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A polytope $${{ \mathsf {P}}}$$ in some euclidean space is called quasi-regular if each facet $${{ \mathsf {F}}}$$ of $${{ \mathsf {P}}}$$ is regular and the symmetry group $${\mathbf {G}}({{ \mathsf {F}}})$$ of $${{ \mathsf {F}}}$$ is a subgroup of the symmetry group $${\mathbf {G}}({{ \mathsf {P}}})$$ of $${{ \mathsf {P}}}$$ . Further, $${{ \mathsf {P}}}$$ is of full rank if its rank and dimension are the same. In this paper, the quasi-regular polytopes of full rank that are not regular are classified. Similarly, an apeirotope of full rank sits in a space of one fewer dimension; the discrete quasi-regular apeirotopes that are not regular are also classified here. One curiosity of the classification is the difference between even and odd dimensions, in that certain families are present in $${\mathbb {E}}^d$$ if d is even, but are absent if d is odd.

It is shown that a full-dimensional polytope P is uniquely determined by its r-dimensional fibre polytopes when $$r \ge 2$$ r ≥ 2 . Further, if $$r \ge 4$$ r ≥ 4 and the r-dimensional fibre polytopes are zonotopes, then P itself must be a zonotope.

This paper gives an almost complete description of the compounds of regular honeycombs with finite vertices and cells in d-dimensional hyperbolic space H d for d = 3 , 4 . It is conjectured that the list is complete.

This note describes six apparently new regular compounds of polytopes in \(\mathbb {E}^4\), that were missed by Coxeter in his systematic faceting procedure. In fact, three have the same symbols as ones in Coxeter’s list (and two are nearly the same); however, their symmetry groups are much smaller. The treatment relies heavily on the use of quaternions.

This paper describes ways that certain regular honeycombs of non-finite type in d-dimensional hyperbolic space \(\mathbb {H}^d\) for \(d = 2,3\) and 5 can be inscribed in others, in particular showing that some can be inscribed properly in copies of themselves.

An abstract regular polytope $\mathcal{P}$ of rank n can only be realized faithfully in Euclidean space $\mathbb{E}^{d}$ of dimension d if d≥n when $\mathcal{P}$ is finite, or d≥n−1 when $\mathcal{P}$ is infinite (that is, $\mathcal{P}$ is an apeirotope). In case of equality, the realization P of $\mathcal{P}$ is said to be of full rank. If there is a faithful realization P of $\mathcal{P}$ of dimension d=n+1 or d=n (as $\mathcal {P}$ is finite or not), then P is said to be of nearly full rank. In previous papers, all the at most four-dimensional regular polytopes and apeirotopes of nearly full rank have been classified. This paper classifies the regular polytopes and apeirotopes of nearly full rank in all higher dimensions.

This note corrects a mistake in the earlier paper with the same title, and gives an example to illustrate the modified theory.

In previous papers, all the four-dimensional (finite) regular polytopes have been classified, as well as the regular apeirotopes of full rank (that is, of rank 5). Of the two problems in $\mathbb{E}^{4}$thus left open (namely, regular apeirotopes of ranks 3 and 4), this paper describes the regular apeirotopes of rank 4. The methods employed here are somewhat different from those in earlier work; while knowledge of the possible dimension vectors (dim R 0,…,dim R 3) of the mirrors R 0,…,R 3 of the generating reflexions of the symmetry groups plays a role, the crystallographic restriction leads to a considerable emphasis being placed on the vertex-figures.

In a recent paper, Karpenkov has classified the lattice polytopes (that is, with vertices in the integer lattice Zd) which are regular with respect to those affinities which preserve the lattice. An alternative approach is adopted in this paper. For each regular polytope P in euclidean space Ed, those lattices Λ are classified which are compatible with P, in the sense that some translate of Λ contains the vertices of P, and this translate is preserved by the symmetries of P.

This paper completes the classification of the four-dimensional (finite) regular polyhedra, of which those with planar faces were—in effect—found by Arocha, Bracho and Montejano. However, the methods employed here are in the same spirit as those used in the description of all three-dimensional regular polytopes by this author and Schulte, and the regular polytopes of full rank by this author. The procedure has two stages. First, the possible dimension vectors (dim R0,dim R1,dim R2) of the mirrors R0,R1,R2 of the generating reflexions of the symmetry groups are determined. Second, all polyhedra with a given dimension vector are found. Most of the polyhedra are related to four-dimensional Coxeter groups, although one class has to be approached using quaternions.

