Equivelar polyhedral manifolds inE 3

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

doi:10.1007/bf02760539

Cited by 40 publications

(38 citation statements)

References 8 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: 93%

“…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: 93%

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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“…Brehm and Schild [8] showed that every triangulation of the torus does have a realization in R 4 . We refer the reader to [4], [18], [21], and [22] for further results on the realizability of certain manifolds.…”

Section: Introductionmentioning

confidence: 99%

How to Exhibit Toroidal Maps in Space

Archdeacon

Bonnington

Ellis-Monaghan

2007

Discrete Comput Geom

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Abstract. Steinitz's theorem states that a graph is the 1-skeleton of a convex polyhedron if and only if it is 3-connected and planar. The polyhedron is called a geometric realization of the embedded graph. Its faces are bounded by convex polygons whose points are coplanar.A map on the torus does not necessarily have such a geometric realization. In this paper we relax the condition that faces are the convex hull of coplanar points. We require instead that the convex hull of the points on a face can be projected onto a plane so that the boundary of the convex hull of the projected points is the image of the boundary of the face. We also require that the interiors of the convex hulls of different faces do not intersect. Call this an exhibition of the map. A map is polyhedral if the intersection of any two closed faces is simply connected. Our main result is that every polyhedral toroidal map can be exhibited. As a corollary, every toroidal triangulation has a geometric realization.

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Semi-Platonic Manifolds

Wills

1983

Convexity and Its Applications

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Equivelar polyhedral manifolds inE 3

Cited by 40 publications

References 8 publications

Convex-Faced Combinatorially Regular Polyhedra of Small Genus

Convex-Faced Combinatorially Regular Polyhedra of Small Genus

How to Exhibit Toroidal Maps in Space

Semi-Platonic Manifolds

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