Regular Skew Polyhedra in Three and Four Dimension, and their Topological Analogues

Coxeter, H. S. M.

doi:10.1112/plms/s2-43.1.33

Cited by 126 publications

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“…There are similar such polyhedral embeddings for Coxeter's regular skew polyhedra {4, 8|3} and {8, 4|3}, a dual pair of genus 73, but they fall outside the genus range under consideration (see [1,2,10,23,36]). …”

Section: Coxeter's Geometric Skew Polyhedramentioning

confidence: 99%

“…The corresponding 4-dimensional polyhedra are two of Coxeter's regular skew polyhedra in E 4 and have (convex) squares or regular hexagons as faces. These polyhedra in E 4 actually trace back to Alicia Boole Stott (see Stott [23] and also p. 45 of Reference [10]), but it was Coxeter who fully explored and popularized them. They are closely related to the regular 4-simplex.…”

Section: Coxeter's Geometric Skew Polyhedramentioning

confidence: 99%

“…For each q 3, the combinatorial automorphism group of either map is C q 2 D q ( ∼ = C 2 D q ), and the genus is 2 q−3 (q − 4) + 1. The two series were first discovered by Coxeter [10,14] in 1937, but have been rediscovered several times since then (see pp. 260, 261 of Reference [6] for more details).…”

Section: Coxeter's Topological Analogues Of Skew Polyhedramentioning

confidence: 99%

“…For g 6, a full classification was first obtained by Sherk [8] (for genus 3) and Garbe [9]. Historically, genus 6 provides an important upper bound on the genera for two reasons: first, two of Coxeter's [10] classical regular skew polyhedra discovered in the early days of the study of maps on surfaces have genus 6; and second, and more importantly, genus 6 turned out to be the threshold for a by-hand enumeration of regular maps by genus.…”

Section: Introductionmentioning

confidence: 99%

“…Only eight regular maps in this range are known to have polyhedral realizations, and two were only discovered quite recently. Some of these polyhedra are quite spectacular and realize famous maps of Klein, Fricke, Dyck, and Coxeter (see [5,10,[14][15][16][17][18][19]). In Section 2 we briefly review these polyhedra and their maps.…”

Section: Introductionmentioning

confidence: 99%

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Convex-Faced Combinatorially Regular Polyhedra of Small Genus

Schulte

Wills

2011

Symmetry

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Combinatorially regular polyhedra are polyhedral realizations (embeddings) in Euclidean 3-space E 3 of regular maps on (orientable) closed compact surfaces. They are close analogues of the Platonic solids. A surface of genus g 2 admits only finitely many regular maps, and generally only a small number of them can be realized as polyhedra with convex faces. When the genus g is small, meaning that g is in the historically motivated range 2 g 6, only eight regular maps of genus g are known to have polyhedral realizations, two discovered quite recently. These include spectacular convex-faced polyhedra realizing famous maps of Klein, Fricke, Dyck, and Coxeter. We provide supporting evidence that this list is complete; in other words, we strongly conjecture that in addition to those eight there are no other regular maps of genus g, with 2 g 6, admitting realizations as convex-faced polyhedra in E 3 . For all admissible maps in this range, save Gordan's map of genus 4, and its dual, we rule out realizability by a polyhedron in E 3 .

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Section: Coxeter's Geometric Skew Polyhedramentioning

confidence: 99%

Section: Coxeter's Geometric Skew Polyhedramentioning

confidence: 99%

Section: Coxeter's Topological Analogues Of Skew Polyhedramentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Convex-Faced Combinatorially Regular Polyhedra of Small Genus

Schulte

Wills

2011

Symmetry

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Arc‐transitive maps with underlying Rose Window graphs

Hubard

Ramos-Rivera

Šparl

2020

Journal of Graph Theory

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In the late 1990s, Graver and Watkins initiated the study of all edge-transitive maps. Recently, Gareth Jones revisited the study of such maps and suggested classifying the maps in terms of either their automorphism groups or their underlying graphs. A natural step towards classifying edge-transitive maps is to study the arc-transitive ones. In this paper, we investigate the connection of a class of arc-transitive maps to consistent cycles of the underlying graph, with special emphasis on maps of smallest possible valence, namely 4. We give a complete classification of arc-transitive maps whose underlying graphs are arc-transitive Rose Window graphs. K E Y W O R D S arc-transitive graph, arc-transitive map, consistent cycle, 2-orbit map maps, based on the local configuration of the flags (incident vertex-edge-face triples) and the orbits they belong to (see also [24]). Two of these 14 classes correspond to rotary maps. Even though most of the papers on edge-transitive maps in the existing literature focus on these two classes of rotary maps, some papers also investigated other classes (see, for instance [19, 23]). In these papers questions like which edge-transitive maps of small genera exist, and which of the 14 types of edge-transitive maps can be realized by maps with an automorphism group abstractly isomorphic to a symmetric group, have been considered. Recently, Jones (see [12]) extended the work of [23] while working on the question of which groups can act as the automorphism group of an edge-transitive map by answering it for some specific classes of groups. This question naturally extends to the question of which graphs can be the underlying graph of an edge-transitive map. Moreover, in both cases one can ask which are the classes of edge-transitive maps that can occur, how many maps of a given class are there and whether one can describe all of them. When studying edge-transitive maps it is natural to restrict to a subset of them: those whose automorphism group acts transitively on the set of arcs. Rotary maps are an example of such maps, but again, there are others. For example, the cuboctahedron (or medial of the cube) seen as a map on the sphere is arc-transitive, but it is not a rotary map. As already mentioned, rotary maps can be divided into two different classes: one that contains maps admitting involutory automorphisms that fix a given arc but interchange the two faces incident to it (maps in this class are called regular or reflexible maps), and the other, which contains the rest of the rotary maps (maps in this class are called chiral). The automorphism group of a reflexible map is transitive on the set of its flags, while the automorphism group of a chiral map has two orbits on the set of its flags. As we shall see, it is not difficult to prove that there are exactly five classes of arc-transitive maps. When one applies the Petrie dual operator to a map, the underlying graph, as well as the automorphism group of the map, remain unchanged. Hence, classes of arc-transitive maps are linked via th...

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Rigidity of Regular Polytopes

McMullen

2014

Rigidity and Symmetry

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Regular Skew Polyhedra in Three and Four Dimension, and their Topological Analogues

Cited by 126 publications

References 0 publications

Convex-Faced Combinatorially Regular Polyhedra of Small Genus

Convex-Faced Combinatorially Regular Polyhedra of Small Genus

Arc‐transitive maps with underlying Rose Window graphs

Rigidity of Regular Polytopes

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