Let V be a finite-dimensional vector space over a square-root closed ordered field F (this restriction permits an inner product with corresponding norm to be imposed on V).

A small family of regular polytopes of nearly full rank was omitted from the earlier paper with this title. This omission is rectified here.

With a slight modification of the original definition by Billera and Sturmfels, it is shown that the theory of fibre polytopes extends to one of mixed fibre polytopes. Indeed, there is a natural surjective homomorphism from the space of tensor weights on polytopes in a euclidean space V to the corresponding space of such weights on fibre polytopes in a subspace of V. Moreover, these homomorphisms compose in the correct way; this is in contrast to the original situation of the fibre polytope construction.

The dimension of a faithful realization of a finite abstract regular polytope in some euclidean space is no smaller than its rank. Similarly, that of a discrete faithful realization of a regular apeirotope is at least one fewer than the rank. Realizations which attain the minimum are said to be of full rank. The regular polytopes and apeirotopes of full rank in two and three dimensions were classified in an earlier paper. In this paper these polytopes and apeirotopes are classified in all dimensions. Moreover, it is also shown that there are no chiral polytopes of full rank.

Mixing is an operation which yields subgroups generated by involutions of a larger group of the same kind. When it is applied to the product of the automorphism groups of two regular polytopes, one talks about the mix of the polytopes. This paper is concerned with conditions under which the mix of two regular polytopes is again a regular polytope. In the important special case of a mix of a polytope with one of its faces, fairly general results about the polytopality of the mix are obtained. In particular, the case when the mix is isomorphic to the original polytope is characterized.

Complex groups generated by involutory reflexions arise naturally in the modern theory of abstract regular polytopes. The paper investigates this relationship and explains how the enumeration of certain finite universal regular polytopes can be accomplished through the enumeration of certain types of finite complex reflexion groups. In particular, the paper enumerates all the finite groups and their diagrams which arise in this context, and describes the corresponding regular polytopes.

A (geometric) regular polygon {p} in a euclidean space can be specified by a fine Schlafli symbol {p}, where $$\displaystyle{p = \frac{r} {s_{1},\ldots,s_{k}}}$$ is a generalized fraction; here, \(0 \leq s_{1} < \cdots < s_{k} \leq \frac{1} {2}r\). This means that {p} projects onto planar polygons \(\{ \frac{r} {s_{j}}\}\) (reduced to lowest terms) in orthogonal planes, with \(\infty = \frac{1} {0}\) giving the linear apeirogon and 2 the digon (line segment). More generally, it may be possible to specify the shape or similarity class of a geometric regular polytope by means of a fine Schlafli symbol, whose data contain information about certain regular polygons occurring among its vertices in terms of generalized fractions. If so, then the fine Schlafli symbol is called rigid. This paper gives various criteria for rigidity; for instance, the classical regular polytopes are rigid. The theory is also illustrated by several examples. It is noteworthy, though, that a combinatorial description of a regular polytope – a presentation of its symmetry group – can differ considerably from its fine Schlafli symbol.

If all the edges of a d -simplex T have the same length, then T is regular. However, if d \geq 3 , then it is clear that the facets of T may have the same (d-1) -volume without T being regular. Here, the question of the extent to which the equality of r -volumes of the r -faces of T implies regularity of T is investigated, the case r = d-2 proving most fruitful.

It is shown here that, for n ⩾ 2, the n-torus is the only n-dimensional compact euclidean space-form which can admit a regular or chiral tessellation. Further, such a tessellation can only be chiral if n = 2.

The family of convex sets in a (finite dimensional) real vector space admits several unary and binary operations – dilatation, intersection, convex hull, vector sum – which preserve convexity. These generalize to convex functions, where there are in fact further operations of this kind. Some of the latter may be regarded as combinations of two such operations, acting on complementary subspaces. In this paper, a general theory of such mixed operations is introduced, and some of its consequences developed.

The regular star-polyhedron is isomorphic to the abstract polyhedron {5, 5,| 3}, where the last entry “3” in its symbol denotes the size of a hole, given by the imposition of a certain extra relation on the group of the hyperbolic honeycomb {5, 5}. Here, analogous formulations are found for the groups of the regular 4-dimensional star-polytopes, and for those of the non-discrete regular 4-dimensional honeycombs. In all cases, the extra group relations to be imposed on the corresponding Coxeter groups are those arising from “deep holes”; thus the abstract description of for k = 1 or 2. The non-discrete quasi-regular honeycombs in 𝔼3, on the other hand, are not determined in an analogous way.

