geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

, a toroidal polyhedron is a

polyhedron In geometry, a polyhedron (: polyhedra or polyhedrons; ) is a three-dimensional figure with flat polygonal Face (geometry), faces, straight Edge (geometry), edges and sharp corners or Vertex (geometry), vertices. The term "polyhedron" may refer ...

which is also a

toroid In mathematics, a toroid is a surface of revolution with a hole in the middle. The axis of revolution passes through the hole and so does not intersect the surface. For example, when a rectangle is rotated around an axis parallel to one of its ...

(a -holed

torus In geometry, a torus (: tori or toruses) is a surface of revolution generated by revolving a circle in three-dimensional space one full revolution about an axis that is coplanarity, coplanar with the circle. The main types of toruses inclu ...

), having a

topological Topology (from the Greek words , and ) is the branch of mathematics concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, wit ...

genus Genus (; : genera ) is a taxonomic rank above species and below family (taxonomy), family as used in the biological classification of extant taxon, living and fossil organisms as well as Virus classification#ICTV classification, viruses. In bino ...

() 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 made up of line segments connected to form a closed polygonal chain. The segments of a closed polygonal chain are called its '' edges'' or ''sides''. The points where two edges meet are the polygon ...

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 N ...

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. 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, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...

that do not cross themselves or each other, from abstract polyhedra, 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, Decagram (geometry)#Related figures, certain notable ones can ...

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 space's ...

being non-positive. The Euler characteristic generalizes to ''V'' − ''E'' + ''F'' = 2 − 2''g'', where ''g'' is its topological genus.

Császár and Szilassi polyhedron

The Császár polyhedron is a seven-vertex toroidal polyhedron with 21 edges and 14 triangular faces. It and the

tetrahedron In geometry, a tetrahedron (: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular Face (geometry), faces, six straight Edge (geometry), edges, and four vertex (geometry), vertices. The tet ...

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

theorem In mathematics and formal logic, a theorem is a statement (logic), statement that has been Mathematical proof, proven, or can be proven. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to esta ...

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.

Conway's toroidal deltahedron

A toroidal

deltahedron A deltahedron is a polyhedron whose faces are all equilateral triangles. The deltahedron was named by Martyn Cundy, after the Greek capital letter delta resembling a triangular shape Δ. Deltahedra can be categorized by the property of convexi ...

was described by John H. Conway in 1997, containing 18 vertices and 36 faces. Some adjacent faces are

coplanar In geometry, a set of points in space are coplanar if there exists a geometric plane that contains them all. For example, three points are always coplanar, and if the points are distinct and non-collinear, the plane they determine is unique. How ...

. Conway suggested that it should be the deltahedral toroid with the fewest possible faces.

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 Equiangular polygon, direct equiangular (all angles are equal in measure) and Equilateral polygon, equilateral (all sides have the same length). Regular polygons may be either ''convex ...

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 Bonnie Madison Stewart (July 10, 1914 – April 15, 1994) was a professor of mathematics at Michigan State University from 1940 to 1980. He earned his Ph.D. from the University of Wisconsin–Madison in 1941, under the supervision of Cyrus Colton ...

, who studied them intensively. They are analogous to the

Johnson solid In geometry, a Johnson solid, sometimes also known as a Johnson–Zalgaller solid, is a convex polyhedron whose faces are regular polygons. They are sometimes defined to exclude the uniform polyhedrons. There are ninety-two Solid geometry, s ...

s in the case of

convex polyhedra In geometry, a polyhedron (: polyhedra or polyhedrons; ) is a three-dimensional figure with flat polygonal faces, straight edges and sharp corners or vertices. The term "polyhedron" may refer either to a solid figure or to its boundary surfa ...

; however, unlike the Johnson solids, there are infinitely many Stewart toroids. They include also toroidal

deltahedra A deltahedron is a polyhedron whose faces are all equilateral triangles. The deltahedron was named by Martyn Cundy, after the Greek capital letter Delta (letter), delta resembling a triangular shape Δ. Deltahedra can be categorized by the prope ...

, 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 hull In geometry, the convex hull, convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space, ...

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

great dodecahedron In geometry, the great dodecahedron is one of four Kepler–Poinsot polyhedra. It is composed of 12 pentagonal faces (six pairs of parallel pentagons), intersecting each other making a pentagrammic path, with five pentagons meeting at each vert ...

Crown polyhedra

A crown polyhedron or stephanoid is a toroidal polyhedron which is also

noble A noble is a member of the nobility. Noble may also refer to: Places Antarctica * Noble Glacier, King George Island * Noble Nunatak, Marie Byrd Land * Noble Peak, Wiencke Island * Noble Rocks, Graham Land Australia * Noble Island, Gr ...

, 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