 k = 3 k = 4 k = 5 w = 0 exo {7} (2,0)-EEB exo {7} (3,0)-EEB exo {7} (4,0)-EEB exo {7} (5,0)-EEB w = 1 exo {7} (3,1)-EEB exo {7} (4,1)-EEB exo {7} (5,1)-EEB w = 2 exo {7} (4,2)-EEB exo {7} (5,2)-EEB w = 3 exo {7} (5,3)-EEB	⭳©

Somespecial cases clearly are the {n/d}-gonal endo (1,0)-EEBs. Those are also known as (prograde) orthobicupola. The special casesn/d = 3/1, 4/1, 5/1 belong to theJohnson solids, in fact those aretobcu (J27),squobcu (J28), resp.pobcu (J30). –The {n/d}-gonal exo (1,0)-EEBs then are the corresponding retrograde orthobicupola, i.e. orthobicupola with top base{n/(n-d)} (withn ≥ 2d).

Other special cases are the {n/d}-gonal endo (2,1)-EEBs. Those are also known as sphenoprisms, i.e. the connection of the bases by (pairs of) triangle-square-triangle sphenoids. The special casen/d=2 here again is a Johnson solid, in fact theesquidpy (J15). But even within the range2≤n/d<3 all those {n/d}-gonal endo (2,1)-EEBs are at least locally convex. (The limiting casen/d = 3 then would become flat.)

One even could extrapolate EEBs toretrograde {n/d}, i.e. to {n/(n-d)} within the so far assumed prograde boundn ≥ 2d. This extension would switch endo- and exo EEBs. For sure, the parameterk is un-affected. But withw' = k-w-1 one gets the identity

{n/d}-gonal exo (k,w)-EEB = {n/(n-d)}-gonal endo (k,k-w-1)-EEB{n/d}-gonal endo (k,w)-EEB = {n/(n-d)}-gonal exo (k,k-w-1)-EEB

which shows, that such retrograde bases not truely produce anything new.

Edge-Expanded Bi-Antiprisms   (up)

©

Just as theEEBs start with the exterior blend of 2 {n/d}-prisms, the EEAs, i.e. edge-expanded antiprisms, are meant to start with the appropriate blend of 2 {n/d}-antiprisms. But, in fact, this would become rather thek=1 cases. We even could start instead by a pair of {n/d}-pyramids, which are mirrored at their tips. This then will become the general {n/d}-gonal (0,0)-EEA. Iterated insertion of triangle pairs at the lacing edges of thatbipyramid produces – similar to the EEBs – a new set of {n/d}-gonal exo/endo (k,w)-EEAs. Here the former prism-squares (of the EEBs) clearly are replaced by the lacing antiprism-triangles. – The EEAs where found by J. McNeill.

exo {5} (2,0) EEA

endo {7} (k,w)-EEA

clicking the drawings provides the corresponding preview in the canvas right
k = 2 k = 3 k = 4 k = 5 w = 0 endo {7} (2,0)-EEA endo {7} (3,0)-EEA endo {7} (4,0)-EEA w = 1 endo {7} (4,1)-EEA endo {7} (5,1)-EEA	⭳©

exo {7} (k,w)-EEA

clicking the drawings provides the corresponding preview in the canvas right
k = 2 k = 3 k = 4 k = 5 w = 0 exo {7} (2,0)-EEA exo {7} (3,0)-EEA exo {7} (4,0)-EEA exo {7} (5,0)-EEA w = 1 exo {7} (3,1)-EEA exo {7} (4,1)-EEA exo {7} (5,1)-EEA w = 2 exo {7} (5,2)-EEA	⭳©

The height can be calculated along the same lines as were shown for the EEBs. The height of the {n/d}-gonal endo (k,w)-EEA reads as follows. (Those for the exo versions could be deduced therefrom by substituting w → w* = k-w-1.)

h( {n/d}-gonal endo (k,w)-EEA ) = sqrt[4 - 1/sin2(π(1+w-d/n)/(k+1))]h( {n/d}-gonal exo (k,w)-EEA )  = sqrt[4 - 1/sin2(π(k-w-d/n)/(k+1))]

Ursatopes   (up)

In 2004 A. Weimholt came up with an interesting set of polytopes which later became known asursatopes (orursulates).The 3D member of this large family, which since got lots of extensions, isteddi.This is why the whole family (which then will be forced to contain it as sub-dimensional boundary) was named ursatopes.

