geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, a toroidal polyhedron is a

polyhedron In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. A convex polyhedron is the convex hull of finitely many points, not all on ...

which is also a toroid (a -holed

torus In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle. If the axis of revolution does not ...

), having a topological

genus Genus ( plural genera ) is a taxonomic rank used in the biological classification of living and fossil organisms as well as viruses. In the hierarchy of biological classification, genus comes above species and below family. In binomial n ...

() of 1 or greater. Notable examples include the Császár and Szilassi polyhedra.

Variations in definition

Toroidal polyhedra are defined as collections of

polygon In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed '' polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two t ...

s that meet at their edges and vertices, forming a

manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a ...

as they do. That is, each edge should be shared by exactly two polygons, and at each vertex the edges and faces that meet at the vertex should be linked together in a single cycle of alternating edges and faces, the link of the vertex. For toroidal polyhedra, this manifold is an

orientable surface In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "counterclockwise". A space is ...

. Some authors restrict the phrase "toroidal polyhedra" to mean more specifically polyhedra topologically equivalent to the (genus 1)

. In this area, it is important to distinguish embedded toroidal polyhedra, whose faces are flat polygons in three-dimensional

Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean sp ...

that do not cross themselves or each other, from

abstract polyhedra In mathematics, an abstract polytope is an algebraic partially ordered set which captures the dyadic property of a traditional polytope without specifying purely geometric properties such as points and lines. A geometric polytope is said to be ...

, topological surfaces without any specified geometric realization. Intermediate between these two extremes are polyhedra formed by geometric polygons or

star polygon In geometry, a star polygon is a type of non- convex polygon. Regular star polygons have been studied in depth; while star polygons in general appear not to have been formally defined, certain notable ones can arise through truncation operation ...

s in Euclidean space that are allowed to cross each other. In all of these cases the toroidal nature of a polyhedron can be verified by its orientability and by its

Euler characteristic In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological spac ...

being non-positive. The Euler characteristic generalizes to ''V'' − ''E'' + ''F'' = 2 − 2''N'', where ''N'' is the number of holes.

Császár and Szilassi polyhedra

Two of the simplest possible embedded toroidal polyhedra are the Császár and Szilassi polyhedra. The Császár polyhedron is a seven-vertex toroidal polyhedron with 21 edges and 14 triangular faces. It and the

tetrahedron In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the ...

are the only known polyhedra in which every possible line segment connecting two vertices forms an edge of the polyhedron. Its dual, the

Szilassi polyhedron In geometry, the Szilassi polyhedron is a nonconvex polyhedron, topologically a torus, with seven hexagonal faces. Coloring and symmetry The 14 vertices and 21 edges of the Szilassi polyhedron form an embedding of the Heawood graph onto the surf ...

, has seven hexagonal faces that are all adjacent to each other, hence providing the existence half of the

theorem In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of ...

that the maximum number of colors needed for a map on a (genus one) torus is seven. The Császár polyhedron has the fewest possible vertices of any embedded toroidal polyhedron, and the Szilassi polyhedron has the fewest possible faces of any embedded toroidal polyhedron.

Stewart toroids

A special category of toroidal polyhedra are constructed exclusively by

regular polygon In Euclidean geometry, a regular polygon is a polygon that is direct equiangular (all angles are equal in measure) and equilateral (all sides have the same length). Regular polygons may be either convex, star or skew. In the limit, a sequence ...

faces, without crossings, and with a further restriction that adjacent faces may not lie in the same plane as each other. These are called Stewart toroids, named after Bonnie Stewart, who studied them intensively. They are analogous to the

Johnson solid In geometry, a Johnson solid is a strictly convex polyhedron each face of which is a regular polygon. There is no requirement that each face must be the same polygon, or that the same polygons join around each vertex. An example of a Johns ...

s in the case of convex polyhedra; however, unlike the Johnson solids, there are infinitely many Stewart toroids. They include also toroidal deltahedra, polyhedra whose faces are all equilateral triangles. A restricted class of Stewart toroids, also defined by Stewart, are the ''quasi-convex toroidal polyhedra''. These are Stewart toroids that include all of the edges of their convex hulls. For such a polyhedron, each face of the convex hull either lies on the surface of the toroid, or is a polygon all of whose edges lie on the surface of the toroid.

Self-crossing polyhedra

A polyhedron that is formed by a system of crossing polygons corresponds to an abstract topological manifold formed by its polygons and their system of shared edges and vertices, and the genus of the polyhedron may be determined from this abstract manifold. Examples include the genus-1

octahemioctahedron In geometry, the octahemioctahedron or allelotetratetrahedron is a nonconvex uniform polyhedron, indexed as . It has 12 faces (8 triangles and 4 hexagons), 24 edges and 12 vertices. Its vertex figure is a crossed quadrilateral. It is on ...

, the genus-3 small cubicuboctahedron, and the genus-4

great dodecahedron In geometry, the great dodecahedron is a Kepler–Poinsot polyhedron, with Schläfli symbol and Coxeter–Dynkin diagram of . It is one of four nonconvex regular polyhedra. It is composed of 12 pentagonal faces (six pairs of parallel penta ...

Crown polyhedra

A crown polyhedron or stephanoid is a toroidal polyhedron which is also noble, being both isogonal (equal vertices) and isohedral (equal faces). Crown polyhedra are self-intersecting and topologically self-dual.. See in particula
p. 60

References

External links

*{{MathWorld, urlname=ToroidalPolyhedron, title=Toroidal polyhedron
Stewart Toroids (Toroidal Solids with Regular Polygon Faces)Stewart toroids