Toroidal polyhedron

Toroidal polyhedra are defined as a set of polygons that have common vertices and edges, forming a manifold . That is, each edge must be common exactly for two polygons, the vertex figure of each vertex must be one cycle of the polygons to which this vertex belongs. For toroidal polyhedra, this manifold will be an oriented surface ^[1] . Some authors restrict the concept of a “toroidal polyhedron” to polytopes that are topologically equivalent to (genus 1) torus ^[2] .

Here it is necessary to distinguish embedded toroidal polyhedra, whose faces are flat non-intersecting polygons in three - dimensional Euclidean space , from , topological surfaces without a specific geometric implementation ^[3] . The middle between these two extremes can be considered submerged toroidal polyhedra, that is, polyhedra formed by polygons or star-shaped polygons in Euclidean space, which are allowed to intersect each other.

In all these cases, the toroidal nature of the polyhedra can be verified by their orientation and Euler characteristic, which is not positive for these polyhedra.

The two simplest possible nested toroidal polyhedra are the polyhedra of Chasar and Silashi.

The Chasar polyhedron is a toroidal polyhedron with seven vertices, 21 edges and 14 triangular faces ^[4] . Only this polyhedron and the tetrahedron (from the known ones) have the property that any segment connecting the vertices of the polyhedron is an edge of the polyhedron ^[5] . The dual polyhedron is the Silashi polyhedron , which has 7 hexagonal faces, each pair of which is adjacent to each other ^[6] , providing half of the theorem that the maximum value of colors for coloring a map on a torus (genus 1) is seven ^[7] .

The Chasar polyhedron has the smallest possible number of vertices that a nested toroidal polyhedron can have, and the Silashi polyhedron has the smallest possible number of faces.

A special category of toroidal polyhedra is constructed exclusively with the help of regular polygonal faces without intersecting them with the additional restriction that adjacent faces do not lie in the same plane. These polyhedra are called Stewart toroids ^[8] by the name of Professor , who investigated their existence ^[9] . They are similar to Johnson bodies in the case of convex polyhedra , but, unlike them, there are infinitely many Stuart toroids ^[10] . These polyhedra also include toroidal deltahedrons , polyhedra whose faces are equilateral triangles.

A limited class of Stuart toroids, also defined by Stuart, are quasiconvex toroidal polyhedra . These are Stuart toroids, which include all the edges of their convex hulls . For these polyhedra, each face of the convex hull either lies on the surface of the toroid or is a polygon whose edges lie on the surface of the toroid ^[11] .

A polyhedron formed by a system of intersecting polygons in space is a multifaceted immersion of an abstract topological manifold formed by its polygons and its system of edges and vertices. Examples include the (genus 1), the (genus 3), and the large dodecahedron (genus 4).

A crowned polyhedron (or stephanoid ) is a toroidal polyhedron, which is a polyhedron, being both isogonal (identical types of vertices) and (identical faces). The crowned polyhedron is self-intersecting and topologically self-dual ^[12] .

• Spherical polyhedron
• Toroid graph

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• Robert Webb. Stella: polyhedron navigator // Symmetry: Culture and Science. - 2000. - T. 11 , no. 1-4 .
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• Günter M. Ziegler. Discrete Differential Geometry / AI Bobenko, P. Schröder, JM Sullivan, GM Ziegler. - Springer-Verlag, 2008 .-- T. 38 . - ISBN 978-3-7643-8620-7 . - DOI : 10.1007 / 978-3-7643-8621-4_10 . - arXiv : math.MG/0412093 .
• Walter Whiteley. Realizability of polyhedra // Structural Topology. - 1979. - Vol. 1 .
• William T. Webber. Monohedral idemvalent polyhedra that are toroids // Geometriae Dedicata . - 1997. - T. 67 , no. 1 . - DOI : 10.1023 / A: 1004997029852 .

• Weisstein, Eric W. Toroidal polyhedron on the Wolfram MathWorld website.
• Stewart Toroids (Toroidal Solids with Regular Polygon Faces)
• Stewart's polyhedra
• Toroidal polyhedra
• Stewart toroids