The general setup here later was mainly elaborated by W. Krieger. In fact, these polytopes derive from a cone where the edges of the bottom base become acute golden triangular sides (i.e. ox&#f) of a pyramid. From that cone one consideres first asection of lacing edge length 2f. The remainder then is a truncation of that pyramid: Chopping off the tip at half of its height,revealing thus the ursatopal bottom base as this crosssection, resp. chopping off the other vertices in such a way,that the remaining lacings get reduced from their f-length down to x-sized edges only, while the pyramidal bottom edges getreduced to zero, i.e. this ursatopal top face will be the rectification of the bottom one. –The new additional lacing edges introduced by the latter choppings by construction will have unit size. But the existence ofthe to be producedrectification as well as the to be obtained unit edge sizes all overraise some restrictions on the possible ursatopal bottom bases. And also the height of that starting pyramid would be required to be positive.

All this can be cast into the following necessary and sufficient conditions:

• Use a quasiregular Dynkin symbol only containing links marked 3 (or 2) incident to the marked node for base layer.
• The polytopal circumradius corresponding to this former Dynkin symbol ought be < f = (1+sqrt(5))/2 = 1.618034 (in edge units).
• Use the same, f-scaled Dynkin symbol for medium layer.
• Use the rectification of the base layer as top layer. (According to the Conway-Hart rule the rectification amountshere just in unmarking that special node, while marking instead all its directly neighbouring nodes.)

These then could be subject toStott expansions within the across subsymmetry.

Thesimplexial ursatopes as well as theorthoplexial ones moreover are also enlisted in a more detailed comparision on the dimensional analogs page.

It happens that ursatopes areorbiform in general(or, if they would contain non-unit edges, at least circumscribable).E.g. for thenon-expanded ursatopes one even can calculate their full-dimensional circumradiusRfrom the subdimensional circumradiusr of the quasiregular base according to: R² = f² (f r² + 1/4) / (f² - r²) ,where f = (1+sqrt(5))/2 = 1.618034 is the golden ratio. That one moreover shows that those axial versions with H₄ across symmetry don't exist, because those become degenerate (having zero heights) – simply because there one hasr = f. This already got reflected within point 2 of the preconditions.

Tutsatopes   (up)

Thetutsatopes (ortutisms) are a quite similar genuinely bistratic polytopic family as theursatopes. They just substitute within their role theteddies bytuts. The accordingly changed necessary and sufficient conditions then reads like this:

• Use a quasiregular Dynkin symbol only containing links marked 3 (or 2) incident to the marked node for base layer.
• The polytopal circumradius corresponding to this former Dynkin symbol ought be <x = 1 (in edge units).
• Use the same, u-scaled Dynkin symbol for medium layer.
• Use thetruncation of the base layer as top layer. (According to the Conway-Hart rule the truncation amountshere just in marking additionally all the directly neighbouring nodes of that special node.)

But this class of polytopes happens to be describable in a closed form. In fact those are nothing but theP-first bistratic caps ofQ, whereP clearly is just that quasiregular base of the tutsatope, andQ can be obtained in Dynkin symbol description from that ofP by adding to that single ringed node a further leg, marked 3, the other end of which has to be ringed as well: E.g. thedot-based tutsatopeooo3oox3xux3oox3ooo&#xtis just thedot-first bistratic cap oftim.

These base tutsatopes could be understood even better. LetP be again that quasiregular base.Then the tutsatope is defined asP || u-scaled P || truncated P. But this in turn is nothing but thetruncation ofpoint || P, i.e. of theP pyramid.

Again those base tutsatopes could be subject toStott expansions within the across subsymmetry. Here the implied additional ringings on the Dynkin diagram of the base layerP can be correspondingly transfered onto that ofQ.

Therefore both, all the base tutsatopes and all their expansions, trivially areorbiform in general.

It happens that those axial versions with F₄ across symmetry don't exist, because those become degenerate (having zero heights). This already got reflected within point 2 of the preconditions.

