Chiral Polytopes Research Articles

Abelian covers of regular hypertopes

Known as thin residually connected incidence geometry, hypertopes extend the framework of abstract polytopes, and can be built from Coxeter groups (not necessarily with linear diagrams). A regular hypertope is a flag-transitive hypertope. In this paper, we present infinite families of regular hypertopes of ranks 5, 6 and 7, in terms of a certain group covering approach (analogous to a method introduced by Conder and the author in an earlier paper on abstract chiral polytopes, but with wider adaptation). Although the illustrative examples within these families are derived from Coxeter groups exhibiting Y-shaped diagrams, this approach is applicable to obtaining regular hypertopes from Coxeter groups with other diagrams, such that the size of the family members grows linearly with entries of their types.

Answers to questions about medial layer graphs of self-dual regular and chiral polytopes

Answers to questions about medial layer graphs of self-dual regular and chiral polytopes

Four infinite families of chiral 3-polytopes of type {4,8} with solvable automorphism groups

We construct four infinite families of chiral 3-polytopes of type {4,8}, with 1024m4, 2048m4, 4096m4 and 8192m4 automorphisms for every positive integer m, respectively. The automorphism groups of these polytopes are solvable groups, and when m is a power of 2, they provide examples with automorphism groups of order 2n where n≥10. (On the other hand, no chiral polytopes of type {4,8} exist for n≤9.) In particular, our families give a partial answer to a problem proposed by Schulte and Weiss (2006) [15] and a problem proposed by Pellicer (2012) [13].

Chiral polytopes whose smallest regular cover is a polytope

We give a criterion for when the smallest regular cover of a chiral polytope P is itself a polytope, using only information about the facets and vertex-figures of P.

More on chiral polytopes of type \{4, 4, …, 4\} with~solvable automorphism groups

Abstract In a 2021 paper, Conder et al. constructed two infinite families of chiral 4-polytopes of type { 4 , 4 , 4 } \{4,4,4\} with solvable automorphism groups. Here we present a general construction for chiral polytopes of type { 4 , 4 , … , 4 } \{4,4,\dots,4\} with rank 4, 5 and 6, which are obtained as Boolean covers of the unique tight regular polytope of the same type.

All polytopes are coset geometries: Characterizing automorphism groups of [formula omitted]-orbit abstract polytopes

Abstract polytopes generalize the classical notion of convex polytopes to more general combinatorial structures. The most studied ones are regular and chiral polytopes, as it is well-known, they can be constructed as coset geometries from their automorphism groups. This is also known to be true for 2- and 3-orbit 3-polytopes. In this paper we show that every abstract n-polytope can be constructed as a coset geometry. This construction is done by giving a characterization, in terms of generators, relations and intersection conditions, of the automorphism group of a k-orbit polytope with given symmetry type graph. Furthermore, we use these results to show that for all k≠2, there exist k-orbit n-polytopes with Boolean automorphism groups, for all n≥3.

On chiral polytopes having a group PSL(3,q)$\mathrm{PSL}(3,q)$ as automorphism group

For each prime power q ⩾ 5 $q\geqslant 5$ , we construct a rank four chiral polytope that has a group PSL ( 3 , q ) $\mathrm{PSL}(3,q)$ as automorphism group and Schläfli type [ q − 1 , 2 ( q − 1 ) ( 3 , q − 1 ) , q − 1 ] $[q-1,\frac{2(q-1)}{(3,q-1)},q-1]$ . We also construct rank five polytopes for some values of q $q$ and we show that there is no chiral polytope of rank at least six having a group PSL ( 3 , q ) $\mathrm{PSL}(3,q)$ or PSU ( 3 , q ) $\mathrm{PSU}(3,q)$ as automorphism group.

On the Schläfli symbol of chiral extensions of polytopes

Given an abstract n-polytope K, an abstract (n+1)-polytope P is an extension of K if all the facets of P are isomorphic to K. A chiral polytope is a polytope with maximal rotational symmetry that does not admit any reflections. If P is a chiral extension of K, then all but the last entry of the Schläfli symbol of P are determined. In this paper we introduce some constructions of chiral extensions P of certain chiral polytopes in such a way that the last entry of the Schläfli symbol of P is arbitrarily large.

A chiral 5-polytope of full rank

The first chiral polytope of full rank known was Roli’s cube, a rank 4 polytope discovered in 2014. Since then, other chiral polytopes of rank 4 have been studied. Here we present the first known chiral 5-polytope of full rank. Its 2-skeleton is the same as that of the Euclidean tessellation {3,4,3,3} with 24-cells.

Tight chiral polytopes

A chiral polytope with Schlafli symbol $$\{p_1, \ldots , p_{n-1}\}$$ has at least $$2p_1 \cdots p_{n-1}$$ flags, and it is called tight if the number of flags meets this lower bound. The Schlafli symbols of tight chiral polyhedra were classified in an earlier paper, and another paper proved that there are no tight chiral n-polytopes with $$n \ge 6$$ . Here we prove that there are no tight chiral 5-polytopes, describe 11 families of tight chiral 4-polytopes, and show that every tight chiral 4-polytope covers a polytope from one of those families.

