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 uniform star polyhedron is a self-intersecting

uniform polyhedron In geometry, a uniform polyhedron has regular polygons as Face (geometry), faces and is vertex-transitive (i.e., there is an isometry mapping any vertex onto any other). It follows that all vertices are congruence (geometry), congruent. Unifor ...

. They are also sometimes called nonconvex polyhedra to imply self-intersecting. Each polyhedron can contain either

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 operatio ...

faces, star polygon

vertex figure In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off. Definitions Take some corner or vertex of a polyhedron. Mark a point somewhere along each connected edge. Draw line ...

s, or both. The complete set of 57 nonprismatic uniform star polyhedra includes the 4 regular ones, called the Kepler–Poinsot polyhedra, 5 quasiregular ones, and 48 semiregular ones. There are also two infinite sets of ''uniform star prisms'' and ''uniform star antiprisms''. Just as (nondegenerate) star polygons (which have polygon density greater than 1) correspond to circular polygons with overlapping

tiles Tiles are usually thin, square or rectangular coverings manufactured from hard-wearing material such as ceramic, Rock (geology), stone, metal, baked clay, or even glass. They are generally fixed in place in an array to cover roofs, floors, wa ...

, star polyhedra that do not pass through the center have polytope density greater than 1, and correspond to

spherical polyhedra 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 co ...

with overlapping tiles; there are 47 nonprismatic such uniform star polyhedra. The remaining 10 nonprismatic uniform star polyhedra, those that pass through the center, are the hemipolyhedra as well as Miller's monster, and do not have well-defined densities. The nonconvex forms are constructed from

Schwarz triangle In geometry, a Schwarz triangle, named after Hermann Schwarz, is a spherical triangle that can be used to tile a sphere ( spherical tiling), possibly overlapping, through reflections in its edges. They were classified in . These can be define ...

s. All the uniform polyhedra are listed below by their

symmetry group In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the amb ...

s and subgrouped by their vertex arrangements. Regular polyhedra are labeled by their

Schläfli symbol In geometry, the Schläfli symbol is a notation of the form \ that defines regular polytopes and tessellations. The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, who generalized Euclidean geometry to mo ...

. Other nonregular uniform polyhedra are listed with their vertex configuration. An additional figure, the

pseudo great rhombicuboctahedron In geometry, the pseudo great rhombicuboctahedron is one of the two pseudo uniform polyhedra, the other being the convex elongated square gyrobicupola or pseudo rhombicuboctahedron. It has the same vertex figure as the nonconvex great rhombicubo ...

, is usually not included as a truly uniform star polytope, despite consisting of regular faces and having the same vertices. Note: For nonconvex forms below an additional descriptor nonuniform is used when the

convex hull In geometry, the convex hull or 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 ...

vertex arrangement has same topology as one of these, but has nonregular faces. For example an ''nonuniform cantellated'' form may have rectangles created in place of the edges rather than squares.

Dihedral symmetry

See Prismatic uniform polyhedron.

Tetrahedral symmetry

There is one nonconvex form, the

tetrahemihexahedron In geometry, the tetrahemihexahedron or hemicuboctahedron is a uniform star polyhedron, indexed as U4. It has 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices. Its vertex figure is a crossed quadrilateral. Its Coxeter–Dynkin dia ...

which has ''

tetrahedral symmetry 150px, A regular tetrahedron, an example of a solid with full tetrahedral symmetry A regular tetrahedron has 12 rotational (or orientation-preserving) symmetries, and a symmetry order of 24 including transformations that combine a reflection ...

'' (with fundamental domain

Möbius triangle In geometry, a Schwarz triangle, named after Hermann Schwarz, is a spherical triangle that can be used to tile a sphere ( spherical tiling), possibly overlapping, through reflections in its edges. They were classified in . These can be define ...

(3 3 2)). There are two

s that generate unique nonconvex uniform polyhedra: one right triangle ( 3 2), and one general triangle ( 3 3). The general triangle ( 3 3) generates the

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 ...

which is given further on with its full

octahedral symmetry A regular octahedron has 24 rotational (or orientation-preserving) symmetries, and 48 symmetries altogether. These include transformations that combine a reflection and a rotation. A cube has the same set of symmetries, since it is the polyhedr ...

Octahedral symmetry

There are 8 convex forms, and 10 nonconvex forms with ''

'' (with fundamental domain

(4 3 2)). There are four

s that generate nonconvex forms, two right triangles ( 4 2), and ( 3 2), and two general triangles: ( 4 3), ( 4 4).

Icosahedral symmetry

There are 8 convex forms and 46 nonconvex forms with '' icosahedral symmetry'' (with fundamental domain

(5 3 2)). (or 47 nonconvex forms if Skilling's figure is included). Some of the nonconvex snub forms have reflective vertex symmetry.

Degenerate cases

Coxeter identified a number of degenerate star polyhedra by the Wythoff construction method, which contain overlapping edges or vertices. These degenerate forms include: *

Small complex icosidodecahedron In geometry, the small complex icosidodecahedron is a degenerate uniform star polyhedron. Its edges are doubled, making it degenerate. The star has 32 faces (20 triangles and 12 pentagons), 60 (doubled) edges and 12 vertices and 4 sharing faces. ...

Great complex icosidodecahedron In geometry, the great complex icosidodecahedron is a degenerate uniform star polyhedron. It has 12 vertices, and 60 (doubled) edges, and 32 faces, 12 pentagrams and 20 triangles. All edges are doubled (making it degenerate), sharing 4 faces, bu ...

Small complex rhombicosidodecahedron In geometry, the small complex rhombicosidodecahedron (also known as the small complex ditrigonal rhombicosidodecahedron) is a degenerate uniform star polyhedron. It has 62 faces (20 triangles, 12 pentagrams and 30 squares), 120 (doubled) edg ...

Great complex rhombicosidodecahedron In geometry, the small complex rhombicosidodecahedron (also known as the small complex ditrigonal rhombicosidodecahedron) is a degenerate uniform star polyhedron. It has 62 faces (20 triangles, 12 pentagrams and 30 squares), 120 (doubled) edge ...

Complex rhombidodecadodecahedron Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...

Skilling's figure

One further nonconvex degenerate polyhedron is the

great disnub dirhombidodecahedron In geometry, the great disnub dirhombidodecahedron, also called ''Skilling's figure'', is a degenerate uniform star polyhedron. It was proven in 1970 that there are only 75 uniform polyhedra other than the infinite families of prisms and antipri ...

, also known as ''Skilling's figure'', which is vertex-uniform, but has pairs of edges which coincide in space such that four faces meet at some edges. It is counted as a degenerate uniform polyhedron rather than a uniform polyhedron because of its double edges. It has I_h symmetry.

References

* * * Brückner, M. ''Vielecke und vielflache. Theorie und geschichte.''. Leipzig, Germany: Teubner, 1900

* * * Har'El, Z
''Uniform Solution for Uniform Polyhedra.''
Geometriae Dedicata 47, 57-110, 1993
Zvi Har’ElKaleido software
*
Mäder, R. E.
''Uniform Polyhedra.'' Mathematica J. 3, 48-57, 1993

*Messer, Peter W
''Closed-Form Expressions for Uniform Polyhedra and Their Duals.''
Discrete & Computational Geometry 27:353-375 (2002). *

External links

* {{MathWorld , urlname=UniformPolyhedron , title=Uniform Polyhedron Uniform polyhedra