At the centre of the theory of abstract regular polytopes lies the amalgamation problem: given two regularn-polytopesP 1 andP 2, when does there exist a regular (n+1)-polytopeP whose facets are isomorphic toP 1 and whose vertex-figures are isomorphic toP 2? The most general circumstances known hitherto which lead to a positive answer involve flat polytopes, which are such that each vertex lies in each facet. The object of this paper is to describe an analogous but wider class of constructions, which generalize the previous results.

The three aims of this paper are to obtain the proof by Dress of the completeness of the enumeration of the Grunbaum—Dress polyhedra (that is, the regular apeirohedra, or apeirotopes of rank 3) in ordinary space E 3 in a quicker and more perspicuous way, to give presentations of those of their symmetry groups which are affinely irreducible, and to describe all the discrete regular apeirotopes of rank 4 in E 3 . The paper gives a complete classification of the discrete regular polytopes in ordinary space.

Theg-theorem describes the possible face-vectors of a simple polytopeP. Much of the author's proof of the necessity of its conditions, while working within the polytope algebra π, in fact only used the spaces of weights onP. Even though this proof was conceptually easier than the original, which employed techniques from algebraic geometry, nevertheless the properties of π which are needed still require some effort to establish, despite a recent simpler approach to π itself. In the earlier paper, doubt was expressed about whether two basic results could be proved directly for weights; later, it appeared that there might also be a possible problem concerning an alternative definition of the product of certain weights. In this paper these questions are settled, in the context of developing an independent theory of an algebra Ω of weights on polytopes. Since the construction of Ω is more approachable than that of π, a yet easier proof of theg-theorem is now available.

In recent years, much work has been done on the classification of abstract regular polytopes by their local and global topological type. Abstract regular polytopes are combinatorial structures which generalize the well-known classical geometric regular polytopes and tessellations. In this context, the classical theory is concerned with those which are of globally or locally spherical type. In a sequence of papers, the authors have studied the corresponding classification of abstract regular polytopes which are globally or locally toroidal. Here, this investigation of locally toroidal regular polytopes is continued, with a particular emphasis on polytopes of ranks 5 and 6. For large classes of such polytopes, their groups are explicitly identified using twisting operations on quotients of Coxeter groups. In particular, this leads to new classification results which complement those obtained elsewhere. The method is also applied to describe certain regular polytopes with small facets and vertex-figures.

In an earlier paper, a theory of realizations of (finite) regular polytopes in euclidean spaces was developed. Here, the analogous problem of realizing regular apeirotopes (infinite polytopes) is investigated. While no complete theory is expounded, several basic results are established. Among these is the curious fact that, if a regular apeirotope has a discrete realization, then it has one with no translations in its symmetry group.

polytopes are combinatorial and geometrical structures with a distinctive topological flavor, which resemble the convex polytopes. C-groups are generalizations of Coxeter groups and are the automorphism groups of abstract polytopes which are regular. We investigate general properties of quotients of abstract polytopes and C-groups.

Pegged tilings localize the defining property of or Laguerre tilings, and, like them, admit a natural duality (corresponding to the Delaunay tilings of tilings). It can thus be shown that the projection method, which is generally used to construct quasi-periodic tilings related to tilings of higher dimensional lattices, applies to this wider class of tilings. Of further importance is that pegged tilings are just those which can be lifted to the graphs of convex functions with a certain strong locally polyhedrality property. The context of convex functions also provides a direct way of viewing the projection method, and leads to alternative pictures of special cases such as various grid methods.

LetP be a simpled-polytope ind-dimensional euclidean space $$\mathbb{E}^d $$ , and let Π(P) be the subalgebra of the polytope algebra Π generated by the classes of summands ofP. It is shown that the dimensions of the weight spacesΞ r(P) of Π(P) are theh-numbers ofP, which describe the Dehn-Sommerville equations between the numbers of faces ofP, and reflect the duality betweenΞ r (P) andΞ d-r (P). Moreover, Π(P) admits a Lefschetz decomposition under multiplication by the element ofΞ 1(P) corresponding toP itself, which yields a proof of the necessity of McMullen's conditions in theg-theorem on thef-vectors of simple polytopes. The Lefschetz decomposition is closely connected with the new Hodge-Riemann-Minkowski quadratic inequalities between mixed volumes, which generalize Minkowski's second inequality; also proved are analogous generalizations of the Aleksandrov-Fenchel inequalities. A striking feature is that these are obtained without using Brunn-Minkowski theory; indeed, the Brunn-Minkowski theorem (without characterization of the cases of equality) can be deduced from them. The connexion found between Π(P) and the face ring of the dual simplicial polytopeP * enables this ring to be looked at in two ways, and a conjectured formulation of theg-theorem in terms of a Gale diagram ofP * is also established.

