Site Map  Polytopes  Dynkin Diagrams  Vertex Figures, etc.  Incidence Matrices  Index 
"Polytope" is the general term of the sequence:
(Plural forms for "polytope" and those terms up to dimension 2 is built in the english way by an suffixing "s"; for the higher terms it is built in the greek way, replacing the suffixing "on" by an "a".)
The pages, provided by the links given above, generally intend to list all possible polytopes of the below following types. From dimension 8 on, only the quasiregulars are listed. – The therein given acronyms usually are given by Bowers. – The given circumradii (for uniform polytopes) respectively heights (of lace prisms or towers) are to be taken for unit edged polytopes.
A corresponding explicite listing of Euclidean tesselations of the first few dimensions can be found separately. Furthermore, hyperbolic ones are listed separately too.
Taken settheoretically, a polytope is a partially ordered set of sets. Each of those latter sets is given an order, the empty set has order 1. Any set of order k+1 contains several sets of order k. This connection is also called incidence. The elements of order k are called
k  Name 
1  Nulloid 
0  Vertex 
1  Edge 
2  Face 
3  Cell 
n2  Ridge 
n1  Facet 
n  Polytope 
In terms of the extremes of this table, together with symmetry, and the restriction to regular polygonal faces only, polytopes can be grouped like this (although the more general group names often are not taken exclusively only, and the ones associated to names derive from the polyhedral case)
equivalent facets? = isohedral 
equivalent vertices? = isogonal 
convex?  Name 
yes  yes  yes  Platonics 
no  yes  yes  Archimedeans & Prismatics 
no  only alike  yes  single addition: J37 
yes  no  yes  generally: Catalan solids (the duals of Archimedeans etc. – but those fail to have regular polygonal faces only) 
only alike  no  yes  Deltahedra, Tetrahedrochora, etc. 
no  no  yes  Johnson solids, etc. 
yes  yes  no  Kepler / Poinsotlike polytopes 
no  yes  no  scaliform polytopes 
yes  no  no  fair dices (isohedral polytopes) 
Euklid, in his book Elements, attributes the five solid shapes tetrahedron (tet), hexahedron (cube), octahedron (oct), dodecahedron (doe), and icosahedron (ike) to Plato, therefore they are known since then as Platonical solids. Archimedes lateron added 13 further shapes, since then known as Archimedean solids, which in Loeb's translation of Pappus description of Archimedes' own lost book reads:
Platonic solids can be else described as regular convex polyhedra. The nonconvex regular ones are known as the KeplerPoinsot solids. From Pappus description one can deduce the Archimedeans to be convex and vertex transitive, even so Archimedes surely did not have any notion of modern symmetry action. Taken that for granted, the 2 infinite series of prisms and antiprisms should also belong here, but are not contained within Archimedes' listing. This is why the additional term of semiregular solids is made up, to describe both, the Archimedeans and the 2 prismlike series. Sometimes the attribute "convex" is added explicitely, sometimes it is subsumed.
Solid is taken here in the sense of polyhedron, i.e. 3dimensional. In 4 dimensions, the convex regular ones were first described by Schläfli using his symbols: {3,3,3} (5Zell or pen), {4,3,3} (8Zell or tes), {3,3,4} (16Zell or hex), {3,4,3} (24Zell or ico), {5,3,3} (120Zell or hi), and {3,3,5} (600Zell or ex). The nonconvex regular ones accordingly now are called SchläfliHess polychora.
The high order symmetry of regular polytopes induces lots of interrelations, like facetings, stars, compounds, symmetries implied to subdimensions, etc. Some of those interrelations will be covered here.
Beyond 3D the above deduced interpretation on semiregular polytopes of Pappus words would simply run as being the convex uniforms. In view of the below given quasiregular polytopes, the adjective "semiregular" is feeled to be ment stronger somehow. Even Pappus words of appear to be regularly formed might be bent to this interpretation. Accordingly there are authors found, which use semiregular polytopes in such a stronger sense, like having both, uniform facets (of possibly different kind), and uniform polytopes for vertex figures – besides being convex (or that even not necessarily).
