Polytopes

"Polytope" is the general term of the sequence:

• (0D)   monon
• (1D)   dyad
• (2D)   polygon
• (3D)   polyhedron
• (4D)   polychoron
• (5D)   polyteron
• (6D)   polypeton
• (7D)   polyexon
• (8D)   polyzetton
• (9D)   polyyotton
• (10D) polyxennon
• (11D) polydakon (or: polywekon)
• (12D) polyhendon (or: polyvendekon)
• (13D) polydokon (or: polyudekon)
• etc.

(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. Above dimension 8 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 explicit listing of Euclidean tesselations of the first few dimensions can be found separately. Furthermore, hyperbolic ones are listed separately too.

Taken set-theoretically, 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

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)

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:

Although many solid figures having all kinds of faces can be conceived, those which appear to be regularly formed are most deserving of attention. These include not only the five figures found in the godlike Plato ... but also the solids, thirteen in number, which were discovered by Archimedes and are contained by equilateral and equiangular, but not similar, polygons.

Platonic solids can be else described as regular convex polyhedra. The non-convex regular ones are known as the Kepler-Poinsot solids. From Pappus description one can deduce the Archimedeans to be convex and vertex transitive, although 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 prism-like series. Sometimes the attribute "convex" is added explicitly, sometimes it is subsumed.

Solid is taken here in the sense of polyhedron, i.e. 3-dimensional. In 4 dimensions, the convex regular ones were first described by Schläfli using his symbols: {3,3,3} (5-Zell or pen), {4,3,3} (8-Zell or tes), {3,3,4} (16-Zell or hex), {3,4,3} (24-Zell or ico), {5,3,3} (120-Zell or hi), and {3,3,5} (600-Zell or ex). The non-convex regular ones accordingly now are called Schläfli-Hess polychora.

The high order symmetry of regular polytopes induces lots of inter-relations, like facetings, stars, compounds, symmetries implied to sub-dimensions, etc. Some of those inter-relations 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 end-node 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.

x-p-o-...-o-q-o  =  o-q-o-...-o-p-x  =  {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.

                               t-o-....-o-u-o
                              /
o-p-o-.....-o-q-x-r-o-..-o-s-o
                              \
                               v-o-...o-w-o

  =

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 symmetry-equivalence 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, although 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 singly-ringed 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} = o-p-x-q-o-r-o. 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:

x-p-x-q-o-r-x  =  t0,1,3{p,q,r}  =  t0,2,3{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 Johnson-style adjectives for higher dimensions, i.e. from "pentellated" on, together with all its derivatives, were added later by Ruen):

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 end-ringed 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 end-decorated 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 end-decorations 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 = t₁{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.

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 sub-diagram would be even lower dimmensional, and so do not contribute to the set of facets.)

              3-o             3-o              .            3-o
             /               /                             /
facets( x-3-x     )  =  x-3-x      &   x-3-x      &   .   x
             \                              \              \
              3-o               .            3-o            3-o

(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 1-dimensional only, the polytope is called a prism, if all components are at least 2-dimensional 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 over-all 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, thus a center is always defined.) Together with his co-authors 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

x-n/d-x = {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 non-elementary fundamental triangles of 3-dimensional reflectional symmetry groups. Using those multy-covered 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:

1. vertex transitive symmetry
2. all edges are of the same length
3. facets are uniform in turn   –   (to be added if facets larger than polygons would exist)

Note that the property 1., i.e. having a "vertex transitive symmetry", as such also is known by the term isogonal. Further, because of the separate property 2., the property 3. might be lowered to asking that the facets have just to be isogonal. And this thus could be combined with property 1., then asking that the polytope and all its elements have to be isogonal. Finally, because polygons, which are both isogonal and have all edges of the same length, already are necessarily regular ones, property 2 in here well may be replaced, instead of speaking about the 1D elements of the polytope now speaking about its 2D elements, by asking that the faces have to be regular.

In dimensions above 3, where for "uniformity" the additional 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 that additional requirement about uniformity of facets completely. 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

1. vertex transitive symmetry
2. all edges are of the same length

(what, if applied to polyhedra, would well 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. Several scaliforms are contained within this website, cf. this special listing subpage on scaliform polytopes. – The most complete listing of so far known 4D ones, which is online accessible, however 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 non-euclidean spaces that one can be deduced from the formers.)

