Ingeometry, atoroidal polyhedron is apolyhedron which is also atoroid (ag-holedtorus), having atopologicalgenus (g) of 1 or greater. Notable examples include theCsászár andSzilassi polyhedra.
Toroidal polyhedra are defined as collections ofpolygons that meet at their edges and vertices, forming amanifold 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, thelink of the vertex. For toroidal polyhedra, this manifold is anorientable surface.[1] Some authors restrict the phrase "toroidal polyhedra" to mean more specifically polyhedra topologically equivalent to the (genus 1)torus.[2]
In this area, it is important to distinguishembedded toroidal polyhedra, whose faces are flat polygons in three-dimensionalEuclidean space that do not cross themselves or each other, fromabstract polyhedra, topological surfaces without any specified geometric realization.[3] Intermediate between these two extremes are polyhedra formed by geometric polygons orstar polygons 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 itsEuler characteristic being non-positive. The Euler characteristic generalizes toV −E +F = 2 − 2g, whereg is its topological genus.
TheCsászár polyhedron is a seven-vertex toroidal polyhedron with 21 edges and 14 triangular faces.[4] It and thetetrahedron are the only known polyhedra in which every possible line segment connecting two vertices forms an edge of the polyhedron.[5] Its dual, theSzilassi polyhedron, has seven hexagonal faces that are all adjacent to each other,[6] hence providing the existence half of thetheorem that the maximum number of colors needed for a map on a (genus one) torus is seven.[7]
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.
A toroidaldeltahedron was described byJohn H. Conway in 1997, containing 18 vertices and 36 faces. Some adjacent faces arecoplanar. Conway suggested that it should be the deltahedral toroid with the fewest possible faces.[8]
A special category of toroidal polyhedra are constructed exclusively byregular polygon faces, without crossings, and with a further restriction that adjacent faces may not lie in the same plane as each other. These are calledStewart toroids,[9] named afterBonnie Stewart, who studied them intensively.[10] They are analogous to theJohnson solids in the case ofconvex polyhedra; however, unlike the Johnson solids, there are infinitely many Stewart toroids.[11] They include also toroidaldeltahedra, polyhedra whose faces are all equilateral triangles.
A restricted class of Stewart toroids, also defined by Stewart, are thequasi-convex toroidal polyhedra. These are Stewart toroids that include all of the edges of theirconvex 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.[12]
| Genus | 1 | 1 |
|---|---|---|
| Image |  |  |
| Polyhedra | 6hexagonal prisms | 8octahedra |
| Vertices | 48 | 24 |
| Edges | 84 | 72 |
| Faces | 36 | 48 |
| Genus | 1 | 3 | 11 | 3 | 5 | 7 | 11 | |
|---|---|---|---|---|---|---|---|---|
| Image |  |  |  |  |  |  |  |  |
| Polyhedra | 4square cupolae 8tetrahedra | 6triangular cupolae 6square pyramids | 4triangular cupolae 6square pyramids | 24triangular prisms 6square pyramids 8tetrahedra | 6square cupolae 4triangular cupolae 12cubes | 8triangular cupolae 12cubes | 6square cupolae 12cubes | 6square cupolae 8triangular cupolae |
| Convex hull | truncated cube | truncated octahedron | truncated octahedron | expanded cuboctahedron | truncated cuboctahedron | truncated cuboctahedron | truncated cuboctahedron | truncated cuboctahedron |
| Vertices | 32 | 30 | 30 | 62 | 72 | 72 | 72 | 72 |
| Edges | 64 | 60 | 72 | 168 | 144 | 168 | 168 | 168 |
| Faces | 32 | 30 | 38 | 86 | 68 | 88 | 84 | 76 |
 Octahemioctahedron |  Small cubicuboctahedron |  Great dodecahedron |
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-1octahemioctahedron, the genus-3small cubicuboctahedron, and the genus-4great dodecahedron.
Acrown polyhedron orstephanoid is a toroidal polyhedron which is alsonoble, being bothisogonal (equal vertices) andisohedral (equal faces). Crown polyhedra are self-intersecting and topologicallyself-dual.[13]