In terms of Dynkin diagrams regular polytopes are those, which have a linear diagram and where a single endnode is marked only. Those polytopes are the ones the Schläfli notation is meant for. It lists the link markings of the Dynkin diagram starting at the ringed node within curly brackets from left to right.
xpo...oqo = oqo...opx = {p,...,q}
Quasiregular polytopes can have any reflectional symmetry group; an arbitrary single node of its diagram may be marked. The Schläfli notation can be extended to such figures (provided their diagrams do not contain loops) by folding the Dynkin diagram at the ringed node, having thus an upper and a lower row of numbers within the curly brackets (either might optionally bifurcate) becoming thereby rather wild. Regular polytopes therefore are just special cases of quasiregulars.
to....ouo / opo.....oqxro..oso \ vo...owo 
= 
{

q,.....,p t,....,u r,..,s v,...,w 
}

= 
{

t,....,u r,...,s v,...,w q,.....,p 
}

Note that Coxeter defined quasiregulars slightly different. His definition amounts into symmetryequivalence of all elements except to the facets. In 3D this is isogonal (vertex transitive) and isotoxal (edge transitive). And this is conform to the above given definition in terms of Dynkin diagrams. But it would extend dimensioanally different: the one given above still sticks to those 2 axioms in any dimesnsions above, esp. the faces of polychora need not to be equivalent, in contrast to Coxeters rule. In fact Coxeters rule is a stricter one. Or stated the other way round, what here is said to be quasiregular encompasses more than only the ones Coxeter would have said.
It should be mentioned further, that either extension of Coxeters definition might extend its application to Dynkin diagram describable polytopes only. So thah clearly is both, isogonal and isotoxal, even so not orientable, and therefore has no Dynkin symbol description. In fact it only can be derived as an hemiation of the (now orientable) x3/2o3x. – As in here the main point are Dynkin diagram applications, the above given definition (by Dynkin diagram type) fully serves.
The quasiregular polytopes, in the above sense of singlyringed Dynkin symbol derivates, and semiregular polytopes, in the stricter sense of having uniform polytopes for vertex figure as well, further are also related hierarchically, at least when convexity is not asked for additionally (or, the other way round, is equally applied to all sets): As shown in the derivation of vertex figures by means of Dynkin diagrams, the edges of the vertex figures of quasiregular polytopes might differ in length. Therefore those generally will not be uniform any more. Semiregularity (in that sense) will thus lie concentrically inbetween the set of regulars and that of quasiregulars. In fact, in terms of Dynkin symbols, those are described by the ones with exactly one node ringed, and additionally having all emanating links (from that single node) being marked by 3 (resp. therefore being not marked at all).
For these quasiregular polytopes the Coxeter notation as a folded Schläfli symbol sometimes can be found unfolded as well, depicting the fold then not as mere comma, but as a "marked one", as a semicolon. I.e. {p;q,r} = opxqoro. This type of notation was brought in by Mrs. Krieger.
(Reflectional) Wythoffian polytopes finally can use any number of marked nodes in their Dynkin diagram. Quasiregulars and regulars are thus contained in turn under the Wythoffians. A classical Schläfli notation for this extended set does not exist, although for symmetry groups with linear Dynkin diagrams a operational truncation notation was introduced by Coxeter. And Mrs. Krieger's notation with marked Schläfli symbols would apply as well:
xpxqorx = t_{0,1,3}{p,q,r} = t_{0,2,3}{r,q,p} = {;p;q,r;}
But, as soon as Dynkin symbols with loops and/or bifurcations are to be considered, both latter linear notations soon fall behind this intentionally graphical device.
