A chiral 4-polytope in R^3

Pellicer, Daniel

doi:10.26493/1855-3974.1055.0f4

Cited by 5 publications

(1 citation statement)

References 11 publications

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“…We conclude this section with the description of the polyhedron P 1 (1, 0) π . The geometric aspects of P 1 (1, 0) were discussed in Pellicer (2017). Here we describe the 1-skeleton in another way that better suits our purposes.…”

Section: Two Orbit Polyhedra Derived From Petrie Dualitymentioning

confidence: 99%

On infinite 2-orbit polyhedra in classes $$2_0$$ and $$2_2$$

Pellicer

Williams

2023

Beitr Algebra Geom

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We describe all 2-orbit skeletal polyhedra in classes $$2_0$$ 2 0 and $$2_2$$ 2 2 in $$\mathbb {E}^3$$ E 3 . Those in class $$2_2$$ 2 2 are continuous deformations of the three Petrie–Coxeter polyhedra and therefore are arranged in three families. Those in class $$2_0$$ 2 0 are the Petrie duals of the chiral polyhedra in $$\mathbb {E}^3$$ E 3 and are classified in six families.

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Section: Two Orbit Polyhedra Derived From Petrie Dualitymentioning

confidence: 99%

On infinite 2-orbit polyhedra in classes $$2_0$$ and $$2_2$$

Pellicer

Williams

2023

Beitr Algebra Geom

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Skeletal Geometric Complexes and Their Symmetries

Schulte

Weiss

2017

Math Intelligencer

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Polyhedra and polyhedron-like structures have been studied since the early days of geometry (Coxeter [5]). The most well-known are the five Platonic solids -the tetrahedron, octahedron, cube, icosahedron, and dodecahedron. Book XIII of Euclid's "Elements" [10] was entirely devoted to their mathematical properties.In modern terminology, the Platonic solids are precisely the regular convex polyhedra in ordinary Euclidean space E 3 . Historically, as the name suggests, these figures were viewed as solids bounded by a collection of congruent regular polygons fitting together in a highly symmetric fashion. With the passage of time, the perception of what should constitute a polyhedron has undergone fundamental changes. The famous Kepler-Poinsot star polyhedra no longer are solids but instead form self-intersecting polyhedral structures that feature regular star polygons as faces or vertex-figures. These four "starry" figures are precisely the regular star polyhedra in E 3 [5], which along with the five Platonic solids might be called the classical regular polyhedra [18] and are shown in Figure 1 [31, 32].Why say that a polyhedron must be finite? In the 1930's, Petrie and Coxeter discovered three more regular polyhedra in E 3 , each forming a non-compact polyhedral surface which is tessellated by regular convex polygons and bounds two unbounded (congruent) "solids" (infinite "handlebodies"). The three polyhedra are shown in Figure 2 [31, 32]. The trick here is to allow the vertex-figures to be skew polygons rather than convex polygons. The vertex-figure of a polyhedron P at a vertex v is the polygon whose vertices are the vertices *

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Quasi-Regular Polytopes of Full Rank

McMullen

2021

Discrete Comput Geom

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A chiral 4-polytope in R^3

Cited by 5 publications

References 11 publications

On infinite 2-orbit polyhedra in classes $$2_0$$ and $$2_2$$

On infinite 2-orbit polyhedra in classes $$2_0$$ and $$2_2$$

Skeletal Geometric Complexes and Their Symmetries

Quasi-Regular Polytopes of Full Rank

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