Gyroelongated pentagonal pyramidIn geometry, the gyroelongated pentagonal pyramid is one of the Johnson solids (J_11). As its name suggests, it is formed by taking a pentagonal pyramid and "gyroelongating" it, which in this case involves joining a pentagonal antiprism to its base. It can also be seen as a diminished icosahedron, an icosahedron with the top (a pentagonal pyramid, J_2) chopped off by a plane. Other Johnson solids can be formed by cutting off multiple pentagonal pyramids from an icosahedron: the pentagonal antiprism and metabidiminished icosahedron (two pyramids removed), and the tridiminished icosahedron (three pyramids removed).
Conway polyhedron notationIn geometry, Conway polyhedron notation, invented by John Horton Conway and promoted by George W. Hart, is used to describe polyhedra based on a seed polyhedron modified by various prefix operations. Conway and Hart extended the idea of using operators, like truncation as defined by Kepler, to build related polyhedra of the same symmetry. For example, tC represents a truncated cube, and taC, parsed as t(aC), is (topologically) a truncated cuboctahedron. The simplest operator dual swaps vertex and face elements; e.
Truncation (geometry)In geometry, a truncation is an operation in any dimension that cuts polytope vertices, creating a new facet in place of each vertex. The term originates from Kepler's names for the Archimedean solids. In general any polyhedron (or polytope) can also be truncated with a degree of freedom as to how deep the cut is, as shown in Conway polyhedron notation truncation operation. A special kind of truncation, usually implied, is a uniform truncation, a truncation operator applied to a regular polyhedron (or regular polytope) which creates a resulting uniform polyhedron (uniform polytope) with equal edge lengths.
Pentagonal pyramidIn geometry, a pentagonal pyramid is a pyramid with a pentagonal base upon which are erected five triangular faces that meet at a point (the apex). Like any pyramid, it is self-dual. The regular pentagonal pyramid has a base that is a regular pentagon and lateral faces that are equilateral triangles. It is one of the Johnson solids (J_2). It can be seen as the "lid" of an icosahedron; the rest of the icosahedron forms a gyroelongated pentagonal pyramid, J_11.
Snub (geometry)In geometry, a snub is an operation applied to a polyhedron. The term originates from Kepler's names of two Archimedean solids, for the snub cube (cubus simus) and snub dodecahedron (dodecaedron simum). In general, snubs have chiral symmetry with two forms: with clockwise or counterclockwise orientation. By Kepler's names, a snub can be seen as an expansion of a regular polyhedron: moving the faces apart, twisting them about their centers, adding new polygons centered on the original vertices, and adding pairs of triangles fitting between the original edges.
IcosahedronIn geometry, an icosahedron (ˌaɪkɒsəˈhiːdrən,-kə-,-koʊ- or aɪˌkɒsəˈhiːdrən) is a polyhedron with 20 faces. The name comes . The plural can be either "icosahedra" (-drə) or "icosahedrons". There are infinitely many non-similar shapes of icosahedra, some of them being more symmetrical than others. The best known is the (convex, non-stellated) regular icosahedron—one of the Platonic solids—whose faces are 20 equilateral triangles. There are two objects, one convex and one nonconvex, that can both be called regular icosahedra.
Square antiprismIn geometry, the square antiprism is the second in an infinite family of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps. It is also known as an anticube. If all its faces are regular, it is a semiregular polyhedron or uniform polyhedron. A nonuniform D4-symmetric variant is the cell of the noble square antiprismatic 72-cell. When eight points are distributed on the surface of a sphere with the aim of maximising the distance between them in some sense, the resulting shape corresponds to a square antiprism rather than a cube.
Dihedral symmetry in three dimensionsIn geometry, dihedral symmetry in three dimensions is one of three infinite sequences of point groups in three dimensions which have a symmetry group that as an abstract group is a dihedral group Dihn (for n ≥ 2). There are 3 types of dihedral symmetry in three dimensions, each shown below in 3 notations: Schönflies notation, Coxeter notation, and orbifold notation. Chiral Dn, [n,2]+, (22n) of order 2n – dihedral symmetry or para-n-gonal group (abstract group: Dihn).
Regular icosahedronIn geometry, a regular icosahedron (ˌaɪkɒsəˈhiːdrən,-kə-,-koʊ- or aɪˌkɒsəˈhiːdrən) is a convex polyhedron with 20 faces, 30 edges and 12 vertices. It is one of the five Platonic solids, and the one with the most faces. It has five equilateral triangular faces meeting at each vertex. It is represented by its Schläfli symbol {3,5}, or sometimes by its vertex figure as 3.3.3.3.3 or 35. It is the dual of the regular dodecahedron, which is represented by {5,3}, having three pentagonal faces around each vertex.
Johnson solidIn 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".