The adjectives of names for specific polytopes are usually given in terms of that truncation notation (the first ones given belong to Bowers, the ones given in the right column (if different) generally belong to Johnson; the provided Johnsonstyle adjectives for higher dimensions, i.e. from "pentellated" on, together with all its derivatives, were added later by Ruen):
Prefix  Bowers adjectives  JohnsonRuen adjectives 
t_{0}  regular (itself)  
t_{1}  rectified  
t_{2}  birectified  
t_{3}  trirectified  
t_{4}  quadrirectified  
t_{5}  quintirectified  
t_{0,1}  truncated  
t_{0,2}  (small) rhombated / rhombi  cantellated 
t_{0,3}  (small) prismated / prismato  runcinated 
t_{0,4}  (small) cellated / celli  stericated 
t_{0,5}  (small) terated / teri  pentellated 
t_{0,6}  (small) petated / peti  hexicated 
t_{0,7}  (small) exated / exi  — ??? — 
t_{0,8}  (small) zettated / zetti  — ??? — 
t_{0,9}  (small) yottated / yotti  — ??? — 
t_{1,2}  bitruncated  
t_{1,3}  (small) birhombated  bicantellated 
t_{1,4}  (small) biprismated  biruncinated 
t_{1,5}  (small) bicellated  bistericated 
t_{2,3}  tritruncated  
t_{2,4}  (small) trirhombated  tricantellated 
t_{2,5}  (small) triprismated  triruncinated 
t_{3,4}  quadritruncated  
t_{3,5}  (small) quadrirhombated  quadricantellated 
t_{4,5}  quintitruncated  
t_{0,1,2}  great rhombated / greatorhombated  cantitruncated 
t_{0,1,3}  prismatotruncated  runcitruncated 
t_{0,1,4}  cellitruncated  steritruncated 
t_{0,1,5}  teratruncated  pentitruncated 
t_{0,2,3}  prismatorhombated  runcicantellated 
t_{0,2,4}  (small) cellirhombated  stericantellated 
t_{0,2,5}  (small) terarhombated  penticantellated 
t_{0,3,4}  celliprismated  steriruncinated 
t_{0,3,5}  teraprismated  pentiruncinated 
t_{0,4,5}  teracellated  pentistericated 
t_{1,2,3}  great birhombated / bigreatorhombated  bicantitruncated 
t_{1,2,4}  biprismatotruncated  biruncitruncated 
t_{1,2,5}  bicellitruncated  bisteritruncated 
t_{1,3,4}  biprismatorhombated  biruncicantellated 
t_{1,3,5}  bicellirhombated  bistericantellated 
t_{1,4,5}  bicelliprismated  bisteriruncinated 
t_{2,3,4}  great trirhombated / trigreatorhombated  tricantitruncated 
t_{2,3,5}  triprismatotruncated  triruncitruncated 
t_{2,4,5}  triprismatorhombated  triruncicantellated 
t_{3,4,5}  great quadrirhombated  quadricantitruncated 
t_{0,1,2,3}  great prismated / greatoprismated  runcicantitruncated 
t_{0,1,2,4}  great cellirhombated / celligreatorhombated  stericantitruncated 
t_{0,1,2,5}  great terarhombated / terigreatorhombated  penticantitruncated 
t_{0,1,3,4}  celliprismatotruncated  steriruncitruncated 
t_{0,1,3,5}  teraprismatotruncated  pentiruncitruncated 
t_{0,1,4,5}  teracellitruncated  pentisteritruncated 
t_{0,2,3,4}  celliprismatorhombated  steriruncicantellated 
t_{0,2,3,5}  teraprismatorhombated  pentiruncicantellated 
t_{0,2,4,5}  (small) teracellirhombated  pentistericantellated 
t_{1,2,3,4}  great biprismated / bigreatoprismated  biruncicantitruncated 
t_{1,2,3,5}  great bicellirhombated / bicelligreatorhombated  bistericantitruncated 
t_{1,2,4,5}  bicelliprismatotruncated  bisteriruncitruncated 
t_{1,3,4,5}  bicelliprismatorhombated  bisteriruncicantellated 
t_{2,3,4,5}  great triprismated / trigreatoprismated  triruncicantitruncated 
t_{0,1,2,3,4}  great cellated / greatocellated  steriruncicantitruncated 
t_{0,1,2,3,5}  great teraprismated / terigreatoprismated  pentiruncicantitruncated 
t_{0,1,2,4,5}  great teracellirhombated / tericelligreatorhombated  pentistericantitruncated 
t_{0,1,3,4,5}  teracelliprismatotruncated  pentisteriruncitruncated 