A new algorithm to classify chiral polytopes with a given automorphism group

We present a new algorithm to compute all the chiral polytopes that have a given group as full automorphism group. This algorithm uses a new set of generators that characterize the group, all of them except one being involutions. It permits to compute all chiral polytopes of groups that were previously unreachable by other known algorithms.

On the rank of Suzuki polytopes: An answer to Hubard and Leemans

In this paper we show that the rank of every chiral polytope having a Suzuki group as automorphism group is 3. This gives a positive answer to a conjecture of Isabel Hubard and Dimitri Leemans.

Two infinite families of chiral polytopes of type {4,4,4} with solvable automorphism groups

We construct two infinite families of locally toroidal chiral polytopes of type {4,4,4}, with 1024m2 and 2048m2 automorphisms for every positive integer m, respectively. The automorphism groups of these polytopes are solvable groups, and when m is a power of 2, they provide examples with automorphism groups of order 2n where n can be any integer greater than 9. (On the other hand, no chiral polytopes of type [4,4,4] exist for n≤9.) In particular, our two families give a partial answer to a problem proposed by Schulte and Weiss in [21].

Chiral polytopes of full rank exist only in ranks 4 and 5

Skeletal chiral polytopes of rank n in a Euclidean space must have as ambient space $${\mathbb {E}}^d$$ for some $$d \ge n$$ if they are finite, or some $$d \ge n-1$$ if they are infinite. If the dimension attains the lower bound just mentioned, we say that the polytope is of full rank. In this article it is proven that a chiral polytope of full rank can only have rank 4 or 5.

Some simplifications of the intersection condition of chiral form for polytopes

To determine if a poset of type [Formula: see text] is a directly regular or chiral polytope, it is necessary to test whether or not its rotation group (as a quotient of the orientation-preserving subgroup of the Coxeter group [Formula: see text]) satisfies the so-called intersection condition of chiral form. However, due to the fact that many cases need to be checked, this process is often very tedious and takes much time. In this paper, under certain circumstances, we give some simplifications for checking the intersection condition, which leads to certain constructions for directly regular or chiral polytopes.

Non-Flat Regular Polytopes and Restrictions on Chiral Polytopes

An abstract polytope is flat if every facet is incident on every vertex. In this paper, we prove that no chiral polytope has flat finite regular facets and finite regular vertex-figures. We then determine the three smallest non-flat regular polytopes in each rank, and use this to show that for $n \geq 8$, a chiral $n$-polytope has at least $48(n-2)(n-2)!$ flags.

Chiral 4-polytopes in ordinary space

In this paper we list three chiral 4-polytopes in Euclidean space, and show that this list is complete. This concludes the classification of chiral polytopes of full rank in 3-space. We also prove that these three chiral 4-polytopes are combinatorially chiral.

Abelian covers of chiral polytopes

Abstract polytopes are combinatorial structures with certain properties drawn from the study of geometric structures, like the Platonic solids, and of maps on surfaces. Of particular interest are the polytopes with maximal possible symmetry (subject to certain natural constraints). Symmetry can be measured by the effect of automorphisms on the ‘flags’ of the polytope, which are maximal chains of elements of increasing rank (dimension). An abstract polytope of rank n is said to be chiral if its automorphism group has precisely two orbits on the flags, such that two flags that differ in one element always lie in different orbits. Examples of chiral polytopes have been difficult to find and construct. In this paper, we introduce a new covering method that allows the construction of some infinite families of chiral polytopes, with each member of a family having the same rank as the original, but with the size of the members of the family growing linearly with one (or more) of the parameters making up its ‘type’ (Schläfli symbol). In particular, we use this method to construct several new infinite families of chiral polytopes of ranks 3, 4, 5 and 6.

A chiral 4-polytope in R^3

In this paper we describe an infinite chiral 4-polytope in the Euclidean 3-space. This builds on previous work of Bracho, Hubard and the author, where a finite chiral 4-polytope in the Euclidean 4-space is constructed. These two polytopes show that there are finite and infinite chiral polytopes of full rank as defined by McMullen.

Projective linear groups as automorphism groups of chiral polytopes

It is already known that the automorphism group of a chiral polyhedron is never isomorphic to \( PSL \left( 2,q\right) \) or \( PGL \left( 2,q\right) \) for any prime power q. In this paper, we show that \( PSL \left( 2,q\right) \) and \( PGL \left( 2,q\right) \) are never automorphism groups of chiral polytopes of rank at least 5. Moreover, we show that \( PGL \left( 2,q\right) \) is the automorphism group of at least one chiral polytope of rank 4 for every \(q\ge 5\). Finally, we determine for which values of q the group \( PSL \left( 2,q\right) \) is the automorphism group of a chiral polytope of rank 4, except when \(q=p^d\equiv 3\pmod {4}\) where \(d>1\) is not a prime power, in which case the problem remains unsolved.