Concept
Johnson solid
Related concepts (111)
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.
Pentagonal orthobicupola
In geometry, the pentagonal orthobicupola is one of the Johnson solids (J_30). As the name suggests, it can be constructed by joining two pentagonal cupolae (J_5) along their decagonal bases, matching like faces. A 36-degree rotation of one cupola before the joining yields a pentagonal gyrobicupola (J_31). The pentagonal orthobicupola is the third in an infinite set of orthobicupolae.
Pentagonal gyrobicupola
In geometry, the pentagonal gyrobicupola is one of the Johnson solids (J_31). Like the pentagonal orthobicupola (J_30), it can be obtained by joining two pentagonal cupolae (J_5) along their bases. The difference is that in this solid, the two halves are rotated 36 degrees with respect to one another. The pentagonal gyrobicupola is the third in an infinite set of gyrobicupolae. The pentagonal gyrobicupola is what you get when you take a rhombicosidodecahedron, chop out the middle parabidiminished rhombicosidodecahedron (J_80), and paste the two opposing cupolae back together.
Elongated triangular orthobicupola
In geometry, the elongated triangular orthobicupola or cantellated triangular prism is one of the Johnson solids (J_35). As the name suggests, it can be constructed by elongating a triangular orthobicupola (J_27) by inserting a hexagonal prism between its two halves. The resulting solid is superficially similar to the rhombicuboctahedron (one of the Archimedean solids), with the difference that it has threefold rotational symmetry about its axis instead of fourfold symmetry.
Toroidal polyhedron
In geometry, a toroidal polyhedron is a polyhedron which is also a toroid (a g-holed torus), having a topological genus (g) of 1 or greater. Notable examples include the Császár and Szilassi polyhedra. Toroidal polyhedra are defined as collections of polygons 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.
Diminished trapezohedron
In geometry, a diminished trapezohedron is a polyhedron in an infinite set of polyhedra, constructed by removing one of the polar vertices of a trapezohedron and replacing it by a new face (diminishment). It has one regular n-gonal base face, n triangle faces around the base, and n kites meeting on top. The kites can also be replaced by rhombi with specific proportions. Along with the set of pyramids and elongated pyramids, these figures are topologically self-dual.
Gyroelongated bipyramid
In geometry, the gyroelongated bipyramids are an infinite set of polyhedra, constructed by elongating an n-gonal bipyramid by inserting an n-gonal antiprism between its congruent halves. Two members of the set can be deltahedra, that is, constructed entirely of equilateral triangles: the gyroelongated square bipyramid, a Johnson solid, and the icosahedron, a Platonic solid. The gyroelongated triangular bipyramid can be made with equilateral triangles, but is not a deltahedron because it has coplanar faces, i.
Hexagonal prism
In geometry, the hexagonal prism is a prism with hexagonal base. Prisms are polyhedrons; this polyhedron has 8 faces, 18 edges, and 12 vertices. Since it has 8 faces, it is an octahedron. However, the term octahedron is primarily used to refer to the regular octahedron, which has eight triangular faces. Because of the ambiguity of the term octahedron and tilarity of the various eight-sided figures, the term is rarely used without clarification. Before sharpening, many pencils take the shape of a long hexagonal prism.
Trigonal trapezohedron
In geometry, a trigonal trapezohedron is a rhombohedron (a polyhedron with six rhombus-shaped faces) in which, additionally, all six faces are congruent. Alternative names for the same shape are the trigonal deltohedron or isohedral rhombohedron. Some sources just call them rhombohedra. Six identical rhombic faces can construct two configurations of trigonal trapezohedra. The acute or prolate form has three acute angle corners of the rhombic faces meeting at the two polar axis vertices.
Elongated pentagonal pyramid
In geometry, the elongated pentagonal pyramid is one of the Johnson solids (J_9). As the name suggests, it can be constructed by elongating a pentagonal pyramid (J_2) by attaching a pentagonal prism to its base. The following formulae for the height (), surface area () and volume () can be used if all faces are regular, with edge length : The dual of the elongated pentagonal pyramid has 11 faces: 5 triangular, 1 pentagonal and 5 trapezoidal. It is topologically identical to the Johnson solid.