Triangular hebesphenorotunda | |
---|---|
Type | Johnson J91 – J92 – J1 |
Faces | 13 triangles 3 squares 3 pentagons 1 hexagon |
Edges | 36 |
Vertices | 18 |
Vertex configuration | 3(33.5) 6(3.4.3.5) 3(3.5.3.5) 2.3(32.4.6) |
Symmetry group | C3v |
Dual polyhedron | - |
Properties | convex |
Net | |
In geometry, the triangular hebesphenorotunda is one of the Johnson solids (J92).
A Johnson solid is one of 92 strictly convex polyhedra that is composed of regular polygon faces but are not uniform polyhedra (that is, they are not Platonic solids , Archimedean solids , prisms , or antiprisms ). They were named by Norman Johnson , who first listed these polyhedra in 1966. [1]
It is one of the elementary Johnson solids, which do not arise from "cut and paste" manipulations of the Platonic and Archimedean solids. However, it does have a strong relationship to the icosidodecahedron, an Archimedean solid. Most evident is the cluster of three pentagons and four triangles on one side of the solid. If these faces are aligned with a congruent patch of faces on the icosidodecahedron, then the hexagonal face will lie in the plane midway between two opposing triangular faces of the icosidodecahedron.
The triangular hebesphenorotunda also has clusters of faces that can be aligned with corresponding faces of the rhombicosidodecahedron: the three lunes, each lune consisting of a square and two antipodal triangles adjacent to the square.
The faces around each (33.5) vertex can also be aligned with the corresponding faces of various diminished icosahedra.
Johnson uses the prefix hebespheno- to refer to a blunt wedge-like complex formed by three adjacent lunes, a lune being a square with equilateral triangles attached on opposite sides. The suffix (triangular) -rotunda refers to the complex of three equilateral triangles and three regular pentagons surrounding another equilateral triangle, which bears structural resemblance to the pentagonal rotunda. [1]
The triangular hebesphenorotunda is the only Johnson solid with faces of 3, 4, 5 and 6 sides.
Cartesian coordinates for the triangular hebesphenorotunda with edge length √5 – 1 are given by the union of the orbits of the points
under the action of the group generated by rotation by 120° around the z-axis and the reflection about the yz-plane. [2] Here, 𝜏 = √5 + 1/2 (sometimes written φ) is the golden ratio. The first point generates the triangle opposite the hexagon, the second point generates the bases of the triangles surrounding the previous triangle, the third point generates the tips of the pentagons opposite the first triangle, and the last point generates the hexagon.
One may then calculate the surface area of a triangular hebesphenorotunda of edge length a as
and its volume as
A second, inverted, triangular hebesphenorotunda can be obtained by negating the second and third coordinates of each point. This second polyhedron will be joined to the first at their common hexagonal face, and the pair will inscribe an icosidodecahedron. If the hexagonal face is scaled by the golden ratio, then the convex hull of the result will be the entire icosidodecahedron.
In geometry, a dodecahedron or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagons as faces, which is a Platonic solid. There are also three regular star dodecahedra, which are constructed as stellations of the convex form. All of these have icosahedral symmetry, order 120.
In geometry, an icosidodecahedron is a polyhedron with twenty (icosi) triangular faces and twelve (dodeca) pentagonal faces. An icosidodecahedron has 30 identical vertices, with two triangles and two pentagons meeting at each, and 60 identical edges, each separating a triangle from a pentagon. As such it is one of the Archimedean solids and more particularly, a quasiregular polyhedron.
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 ; it has 1 square face and 4 triangular faces. Some authors require that the solid not be uniform before they refer to it as a "Johnson solid".
In geometry, a truncated icosidodecahedron, rhombitruncated icosidodecahedron, great rhombicosidodecahedron, omnitruncated dodecahedron or omnitruncated icosahedron is an Archimedean solid, one of thirteen convex, isogonal, non-prismatic solids constructed by two or more types of regular polygon faces.
In geometry, the snub dodecahedron, or snub icosidodecahedron, is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids constructed by two or more types of regular polygon faces.
In geometry, the triangular bipyramid is a type of hexahedron, being the first in the infinite set of face-transitive bipyramids. It is the dual of the triangular prism with 6 isosceles triangle faces.
In geometry, the pentagonal bipyramid is third of the infinite set of face-transitive bipyramids, and the 13th Johnson solid. Each bipyramid is the dual of a uniform prism.
In geometry, a pentagonal pyramid is a pyramid with a pentagonal base upon which are erected five triangular faces that meet at a point. Like any pyramid, it is self-dual.
In geometry, the triangular cupola is one of the Johnson solids. It can be seen as half a cuboctahedron.
In geometry, the pentagonal cupola is one of the Johnson solids. It can be obtained as a slice of the rhombicosidodecahedron. The pentagonal cupola consists of 5 equilateral triangles, 5 squares, 1 pentagon, and 1 decagon.
In geometry, the snub square antiprism is one of the Johnson solids . A Johnson solid is one of 92 strictly convex polyhedra that is composed of regular polygon faces but are not uniform polyhedra. They were named by Norman Johnson, who first listed these polyhedra in 1966.
In geometry, the sphenomegacorona is one of the Johnson solids. It is one of the elementary Johnson solids that do not arise from "cut and paste" manipulations of the Platonic and Archimedean solids.
In geometry, the sphenocorona is one of the Johnson solids. It is one of the elementary Johnson solids that do not arise from "cut and paste" manipulations of the Platonic and Archimedean solids.
In geometry, the bilunabirotunda is one of the Johnson solids. A Johnson solid is one of 92 strictly convex polyhedra that is composed of regular polygon faces but are not uniform polyhedra. They were named by Norman Johnson, who first listed these polyhedra in 1966.
In geometry, the augmented sphenocorona is one of the Johnson solids, and is obtained by adding a square pyramid to one of the square faces of the sphenocorona. It is the only Johnson solid arising from "cut and paste" manipulations where the components are not all prisms, antiprisms or sections of Platonic or Archimedean solids.
In geometry, the elongated triangular bipyramid or triakis triangular prism is one of the Johnson solids, convex polyhedra whose faces are regular polygons. As the name suggests, it can be constructed by elongating a triangular bipyramid by inserting a triangular prism between its congruent halves.
In geometry, the gyrobifastigium is the 26th Johnson solid. It can be constructed by joining two face-regular triangular prisms along corresponding square faces, giving a quarter-turn to one prism. It is the only Johnson solid that can tile three-dimensional space.
In geometry, the elongated triangular cupola is one of the Johnson solids. As the name suggests, it can be constructed by elongating a triangular cupola by attaching a hexagonal prism to its base.
In geometry, the gyroelongated triangular cupola is one of the Johnson solids (J22). It can be constructed by attaching a hexagonal antiprism to the base of a triangular cupola (J3). This is called "gyroelongation", which means that an antiprism is joined to the base of a solid, or between the bases of more than one solid.
In geometry, chamfering or edge-truncation is a topological operator that modifies one polyhedron into another. It is similar to expansion, moving faces apart and outward, but also maintains the original vertices. For polyhedra, this operation adds a new hexagonal face in place of each original edge.
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