Concept

Rhombicosidodecahedron

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Uniform polyhedron

In geometry, a uniform polyhedron has regular polygons as faces and is vertex-transitive (i.e., there is an isometry mapping any vertex onto any other). It follows that all vertices are congruent. Uniform polyhedra may be regular (if also face- and edge-transitive), quasi-regular (if also edge-transitive but not face-transitive), or semi-regular (if neither edge- nor face-transitive). The faces and vertices need not be convex, so many of the uniform polyhedra are also star polyhedra.

Johnson solid

In geometry, a Johnson solid is a strictly convex polyhedron each face of which is a regular polygon. There is no requirement that each face must be the same polygon, or that the same polygons join around each vertex. An example of a Johnson solid is the square-based pyramid with equilateral sides (J_1); it has 1 square face and 4 triangular faces. Some authors require that the solid not be uniform (i.e., not Platonic solid, Archimedean solid, uniform prism, or uniform antiprism) before they refer to it as a "Johnson solid".

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 of the icosahedron) and the rhombic triacontahedron. Every polyhedron with icosahedral symmetry has 60 rotational (or orientation-preserving) symmetries and 60 orientation-reversing symmetries (that combine a rotation and a reflection), for a total symmetry order of 120.

Diminished rhombicosidodecahedron

In geometry, the diminished rhombicosidodecahedron is one of the Johnson solids (J_76). It can be constructed as a rhombicosidodecahedron with one pentagonal cupola removed. Related Johnson solids are: J_80: parabidiminished rhombicosidodecahedron with two opposing cupolae removed, and J_81: metabidiminished rhombicosidodecahedron with two non-opposing cupolae removed, and J_83: tridiminished rhombicosidodecahedron with three cupola removed.

Parabidiminished rhombicosidodecahedron

In geometry, the parabidiminished rhombicosidodecahedron is one of the Johnson solids (J_80). It is also a canonical polyhedron. It can be constructed as a rhombicosidodecahedron with two opposing pentagonal cupolae removed. Related Johnson solids are the diminished rhombicosidodecahedron (J_76) where one cupola is removed, the metabidiminished rhombicosidodecahedron (J_81) where two non-opposing cupolae are removed, and the tridiminished rhombicosidodecahedron (J_83) where three cupolae are removed.

Vertex configuration

In geometry, a vertex configuration is a shorthand notation for representing the vertex figure of a polyhedron or tiling as the sequence of faces around a vertex. For uniform polyhedra there is only one vertex type and therefore the vertex configuration fully defines the polyhedron. (Chiral polyhedra exist in mirror-image pairs with the same vertex configuration.) A vertex configuration is given as a sequence of numbers representing the number of sides of the faces going around the vertex. The notation "a.

Metabidiminished rhombicosidodecahedron

In geometry, the metabidiminished rhombicosidodecahedron is one of the Johnson solids (J_81). It can be constructed as a rhombicosidodecahedron with two non-opposing pentagonal cupolae (J_5) removed. Related Johnson solids are: The diminished rhombicosidodecahedron (J_76) where one cupola is removed, The parabidiminished rhombicosidodecahedron (J_80) where two opposing cupolae are removed, The gyrate bidiminished rhombicosidodecahedron (J_82) where two non-opposing cupolae are removed and a third is rotated 36 degrees, And the tridiminished rhombicosidodecahedron (J_83) where three cupolae are removed.

Tridiminished rhombicosidodecahedron

In geometry, the tridiminished rhombicosidodecahedron is one of the Johnson solids (J_83). It can be constructed as a rhombicosidodecahedron with three pentagonal cupolae removed. Related Johnson solids are: J_76: diminished rhombicosidodecahedron with one cupola removed, J_80: parabidiminished rhombicosidodecahedron with two opposing cupolae removed, and J_81: metabidiminished rhombicosidodecahedron with two non-opposing cupolae removed.

Trigyrate rhombicosidodecahedron

In geometry, the trigyrate rhombicosidodecahedron is one of the Johnson solids (J_75). It contains 20 triangles, 30 squares and 12 pentagons. It is also a canonical polyhedron. It can be constructed as a rhombicosidodecahedron with three pentagonal cupolae rotated through 36 degrees. Related Johnson solids are: The gyrate rhombicosidodecahedron (J_72) where one cupola is rotated; The parabigyrate rhombicosidodecahedron (J_73) where two opposing cupolae are rotated; And the metabigyrate rhombicosidodecahedron (J_74) where two non-opposing cupolae are rotated.

Rectification (geometry)

In Euclidean geometry, rectification, also known as critical truncation or complete-truncation, is the process of truncating a polytope by marking the midpoints of all its edges, and cutting off its vertices at those points. The resulting polytope will be bounded by vertex figure facets and the rectified facets of the original polytope. A rectification operator is sometimes denoted by the letter r with a Schläfli symbol. For example, r{4,3} is the rectified cube, also called a cuboctahedron, and also represented as .