![Hexagonal torus](https://upload.wikimedia.org/wikipedia/commons/4/4b/Hexagonal_torus.png)
In
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 ...
, 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
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 (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 tou ...
), having a
topological
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
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 nom ...
() of 1 or greater. Notable examples include the
Császár
Császár is a village in Komárom-Esztergom county, Hungary
Hungary ( hu, Magyarország ) is a landlocked country in Central Europe. Spanning of the Pannonian Basin, Carpathian Basin, it is bordered by Slovakia to the north, Ukraine to ...
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 to ...
s that meet at their edges and vertices, forming a
manifold 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)
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 tou ...
.
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 ...
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 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 its
Euler characteristic 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
In geometry, the Császár polyhedron () is a nonconvex toroidal polyhedron with 14 triangular faces.
This polyhedron has no diagonals; every pair of vertices is connected by an edge. The seven vertices and 21 edges of 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 th ...
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, 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 t ...
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
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 M ...
, who studied them intensively. They are analogous to the
Johnson solids in the case of
convex polyhedra; however, unlike the Johnson solids, there are infinitely many Stewart toroids. They include also toroidal
deltahedra
In geometry, a deltahedron (plural ''deltahedra'') is a polyhedron whose faces are all equilateral triangles. The name is taken from the Greek upper case delta (Δ), which has the shape of an equilateral triangle. There are infinitely many del ...
, 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 one of n ...
, the genus-3
small cubicuboctahedron
In geometry, the small cubicuboctahedron is a uniform star polyhedron, indexed as U13. It has 20 faces (8 triangles, 6 squares, and 6 octagons), 48 edges, and 24 vertices. Its vertex figure is a crossed quadrilateral.
The small cubicuboctahedro ...
, 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 pentagon ...
.
Crown polyhedra
![Stephanoid5](https://upload.wikimedia.org/wikipedia/commons/f/fd/Stephanoid5.png)
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
In geometry, a tessellation of dimension (a plane tiling) or higher, or a polytope of dimension (a polyhedron) or higher, is isohedral or face-transitive if all its faces are the same. More specifically, all faces must be not merely congruent ...
(equal faces). Crown polyhedra are self-intersecting and topologically
self-dual.
[. See in particula]
p. 60
See also
*
Projective polyhedron
In geometry, a (globally) projective polyhedron is a tessellation of the real projective plane. These are projective analogs of spherical polyhedra – tessellations of the sphere – and toroidal polyhedra – tessellations of the toroids.
Projec ...
*
Skew apeirohedron
In geometry, a skew apeirohedron is an infinite skew polyhedron consisting of nonplanar faces or nonplanar vertex figures, allowing the figure to extend indefinitely without folding round to form a closed surface.
Skew apeirohedra have also been ...
(infinite skew polyhedron)
*
Spherical polyhedron
In geometry, a spherical polyhedron or spherical tiling is a tiling of the sphere in which the surface is divided or partitioned by great arcs into bounded regions called spherical polygons. Much of the theory of symmetrical polyhedra is most ...
*
Toroidal graph
In the mathematical field of graph theory, a toroidal graph is a graph that can be embedded on a torus. In other words, the graph's vertices can be placed on a torus such that no edges cross.
Examples
Any graph that can be embedded in a plane ...
References
External links
*{{MathWorld, urlname=ToroidalPolyhedron, title=Toroidal polyhedron
Stewart Toroids (Toroidal Solids with Regular Polygon Faces)Stewart toroids