Polyhedral 2-manifolds inE 3 with unusually large genus

McMullen, Peter; Schulz, Ch.; Wills, J. M.

doi:10.1007/bf02760627

Cited by 49 publications

(165 citation statements)

References 5 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: 94%

“…Their underlying maps inherit their combinatorial symmetries from two sources, the symmetries and the dualities of the (self-dual) 4-simplex; in fact, the combinatorial automorphism group is isomorphic to S 5 × C 2 . It was observed by McMullen, Schulz and Wills [1,2] that, via projection from E 4 , they can be polyhedrally embedded with convex faces in E 3 . For illustrations of these polyhedra in E 3 see [27,36].…”

Section: Coxeter's Geometric Skew Polyhedramentioning

confidence: 99%

“…The map of type {4, q} can be realized (with square faces) in the 2-skeleton of the q-dimensional cube in E q . In McMullen, Schulz and Wills [1,2], two remarkable infinite series of polyhedra in E 3 of types {4, q} and {q, 4} were described, and in [37] were proved to be polyhedral embeddings of Coxeter's regular maps of these types. Thus all of Coxeter's maps can be realized as polyhedra in E 3 .…”

Section: Coxeter's Topological Analogues Of Skew Polyhedramentioning

confidence: 99%

“…A polyhedron P , for the purpose of this paper, is a closed compact surface in Euclidean 3-space E 3 made up from finitely many convex polygons, called the faces of P , such that any two polygons intersect, if at all, in a common vertex or a common edge (see McMullen, Schulz and Wills [1,2], Brehm and Wills [3], and Brehm and Schulte [4]). The vertices and edges of P are the vertices and edges of the faces of P , respectively.…”

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: 94%

Section: Coxeter's Geometric Skew Polyhedramentioning

confidence: 99%

Section: Coxeter's Topological Analogues Of Skew Polyhedramentioning

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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“…An example where all the h j are required for the specification is provided by Coxeter's polyhedron {4, n|4 n/2−1 } (see [2]; in [14]- [16] embeddings in E 3 are found of this polyhedron, but of course without full symmetry).…”

Section: Facettingmentioning

confidence: 99%

Regular Polytopes in Ordinary Space

McMullen¹,

Schulte²

1997

Discrete Comput Geom

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Abstract. The three aims of this paper are to obtain the proof by Dress of the completeness of the enumeration of the Grünbaum-Dress polyhedra (that is, the regular apeirohedra, or apeirotopes of rank 3) in ordinary space E 3 in a quicker and more perspicuous way, to give presentations of those of their symmetry groups which are affinely irreducible, and to describe all the discrete regular apeirotopes of rank 4 in E 3 . The paper gives a complete classification of the discrete regular polytopes in ordinary space.

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