A display of uniform polyhedra at the Science Museum in LondonThe small snub icosicosidodecahedron is a uniform star polyhedron, with vertex figure 35.5/2
In geometry, a uniform star polyhedron is a self-intersecting uniform polyhedron. They are also sometimes called nonconvex polyhedra to imply self-intersecting. Each polyhedron can contain either star polygon faces, star polygon vertex figures, 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, 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 groups and subgrouped by their vertex arrangements.
Regular polyhedra are labeled by their Schläfli symbol. Other nonregular uniform polyhedra are listed with their vertex configuration.
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 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.
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In geometry, a starpolyhedron is a polyhedron which has some repetitive quality of nonconvexity giving it a star-like visual quality. There are two general...
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to recover uniformity when changing a spherical polyhedron to its planar counterpart can push faces through the centre of the polyhedron and back out...
star Pentagramma mirificum Regular star 4-polytope Rub el Hizb Star (glyph) Starpolyhedron, Kepler–Poinsot polyhedron, and uniformstarpolyhedron Starfish...
and Coxeter diagram . Nonconvex great rhombicuboctahedron - a uniformstarpolyhedron, with Schläfli symbol r{4,3/2}, and Coxeter diagram . This disambiguation...
icosicosidodecahedron, or retroholosnub icosahedron) is a nonconvex uniformpolyhedron, indexed as U72. It has 112 faces (100 triangles and 12 pentagrams)...
In geometry, an n-gonal antiprism or n-antiprism is a polyhedron composed of two parallel direct copies (not mirror images) of an n-sided polygon, connected...
quasitruncated square, inverted as {4/3}, is an octagram, t{4/3}={8/3}. The uniformstarpolyhedron stellated truncated hexahedron, t'{4,3}=t{4/3,3} has octagram faces...
geometry, the great complex icosidodecahedron is a degenerate uniformstarpolyhedron. It has 12 vertices, and 60 (doubled) edges, and 32 faces, 12 pentagrams...
A regular polyhedron is a polyhedron whose symmetry group acts transitively on its flags. A regular polyhedron is highly symmetrical, being all of edge-transitive...
complex icosidodecahedron is a degenerate uniformstarpolyhedron. Its edges are doubled, making it degenerate. The star has 32 faces (20 triangles and 12 pentagons)...
icosidodecahedron) is a uniformstarpolyhedron, indexed as U69. It is given a Schläfli symbol sr{5⁄3,3}, and Coxeter-Dynkin diagram . In the book Polyhedron Models by...
dirhombicosidodecahedron (or great snub disicosidisdodecahedron) is a nonconvex uniformpolyhedron, indexed last as U75. It has 124 faces (40 triangles, 60 squares,...
This is an indexed list of the uniform and stellated polyhedra from the book Polyhedron Models, by Magnus Wenninger. The book was written as a guide book...
geometry, the nonconvex great rhombicosidodecahedron is a nonconvex uniformpolyhedron, indexed as U67. It has 62 faces (20 triangles, 30 squares and 12...
transitive on its vertices; today, this is more commonly referred to as a uniformpolyhedron (this follows from Thorold Gosset's 1900 definition of the more general...
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