Trippescu series   (up)

Liketeddi = ofx3xoo&#xt gave rise for the familly of convex bistratic orbiformursatopes andtut = xux3xoo&#xt gave rise for the familly of convex bistratic orbiformtutsatopes,trippescu = reduced( ofx3/2oxx&#xt by {6/2} ) too provides an according familly ofnon-convex bistratic orbiform polytopes.

Peratopes   (up)

Liketeddi = ofx3xoo&#xt gave rise for the familly of convex bistratic orbiformursatopes andtut = xux3xoo&#xt gave rise for the familly of convex bistratic orbiformtutsatopes, and just abovetrippescu = reduced( ofx3/2oxx&#xt by {6/2} ) gave rise for the familly of generallynon-convex bistratic orbiforms of thetrippescu series,so toopero = ofx5xox&#xt provides an according familly of bistratic orbiform polytopes.Although this series well has convex members too, because of the large circumradius ofpero itself, it naturally also gives rise to re-entrant members as well.

Elongated Bipyramids   (up)

These can be considered as dissectedbipyramids with a squeezed in equatorial prism.Therefore, whenever the bipyramid was Dynkin describable, the elongated one will be too, just double up the medial vertex layer symbol at eachnode position. I.e. whenever the (sectional) base had anx, then the pyramid would have anox, the bipyramid anoxo,and the elongated bipyramid would require anoxxo node. (Similar foro nodes.)

Forconvex elongated bipyramids, which generally are external blends of segmentotopes, we have:

Supersemicupola   (up)

In the research forn/d,n/d,3-acrohedra –an acrohedron is a polyhedron containing acrons (or vertices), whereacron stems from Greek ακροσ (acros, i.e. summit), as in Acropolis –M. Green found in October 2005 an 7,7,3-acrohedron, which he called asupersemicupola.Based on that finding a small family of n/d,n/d,3-acrohedra was set up according the generalized building rules thereof:

1. Use just the edges of a pseudo {n/d} polygon (resp. polygram) for base.
2. At each such edge both a triangle and an {n/d} will be attached.
3. Any other side of the triangles will cross-attach to the neighbouring {n/d}
   (thus producing [3,n/d,3/2,n/(n-d)] vertex figures at the first vertex level).
4. The next open edges of those {n/d} then will be joined to other {n/d}.
   (This now produces the desired n/d,n/d,3-acrons, i.e. [3,n/d,n/d] vertex figures, at the second vertex level.)
5. If further open sides of the {n/d} exist, again a triangle will be introduced next
   (then resulting in further [3,n/d,n/d] vertex figures).
6. Repeat this process of adjoining alternate triangles and {n/d} –or try to close the polyhedron in any other appropriate way.

Clearly n/d only ranges according to 12/5 < n/d < 12, as in those extremal values the acrons would become flat.

For cases with n being even there generaly would be an easier acrohedron too. In fact, those according to Green's rule then just describes a gyratedblend of 2 such easier ones. (This is quite similar as for thecupolaic blends.)

The following list provides the known n/d,n/d,3-acrohedra which follow thatGreen's rule.

{n/d}	Name		(related easier acrohedron)
3/1	(degenerate): just the Grünbaumian double-covered skin-surface of the 3-fold pyramid		tetrahedron
4/1	tutrip, "Phillips head"		trigonal prism (as digonalcupola)
5/1	ike-faceting ike-5-5		(none)
5/2	sissid-faceting sissid-5-5		(none)
6/1	tutut (has amembrane)		truncated tetrahedron
7/1	(small) supersemicupola (has amembrane)		(none)
7/2	great supersemicupola		(none)
8/1	tutic (is a tube)		truncated cube
8/3	tuquith		quasitruncated cube
10/1	tutid (is a tube)		truncated dodecahedron
10/3	tuquit gissid		quasitruncated great stellated dodecahedron

Cuneprisms   (up)

Only in 2025 a discord member calling himself "Harsin sinquin" came up with this series of chiral, regular faced polyhedra.In fact those are based by concept on a central (chiral isogonal variant of an)antiprism plus lateral, chirally attached wedges.Here the lacing zig-zag of the antiprism variant alternates with x-sized (for up) and y-sized (for down) edges, where the x-edges have the same size as the base edges,while the length of the y-edges is free to be adjusted secondarily. The wedges now burry those y-edges underneath, so that the result becomes all unit-edged again. In fact, the requirement that those wedge-triangles all are regular ones as well, thereby settles the size of those (internal) y-edgesafter all.

