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 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—there is an isometry mapping any vertex onto any other. It follows that all vertices are congruence (geometry), congruent. Uniform po ...

. 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, Decagram (geometry)#Related figures, certain notable ones can ...

faces, star polygon

vertex figure In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a general -polytope is sliced off. Definitions Take some corner or Vertex (geometry), vertex of a polyhedron. Mark a point somewhere along each connected ed ...

s, or both. The complete set of 57 nonprismatic uniform star polyhedra includes the 4 regular ones, called the Kepler–Poinsot polyhedra, 14 quasiregular ones, and 39 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, stone, metal, baked clay, or even glass. They are generally fixed in place in an array to cover roofs, floors, walls, edges, or ot ...

, star polyhedra that do not pass through the center have polytope density greater than 1, and correspond to spherical polyhedra 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 triangles. 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 List of regular polytopes and compounds, regular polytopes and tessellations. The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, wh ...

. Other nonregular uniform polyhedra are listed with their

vertex configuration In geometry, a vertex configuration is a shorthand notation for representing a polyhedron or Tessellation, tiling as the sequence of Face (geometry), faces around a Vertex (geometry), vertex. It has variously been called a vertex description, vert ...

. An additional figure, the pseudo great rhombicuboctahedron, 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, 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 In geometry, a square is a regular polygon, regular quadrilateral. It has four straight sides of equal length and four equal angles. Squares are special cases of rectangles, which have four equal angles, and of rhombuses, which have four equal si ...

Dihedral symmetry

See Prismatic uniform polyhedron.

Tetrahedral symmetry

There is one nonconvex form, the tetrahemihexahedron which has ''

tetrahedral symmetry image:tetrahedron.svg, 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 co ...

'' (with fundamental domain Möbius triangle (3 3 2)). There are two Schwarz triangles 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 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 Möbius triangle (4 3 2)). There are four Schwarz triangles 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 In mathematics, and especially in geometry, an object has icosahedral symmetry if it has the same symmetries as a regular icosahedron. Examples of other polyhedra with icosahedral symmetry include the regular dodecahedron (the dual polyhedr ...

'' (with fundamental domain Möbius triangle (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 * Great complex icosidodecahedron * Small complex rhombicosidodecahedron * Great complex rhombicosidodecahedron * Complex rhombidodecadodecahedron

Skilling's figure

One further nonconvex degenerate polyhedron is the great disnub dirhombidodecahedron, 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