t_{0,2,3,4,5}  teracelliprismatorhombated  pentisteriruncicantellated 
t_{1,2,3,4,5}  great bicellated / bigreatocellated  bisteriruncicantitruncated 
t_{0,1,2,3,4,5}  great terated  pentisteriruncicantitruncated 
t_{0,last}  expanded  
t_{all}  omnitruncated 
It should be noted, that even the stem, to which those adjectives are applied, varies slightly between Bowers and Johnson. So Johnson applies those adjectives for polytopal names to the singly endringed described polytope (i.e. for linear Dynkin diagrams these being the regular ones) in a completely operational sense. Therefore this stem remains unchanged throughout all decorations. In contrast Bowers relates to the overall symmetry of the specific decoration applied. That is, whenever the decoration is asymmetrical, the stem is the same as for Johnson. But if not, he uses instead all relevant singly enddecorated versions simultaneously. For instance, Bowers would use "small rhomb()icosi()dodecahedron" (srid = t_{0,2}{3,5}), while Johnson merely would use either "cantellated icosahedron" or equivalently "cantellated dodecahedron". In fact the Bowers names are designed to continue the namings of Kepler. – Further, for the simplex groups of the various dimensions, when both enddecorations designate the same polytope (only being situated in dual orientations), Bowers generally does not use doubled up number prefixes, but simply adds those; therefore a rectified tetrahedron (oct = t_{1}{3,3}) becomes an octahedron instead of a tetra()tetrahedron, and a "runcinated pentachoron" similarily becomes a "small prismato()decachoron" (spid = t_{0,3}{3,3,3}).
As applications to linear Dynkin diagrams – with first link being even – Johnson considered hemiations within that first node too, what later led to the more general alternated facettings. Here he provides first general namings too, which for higher dimensions then were extended by Ruen. Thereby generally the final extension ated of the last column of the former table just becomes an ic in the table below.
Prefix  JohnsonRuen adjectives 
h_{0}  half (hemiation) 
h_{0}t_{2}  cantic 
h_{0}t_{3}  runcic 
h_{0}t_{4}  steric 
h_{0}t_{5}  pentic 
h_{0}t_{6}  hexic 
h_{0}t_{2,3}  runcicantic 
h_{0}t_{2,4}  stericantic 
h_{0}t_{3,4}  steriruncic 
h_{0}t_{2,3,4}  steriruncicantic 
Facets of quasiregular polytopes and more generally of all Wythoffian polytopes are easily derived from their Dynkin diagrams by omitting any possible node of the diagram. (Cases with no ringed node in a connected subdiagram would be even lower dimmensional, and so do not contribute to the set of facets.)
3o 3o . 3o / / / facets( x3x ) = x3x & x3x & . x \ \ \ 3o . 3o 3o
(Note, that this rule of the removel of a single node is due to the simplicial form of the fundamental domain of symmetry. Cf. also the more general derivation of the related facets in case of more general Coxeter domains, possible in hyperbolic geometry.)
The topology of vertex figures of quasiregular polytopes is likewise easily derived by omitting the marked node and marking in the reduced diagram all neighbouring nodes of the missing node.
From this it follows that facets and vertex figures of regular polytopes are regular polytopes again. Facets of quasiregular polytopes are similarly quasiregular polytopes. But the vertex figures of quasiregular polytopes may be (multi)prisms of quasiregular polytopes (and this possibly only in a topological sense). Facets of Wythoffians are always Wythoffian polytopes again. But the vertex figures of Wythoffians in general do not belong to any of those classes.