3. the vertices of any facet (resp. tile) are on a single (hyper-)sphere
   (in view of the next section this could also be given as:   facets have to be orbiform)

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 Conway-Thurston 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

• all vertices on a single (hyper-)sphere
• all edges of the same length

Thus an unique circumradius does still exists for those polytopes, although 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 and diminishings therof, which therefore all are orbiform by construction, only very few other bistratic polychora are known (up to now) to be orbiform: sidrebcu and 2 out of 3 ursachora (the third then again is a diminishing).

In 2013 a further class of polytopes gained a vivid research interest, although 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 (2D-boundaries).

Although 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 non-regular face, are considered mainly.

The long-known 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 half-space 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.

An ongoing current interest is the research for skeletal polytopes. Here the abstract incidence structures are the same as for other polytopes. But polygons no longer are restricted to have flat representations, rather they just are described by their finite or infinite edge sequences. The only restriction then is that any compact region of embedding space only meets a finite amount of edges, that is, those are not allowed to become dense. Again such a generalized polygon can be considered regular, when there is a finite or infinite group which acts transitively on the flags, i.e. on the geometric representation of the incidence structure of its vertices and edges. Polygons here can be planar (convex or starry), non-planar (skew), or infinite (planar zigzags, linear, or helical).

Extending to skeletal polyhedra and larger polytopes, the local discreteness requirement here is applied to the vertex figures, which thus implies that those represent finite polygons only. And for sure, just as for usual polytopes, just 2 polygons are required to be incident at each edge. The mere edge graph of such a polytope then is called its 1-skeleton. This is where its naming derives from. Again we have flags, e.g. for polyhedra a triple of vertex, edge, and face, all mutually incident. And flags are called adjacent when they differ in precisely a single element. Again the same flag based regularity definition can be applied. For skeletal regular polytopes the facet polytopes likewise have to be skeletal regular polytopes.

Quite close to regular polytopes are the chiral ones. Here the action of symmetry on the flags falls apart into exactly 2 classes, such that any pair of adjacent flags always belongs to different orbits. None the less the symmetry clearly acts transitively on the vertices and the polygons still are bound to be skeletal regular ones. More general, a skeletal polyhedron is said to be uniform, whenever it has a vertex transitive symmetry and (skeltal) regular faces. The research on (skeletal) uniform polyhedra with planar faces only, already has some history. Higher dimensional skaletal uniform polytopes then hierarchically would require (skeletal) uniform ones for facets.

It shall be noted that the fundamental regions used for Wythoff's construction usually are simplices. In this setup however, they are more general. Consider any point and its images under the to be used symmetry. Whenever those are not identical points one collects the set of such images. Then the fundamental regions can be obtained as the Voronoi domain of the original point wrt. to its image points, i.e. the intersection of half spaces where the respective hyperplane bisects each image point and the primary point.

It is this setup of skeletal polytopes, where Petrie polygons are defined. A Petrie polygon of some regular polyhedron runs 2 consecutive edges along one face polygon. This second edge then is already the first for the adjacent face. There again 2 edges, etc. always making sharpest turns. Petrie polygons of spherical or euclidean regular polyhedra never are flat. Locally they are skew zig-zags. Petrie polygons of spherical polyhedra come in finite cycles, those of euclidean regular polyhedra are infinite. For general n-dimensional regular polytopes Petrie polygons are defined by following a facet along exactly n-1 edges, and to any of its corresponding subelements according to the respective dimensionality.

Further interesting polytopes

• Snubs & Alternated Facetings – (PDF)
  published as: "Snubs, Alternated Facetings, and Stott-Coxeter-Dynkin Diagrams", by Dr. R. Klitzing, Symmetry: Culture and Science, vol. 21, no.4, 329-344, 2010
• Edge Facetings of Uniform Polyhedra – (PDF)
  published as: "Axial-Symmetrical Edge-Facetings of Uniform Polyhedra", by Dr. R. Klitzing, Symmetry: Culture and Science, vol. 13, nos.3-4, 241-258, 2002
  external link to work of the author, hosted by U. Mikloweit
• Convex Segmentochora – (PDF)
  published as: "Convex Segmentochora", by Dr. R. Klitzing, Symmetry: Culture and Science, vol. 11, 139-181, 2000
  external link to work of the author, hosted by J. McNeill
• Regular polytopes of elliptical space
• Grünbaum-Coxeter Polytopes   (11-cell, 57-cell, and relatives)
• Uniform Compounds
• Convex Regular Faced (CRF) Polytopes
• Partial Stott Expansions (resp. Contractions)