For an integer m > 2, let L(-1)2 ( Z m) be the group of 2 × 2 matrices over Z m with determinants ±1. Then for each subgroup H with {±1} ⩽ H ⩽ centre (L(-1)2( Z m)) there exists a regular map of type {3, m} whose group is L(-1)2 ( Z m)/H. Ways are determined in which one such map can cover another, and in which a map decomposes as a blend of simpler maps. In all this, the prime factorization of m plays the key role. In some cases the structural results enable a simple description of the corresponding automorphism groups. Also described are realizations for the maps based on the 'plane' Z 2m.

A method is given for constructing new families of quasi-periodic tilings in ordinary space. These admit inflation and deflation with a factor τ, like those recently constructed by Danzer, and in contrast to the factor τ3 of the Ammann tilings. One of these families is considered in detail, and various of its properties are described. In particular, each tiling in this family possesses six infinite sets of planar sections, which are analogous 2-dimensional tilings. Analogous 4-dimensional tilings are also briefly discussed.

The paper studies various relationships between locally toroidal regular 4-polytopes of types {6, 3,p} and {3, 6, 3}. These relationships are based on corresponding relationships between the regular honeycombs with the same Schlafli-symbol in hyperbolic 3-space. Also the paper discusses regular tessellations (secretions of rank 3) which are locally inscribed into regular 4-polytopes. In particular, this leads to local criteria for the finiteness of the polytopes.

regular polytopes generalize the classical concept of a regular polytope and regular tessellation to more complicated combinatorial structures with a distinctive geometrical and topological flavour. In this paper the authors give an almost complete classification of the (universal) locally toroidal regular 4-polytopes of Schlafli types {4,4,3} and {4,4,4}.

Removing the dependence on dimension of the inequalities between quermassintegrals resulting from the Aleksandrov-Fenchel inequalities leads to universal quadratic inequalities between intrinsic volumes, and to an inequality for the Wills functional. The inequalities correspond to equations which hold in the polytope algebra.

In the classical theory of regular polytopes the structure of a finite or infinite polytope (honeycomb) is governed by a real quadratic form. This form determines a geometry into which the polytope is embedded in such a way that all its symmetries are realized by isometries. The symmetry group is the Coxeter group associated with the quadratic form (cf. Coxeter [8]). These facts have important consequences for other fields in mathematics; see, for example, Tits [29, 303. The purpose of this paper is to show that much of the correspondence between polytopes and forms remains true for the theory of abstract regular polytopes, that is, regular incidence-polytopes. The concept of regular incidence-polytopes provides a suitable setting for combinatorial structures resembling the classical regular polytopes (cf. Danzer-Schulte [ 131); for related notions see also McMullen [ 171, Griinbaum [ 161, Dress [14], Buekenhout [2], and Tits [29]. In [I163 Griinbaum suggested studying abstract regular polytopes whose faces and vertex-figures are not necessarily of spherical type. He posed the problem of classifying all finite universal (or, in his notation, naturally generated) abstract regular polytopes {Y,, p see also Coxeter-Shephard [ll], Weiss [31, 321, and [24,25]. In [20,21] this problem was attacked by twisting operations on Coxeter groups and unitary reflexion groups, leading to the explicit recognition of the universal I$, $p,) for many choices of 9, and P>. In this paper we completely classify all the finite universal polytopes of

Forv>d≧3, letm(v, d) be the smallest numberm, such that every convexd-polytope withv vertices has a facet with at mostm vertices. In this paper, bounds form(v, d) are found; in particular, for fixedd≧3, $$\frac{{r - 1}}{r} \leqslant \mathop {\lim \inf }\limits_{\upsilon \to \infty } \frac{{m(\upsilon ,d)}}{\upsilon } \leqslant \mathop {\lim \sup }\limits_{\upsilon \to \infty } \frac{{m(\upsilon ,d)}}{\upsilon } \leqslant \frac{{d - 3}}{{d - 2}}$$ , wherer=[1/3(d+1)].

The paper discusses a general method for constructing regular incidence-polytopes P from certain operations on the generators of a group W, which is generated by involutions. When W is the group of a regular incidence-polytope L , this amounts to constructing the regular skew polytopes (or skew polyhedra) P associated with L .