(Multi)prisms are orthogonal products of lower dimensional polytopes. That is, every element of one factor is completely orthogonal to any of the other. Thus the Dynkin diagram will have those components connected by links marked 2, or, as those are usually not drawn at all, it will fall into some disconnected subgraphs. If one such component is 1dimensional only, the polytope is called a prism, if all components are at least 2dimensional it is called a multiprism. In case "multi" could be specified further: duoprisms (2 component diagrams), triprisms (3 component diagrams), etc.
Coxeter has driven the research for even larger sets of polyhedra up to the uniform polyhedra. They bow to the restriction that their over all symmetry group act transitively on the vertices (i.e. there will be just one equivalence class of vertices only), and faces have to be regular polygons. And, for sure, in order to be a valid polyhedron, all edges have to have exactly 2 incident faces. A flag is defined as a set of incident elements of a polytope, one of each dimension. Thus, except for the equal edge length, uniform polyhedra just ask that the overall symmetry has to act transitively on each class of possibly different flags. In contrast to the set of Archemedeans, neither for the faces (polygons), nor for the solids themselves, any convexity is required in here; so edges may cycle around a face several times (polygrams), and even the faces may wrap around the center more than once. (Note that the vertices are bound to lie on a circumsphere, so a center is always defined.) Together with his coauthors Coxeter listed them all, and later that list was proven to be complete.
Grünbaum reconsidered this result under a slightly different viewpoint: what about "uniform" figures, he asked, which are abstractly resolvable as true polyhedra (i.e. not compounds), but where the multiple wrapping just by accident amounts to the fact that some polyhedral elements come to lie completely coincident.  This includes degeneracies like polygrams
xn/dx = {2n/d}
with even denominator d, because those will be wrapped in such a way that both, vertices and edges, coincide by two, looking just as if it would be a double cover of {n/(d:2)}. Or using edges with an even number greater than 2 of incident faces, which will be resolvable as being just some number of normal edges, each being coincident with only 2 faces, which accidentally coincide and therefore only look like a single edge; and other degeneracies more.  The such enlarged set of "new" uniform polyhedra still is not enumerated.
Both these setups of uniform polyhedra don't bow freely under Dynkin diagrams, because the setups were chosen indepently of Wythoff's kaleidoscopical construction; the point here is, that there might be faces used, although being all regular, which cannot be produced by the mirrors of that symmetry group. On an slightly other thread Schwarz looked for and enumerated all nonelementary fundamental triangles of 3dimensional reflectional symmetry groups. Using those multycovered groups within the Dynkin notation, quite a lot of the (possibly Grünbaumian) uniform polyhedra are derivable. From the set of uniform polyhedra in the sense of Coxeter (no completely coincident elements) just a single one refuses to bow to that description by Dynkin symbols (even if snubs would be included, see Miller's Monster).
An equivalent research in 4D for uniform polychora by usage of the Goursat tetrahedra (the higher dimensional equivalent of Schwarz triangles) was started by Olshevsky. But soon it was abandoned in favor of an other, more direct listing project by Bowers, which took over the techniques used in the work of Coxeter et al. directly into 4D: He considered all possible facetings of the vertex figure polyhedra of any convex Wythoffian polychoron. – The adjective "uniform" in dimensions greater than 3 usually contains as additive requirement the hierarchicality, i.e. any facet of such a polytope has to be in turn uniform. Even then the current count is as high as 8190 uniform polychora, including some exotic figures with coincident or compounded elements. Note, in this enumeration coincident elements never are directly connected, and alike facets never coincide completely. Even by exclusion of the exotic ones and those with compound vertex figures the count is still as high as 1849. This research and especially the full listing is still unpublished in printed literature (so far Bowers lists parts only on his website).
