共查询到10条相似文献,搜索用时 109 毫秒
1.
Hartley 《Discrete and Computational Geometry》2008,29(3):435-443
Abstract. This paper classifies the quotients of a finite and locally projective polytope of type {4,3,5} . Seventy quotients are found, including three regular polytopes, and nine other section regular polytopes which are themselves
locally projective. The classification is done with the assistance of GAP, a computer system for algebraic computation. The
same techniques are also applied to two finite locally projective polytopes respectively of type {3,5,3} and {5,3,5} . No nontrivial quotients of the latter are found. 相似文献
2.
Michael I. Hartley 《Combinatorica》2008,28(3):299-314
This article completes the classification of finite universal locally projective regular abstract polytopes, by summarising
(with careful references) previously published results on the topic, and resolving the few cases that do not appear in the
literature. In rank 4, all quotients of the locally projective polytopes are also noted. In addition, the article almost completes
the classification of the infinite universal locally projective polytopes, except for the {{5,3,3,},{3,3,5}15} and its dual. It is shown that this polytope cannot be finite, but its existence is not established. The most remarkable
feature of the classification is that a nondegenerate universal locally projective polytope is infinite if and only if the rank of is 5 and the facets of or its dual are the hemi-120-cell {5,3,3}15. 相似文献
3.
There are only finitely many locally projective regular polytopes of type {5, 3, 5}. They are covered by a locally spherical polytope whose automorphism group is J1×J1×L2(19), where J1 is the first Janko group, of order 175560, and L2(19) is the projective special linear group of order 3420. This polytope is minimal, in the sense that any other polytope that covers all locally projective polytopes of type {5, 3, 5} must in turn cover this one. 相似文献
4.
Locally projective polytopes of type 总被引:1,自引:0,他引:1
Michael I. Hartley 《Journal of Algebra》2005,290(2):322-336
This paper attempts to classify the locally projective section regular n-polytopes of type {4,3,…,3,p}, that is, to classify polytopes whose facets are cubes or hemicubes, and the vertex figures are spherical or projective polytopes of type {3,…,3,p}, with the facets and vertex figures being not both spherical. Spherical or projective (n−1)-polytopes of type {3,…,3,p} only exist when p4, or p=5 and n−14, or n−1=2. However, some existence and non-existence results are obtained for other values of p and n. In particular, a link is derived between the existence of polytopes of certain types, and vertex-colourability of certain graphs. The main result of the paper is that locally projective section regular n-polytopes exist only when p=4, or when p=5 and n=4 or 5. 相似文献
5.
Abstract polytopes are partially ordered structures which generalize the notion of polyhedra in a combinatorial sense. This
concept includes all of the classical regular polytopes as well as many other well-known configurations. Chiral polytopes
are abstract polytopes with maximal rotational symmetry which lack reflexive symmetry. We employ hyperbolic geometry to derive
toroidal abstract polytopes of type {6, 3, 4} and {6, 3, 5} which are either regular or chiral. Their rotation groups are
projective linear groups over finite rings. 相似文献
6.
Using modular quotients of linear groups defined over the Eisenstein ring Z[], we construct infinite families of finite regular or chiral polytopes of types {3,3,6}, {3,6,3} and {6,3,6}. 相似文献
7.
In ``Quotients of a Universal Locally Projective Polytope' (Math. Z. 247 663–674, DOI: 10.1007/s00209-003-0625-9), the authors analyse the a certain locally projective universal polytope, showing
it to be finite, and enumerating its quotients. The authors have since discovered some errors in the enumeration of the quotients.
This note corrects these errors.
The online version of the original article can be found at 相似文献
8.
Michael I. Hartley 《Annals of Combinatorics》2006,10(3):343-352
The main results of this article facilitate the search for quotients of regular abstract polytopes. A common approach in the
study of abstract polytopes is to construct polytopes with specified facets and vertex figures. Any nonregular polytope
may be constructed as a quotient of a regular polytope
by a (so-called) semisparse subgroup of its automorphism group W (which will be a string C-group). It becomes important, therefore, to be able to identify whether or not a given subgroup N of a string C-group W is semisparse. This article proves a number of properties of semisparse subgroups. These properties may be used to test for
semisparseness in a way which is computationally more efficient than previous methods. The methods are used to find an example
of a section regular polytope of type {6, 3, 3} whose facets are Klein bottles.
Received February 15, 2005 相似文献
9.
Peter McMullen Egon Schulte 《Transactions of the American Mathematical Society》1996,348(4):1373-1410
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 and . 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.
10.
M. I. Hartley 《Aequationes Mathematicae》1999,57(1):108-120
Summary. In an earlier paper, it was shown that every abstract polytope is a quotient Q = M(W)/N {\cal Q} = {\cal M}(W)/N of some regular polytope M(W) {\cal M}(W) whose automorphism group is W, by a subgroup N of W. In this paper, attention is focussed on the quotient Q {\cal Q} , and various important structures relating to polytopes are described in terms of N ', the stabilizer of a flag of the quotient under an action of W (the 'flag action'). It is pointed out how N ' may be assumed without loss of generality to equal N. The paper also shows what properties of N ' yield polytopes which are regular, section regular, chiral, locally regular, or locally universal. The aim is to make it more practical to study non-regular polytopes in terms of group theory. 相似文献