The Greek root for wedge, being "spheno" is already getting used, esp. withinJohnson solids, in fact then refering to lune-based ones.Moreover, in the context ofpolychora that Greek term has already been used in a different wedge-sense meaning as well.Thence, in order not to further mix up that reading, in here the Latin root for wedge, being "cune" has been taken over.

These cuneprisms generally have a singleconcave edge type and always are biform (i.e. same asuniform,except that there are 2 different vertex types).

©

n/d-cuneprism2n  * |  2 1  1  1  1 0 | 1  2  2  1  [n,35] * 2n |  0 0  1  1  1 1 | 0  1  1  2  [34]------+-----------------+----------- 2  0 | 2n *  *  *  * * | 1  1  0  0  ry 2  0 |  * n  *  *  * * | 0  0  2  0  gg (concave) 1  1 |  * * 2n  *  * * | 0  1  1  0  yg 1  1 |  * *  * 2n  * * | 0  1  0  1  yb 1  1 |  * *  *  * 2n * | 0  0  1  1  gb 0  2 |  * *  *  *  * n | 0  0  0  2  bb------+-----------------+----------- n  0 |  n 0  0  0  0 0 | 2  *  *  *  (red) 2  1 |  1 0  1  1  0 0 | * 2n  *  *  (yellow) 2  1 |  0 1  1  0  1 0 | *  * 2n  *  (green) 1  2 |  0 0  0  1  1 1 | *  *  * 2n  (blue)

or (as tower):n * * * | 2 1 1 1 1 0 0 0 0 0 | 1 2 1 1 1 0 0 0  [n,35]* n * * | 0 1 1 0 0 1 1 0 0 0 | 0 1 1 1 0 1 0 0  [34]* * n * | 0 0 1 1 0 1 0 1 1 0 | 0 0 1 0 1 1 1 0  [34]* * * n | 0 0 0 0 1 0 1 1 1 2 | 0 0 0 1 1 1 2 1  [n,35]--------+---------------------+----------------2 0 0 0 | n * * * * * * * * * | 1 1 0 0 0 0 0 0  ry1 1 0 0 | * n * * * * * * * * | 0 1 0 1 0 0 0 0  yg1 1 0 0 | * * n * * * * * * * | 0 1 1 0 0 0 0 0  yb1 0 1 0 | * * * n * * * * * * | 0 0 1 1 0 0 0 0  gb1 0 0 1 | * * * * n * * * * * | 0 0 0 1 1 0 0 0  gg (concave)0 1 1 0 | * * * * * n * * * * | 0 0 1 0 0 1 0 0  bb0 1 0 1 | * * * * * * n * * * | 0 0 0 0 1 1 0 0  gb0 0 1 1 | * * * * * * * n * * | 0 0 0 0 1 0 1 0  yg0 0 1 1 | * * * * * * * * n * | 0 0 0 0 0 1 1 0  yb0 0 0 2 | * * * * * * * * * n | 0 0 0 0 0 0 1 1  ry--------+---------------------+----------------n 0 0 0 | n 0 0 0 0 0 0 0 0 0 | 1 * * * * * * *  (red)2 1 0 0 | 1 1 1 0 0 0 0 0 0 0 | * n * * * * * *  (yellow)1 1 1 0 | 0 0 1 1 0 1 0 0 0 0 | * * n * * * * *  (blue)1 1 0 1 | 0 1 0 1 1 0 0 0 0 0 | * * * n * * * *  (green)1 0 1 1 | 0 0 0 0 1 0 1 1 0 0 | * * * * n * * *  (green)0 1 1 1 | 0 0 0 0 0 1 1 0 1 0 | * * * * * n * *  (blue)0 0 1 2 | 0 0 0 0 0 0 0 1 1 1 | * * * * * * n *  (yellow)0 0 0 n | 0 0 0 0 0 0 0 0 0 n | * * * * * * * 1  (red)

When re-considering the construction-wise re-adjustment of these burried pseudo y-edges, it becomes apparent, that this series of cuneprismsafter all even can be generalised towardscune twisters as well.