Thus after all we have the following definition of uniformity:
In dimensions above 3, where for "uniformity" the second requirement of hierarchicality is added, an even larger class of polytopes can be considered. Those are usually called weakly uniform polytopes. This weak uniformity just neglects this additional second requirement of uniformity of facets. But because of that mere relaxing adjective ("weakly") this class more recently was positively renamed into scaliform polytopes. It is, the according polytope still has to have a
(what applied to polyhedra would be equivalent to uniformity, but not so for higher dimensional polytopes). Esp., as cells of scaliform polychora the Johnson solids would be allowed as well. Some scaliforms are contained within this website, as far as they occur as segmentochora or as Klitzing snubs. Further ones are contained here in the CRF context. – The most complete listing of so far known ones, which is online accessible, can be found on the Bowers website.
One could argue whether the second part of that definition should rather be replaced by a requirement about the regularity of used polygons (which thus implies the single edge length as well). This is because in flat euclidean spaces that more restrictive axiom cannot be deduced right from the so far given ones together with the subsumed planarity of polygons, in contrast to uniformly curved spaces. Infact, a planar intersection of a uniformly curved space defines a circle. And edges with equal length placed with vertices on it are bound to define regular polygons (final closure assumed). That is, scaliform tilings (if defined as above) would allow all kind of rhombs etc. as well, they only need to have a vertex transitive symmetry. This euclidean anomaly of (strict) scaliformity was not in the focus, as so far the main interest in that direction were scaliform polychora only.
As this same argument could be applied not only for polygons, but for all subdimensions 2≤n<D, most recently it was added a third axiom for that exceptional euclidean case. (As described above already, for noneuclidean spaces that one can be deduced from the formers.)
Weimholt has introduced an still broader adjective for polytopes: orbiform. (This term is not to be mixed up with the orbifold notation for polytopes, the term Conway himself is using for what elsewhere is called the ConwayThurston symbol.) Orbiformity transcends the restriction of vertex transity. But replaces it with that (newly added) third axiom of scaliformity, lending thereby the name: Orbiform polytopes do have
Thus an unique circumradius does still exists for those polytopes, even so not all vertices are forced to be symmetry equivalent.
Note that segmentotopes are exactly the monostratic orbiforms. And, furtheron, right by their definition, for bases of segmentotopes exactly the orbiform polytopes can be used. – Although multistratic stacks of segmentotopes easily can be piled up (cf. lace towers), these generally would not be bound to be orbiform any longer. Besides of all the multistratic segments of uniform polytopes, which therefore are orbiform by construction, only a single other bistratic polychoron is known (up to now) to be orbiform: sidrebcu.
In 2013 a further class of polytopes gained a vivid research interest, even so it was considered already long before. It is the equivalent of Johnson solids for higher dimensions: the so called CRF polytopes, asking only for a global convexity and regular shaped faces (2Dboundaries).
Even so it is known that this set already could not be reasonably exhausted, an even wider class was envisaged from different researching groupes once more in 2014: the then so called CUE polytopes. These ask again for a global convexity, but moreover for unit edges only. Thus esp. the exclusive subset of those, which use at least one nonregular face, are considered mainly.
The longknown general set of all equilateral zonohedra, and generally equilateral zonotopes then would be CUE. Any n generating unit vectors, not necessarily all in general, i.e. independent position, would span the according equilateral zonotope. The hull of those points, which are derived by all {0, 1}arrays applied to this vector set then defines the Minkowski sum of those. E.g. the best known zonohedra are derived by the edges of platonic solids, being parallelly translated onto a single vertex (pointing then in a halfspace only), and therefrom further omitting dublicates. – Note that 2D zonotopes (zonogons) always have an even number of sides, and opposite ones will be parallel. (This observation shows that CUE polytopes are not bound to zonotopes only: e.g triangles there would be clearly valid faces too.)
Obviously in such uncountable sets larger or smaller subfamilies are of special interest, i.e. whenever they can be grouped by some common property, just as these zonotopes already did. But every now and then also some individuals, e.g. derived by relaxation towards equal edge size only, while keeping the flatness of faces (or even by obtaining that thereby), do show up.
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