All Polytopes Are Quotients, and Isomorphic Polytopes Are Quotients by Conjugate Subgroups

Hartley, Michael I.

doi:10.1007/pl00009422

Cited by 44 publications

(67 citation statements)

References 8 publications

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“…Given a 4-orbit toroidal map, the minimal regular cover and the minimal rotary cover may or may not coincide; this depends on γ. For example, the minimal regular and minimal rotary covers of the map {4, 4} (2,1), (7,0) coincide. On the other hand, the covers of {4, 4} (4,3),(5,0) = {4, 4} (4,3),(−1,3) are distinct (see Figure 2 (a)).…”

Section: Maps Of Type {4 4}mentioning

confidence: 99%

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Minimal covers of equivelar toroidal maps

Drach¹,

Mixer

2014

Ars Math. Contemp.

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Given any equivelar map on the torus, it is natural to consider its covering maps. The most basic of these coverings are finite toroidal maps or infinite tessellations of the Euclidean plane. In this paper, we prove that each equivelar map on the torus has a unique minimal toroidal rotary cover and also a unique minimal toroidal regular cover. That is to say, of all the toroidal rotary (or regular) maps covering a given map, there is a unique smallest. Furthermore, using the Gaussian and Eisenstein integers, we construct these covers explicitly.

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Section: Maps Of Type {4 4}mentioning

confidence: 99%

“…However, equivelar toroidal maps can be covered by smaller maps that also have high degrees of symmetry. Notably, each equivelar toroidal map has a unique finite minimal regular cover (see [7], [14]). …”

Section: Introductionmentioning

confidence: 99%

Minimal covers of equivelar toroidal maps

Drach¹,

Mixer

2014

Ars Math. Contemp.

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“…In [9] it is shown that any polytope Q may be written in the form P/N for some regular polytope P and some subgroup N of the automorphism group of P. These subgroups N of Γ(P) = ρ 0 , . .…”

Section: Chiral Polytopes As Quotients Of Regular Onesmentioning

confidence: 99%

Two atlases of abstract chiral polytopes for small groups

2012

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We construct chiral abstract polytopes in two different ways. Firstly we seek them as quotients of regular polytopes arising from the Atlas of Small Regular Polytopes (http: //www.abstract-polytopes.com/atlas/); the resulting atlas of chiral polytopes atlas is available on the website http://www.abstract-polytopes.com/ chiral/. Secondly, for each almost simple group Γ such that S ≤ Γ ≤ Aut(S) where S is a simple group and Γ is a group of order less than 900,000 listed in the Atlas of Finite Groups, we give, up to isomorphism, the number of abstract chiral polytopes on which Γ acts regularly. The results have been obtained using a series of MAGMA programs. All these polytopes are made available on the third author's website, at http: //math.auckland.ac.nz/˜dleemans/CHIRAL/.

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“…The polytope K is isomorphic to the quotient P/N where N is the stabilizer in Γ(P) of a flag of K. Here P/N can be thought of as the combinatorial structure resulting from identifying the image Φw of the base flag Φ with all flags of the form Φhw where h ∈ N , for every w in the monodromy group. We refer to [22] and [37, Section 2D] for more information on flag actions and quotients.…”

Section: Group Structuresmentioning

confidence: 99%