On locally spherical polytopes of type {5, 3, 5}

Hartley, Michael I.; Leemans, Dimitri

doi:10.1016/j.disc.2007.12.084

Cited by 8 publications

(10 citation statements)

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“…Coxeter followed this with the discovery of the self-dual, locally projective 57-cell, of type {5, 3, 5}, in [2]. Work by Hartley and Leemans [7][8][9] gave some sporadic examples of polytopes of type {5, 3, 5} related to the 57-cell and its locally projective covers. Some other locally spherical polytopes of these types, including the polytopes P 1 and P 2 defined below, appeared in [5].…”

Section: Applications Of the Constructionmentioning

confidence: 96%

“…Its facets and vertex figures are tesselations of tori by squares, 15 2 squares per facet, 9 2 squares per vertex figure. More succinctly, using the notation of Section 10C of [11], P 4 is a quotient of the universal 2 T (15,0), (9,0) Note that the universal polytopes of type {{4, 4} (a,b) , {4, 4} (c,d) } have been classified, except for those where b = d = 0, and a and c are distinct odd integers (see [11] for details). This theorem therefore addresses one of the exceptional cases, as does the next.…”

Section: Applications Of the Constructionmentioning

confidence: 99%

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Covers ℘ for Abstract Regular Polytopes $\mathcal {Q}$ such that $\mathcal{Q}=\mathcal{P}/\mathbf{Z}_{p}^{k}$

Hartley¹

2009

Discrete Comput Geom

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Covers P for Abstract Regular Polytopes Q such that Q = P/Z k p Michael I. HartleyAbstract One key problem in the theory of abstract polytopes is the so-called amalgamation problem. In its most general form, this is the problem of characterising the polytopes with given facets K and vertex figures L. The first step in solving it for particular K and L is to find the universal such polytope, which covers all the others. This article explains a construction that may be attempted on an arbitrary polytope P, which often yields an infinite family of finite polytopes covering P and sharing its facets and vertex figures. The existence of such an infinite family proves that the universal polytope is infinite; alternatively, the construction can produce an explicit example of an infinite polytope of the desired type. An algorithm for attempting the construction is explained, along with sufficient conditions for it to work. The construction is applied to a few K and L for which it was previously not known whether or not the universal polytope was infinite, or for which only a finite number of finite polytopes was previously known. It is conjectured that the construction is quite broadly applicable.Keywords Abstract regular polytope · Covers of polytopes · Elementary Abelian group

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Section: Applications Of the Constructionmentioning

confidence: 96%

Section: Applications Of the Constructionmentioning

confidence: 99%

Covers ℘ for Abstract Regular Polytopes $\mathcal {Q}$ such that $\mathcal{Q}=\mathcal{P}/\mathbf{Z}_{p}^{k}$

Hartley¹

2009

Discrete Comput Geom

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“…, 3, 4}. Polytopes of this type were considered in Theorem 4.5 of [10]. By duality, we may assume that the vertex figures are projective, that is, are hemi-cross-polytopes.…”

Section: Eulerian Section Regular Polytopes In Rank D ≥mentioning

confidence: 99%

“…By duality, we may assume that the vertex figures are projective, that is, are hemi-cross-polytopes. Then (from [10]) there are only two locally projective polytopes to consider in each rank, depending on whether the facets are cubes or hemicubes. These polytopes are flat, meaning each facet has the same number of vertices as the whole polytope, that is, respectively 2 d−1 or 2 d−2 .…”

Section: Eulerian Section Regular Polytopes In Rank D ≥mentioning

confidence: 99%

Eulerian abstract polytopes

Hartley¹

2011

Aequat. Math.

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The classical theory of regular convex polytopes has inspired many combinatorial analogues. In this article, we examine two of them, the eulerian posets and the abstract regular polytopes, and see what the overlap between the concepts is. It is shown that a section regular polytope is eulerian if and only if it is spherical, or it has even rank and is locally spherical. Equivelar polytopes of rank less than 4 are eulerian, and some progress is made towards a characterisation of equivelar eulerian posets in higher rank. In particular, necessary conditions are given for an equivelar quotient of a cube or a torus to be eulerian. Mathematics Subject Classification (2000). 51M20, 20F65, 52B15.

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“…There are many other possible choices for Q (for example, see [24,46]); however, we cannot choose Coxeter's 57-cell {{5, 3} 5 , {5, 3} 5 }, since this is not directly regular (see [15]). [3,5,3], whose quotient is isomorphic to L 2 (p).…”

Section: Locally Spherical Polytopesmentioning

confidence: 99%