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 a Johnson solid with 13 equilateral triangles, 3 squares, 3 regular pentagons, and 1 regular hexagon, making the total of its faces is 20.
The triangular hebesphenorotunda is named by Johnson (1966), with the prefix hebespheno- referring to a blunt wedge-like complex formed by three adjacent lunes—a figure where two equilateral triangles are attached at the opposite sides of a square. The suffix (triangular) -rotunda refers to the complex of three equilateral triangles and three regular pentagons surrounding another equilateral triangle, which bears a structural resemblance to the pentagonal rotunda. [1] Therefore, the triangular hebesphenorotunda has 20 faces: 13 equilateral triangles, 3 squares, 3 regular pentagons, and 1 regular hexagon. [2] The faces are all regular polygons, categorizing the triangular hebesphenorotunda as the Johnson solid, enumerated the last one . [3] It is one of the elementary Johnson solids, which do not arise from "cut and paste" manipulations of the Platonic and Archimedean solids. [4]
The surface area of a triangular hebesphenorotunda of edge length as: [2]
The triangular hebesphenorotunda with edge length can be constructed by the union of the orbits of the Cartesian coordinates:
under the action of the group generated by rotation by 120° around the z-axis and the reflection about the yz-plane. Here, is denoted as the golden ratio. [5]
In geometry, an octahedron is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex.
The triaugmented triangular prism, in geometry, is a convex polyhedron with 14 equilateral triangles as its faces. It can be constructed from a triangular prism by attaching equilateral square pyramids to each of its three square faces. The same shape is also called the tetrakis triangular prism, tricapped trigonal prism, tetracaidecadeltahedron, or tetrakaidecadeltahedron; these last names mean a polyhedron with 14 triangular faces. It is an example of a deltahedron and of a Johnson solid.
In geometry, the elongated pentagonal pyramid is one of the Johnson solids. As the name suggests, it can be constructed by elongating a pentagonal pyramid by attaching a pentagonal prism to its base.
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 pentagonal rotunda is one of the Johnson solids. It can be seen as half of an icosidodecahedron, or as half of a pentagonal orthobirotunda. It has a total of 17 faces.
In geometry, the gyroelongated pentagonal rotunda is one of the Johnson solids (J25). As the name suggests, it can be constructed by gyroelongating a pentagonal rotunda (J6) by attaching a decagonal antiprism to its base. It can also be seen as a gyroelongated pentagonal birotunda (J48) with one pentagonal rotunda removed.
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 disphenoid is a convex polyhedron with 12 equilateral triangles as its faces. It is an example of deltahedron and Johnson solid. It can be constructed in different approaches. This shape also has alternative names called Siamese dodecahedron, triangular dodecahedron, trigonal dodecahedron, or dodecadeltahedron; these names mean the 12-sided polyhedron.
In geometry, the hebesphenomegacorona 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. It has 21 faces, 18 triangles and 3 squares, 33 edges, and 14 vertices.
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 a Johnson solid with faces of 8 equilateral triangles, 2 squares, and 4 regular pentagons.
In geometry, the augmented sphenocorona is the Johnson solid that can be constructed by attaching an equilateral 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 gyroelongated pentagonal bicupola is one of the Johnson solids. As the name suggests, it can be constructed by gyroelongating a pentagonal bicupola by inserting a decagonal antiprism between its congruent halves.
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, the pentagonal orthocupolarotunda is one of the Johnson solids. As the name suggests, it can be constructed by joining a pentagonal cupola and a pentagonal rotunda along their decagonal bases, matching the pentagonal faces. A 36-degree rotation of one of the halves before the joining yields a pentagonal gyrocupolarotunda.
In geometry, the elongated pentagonal gyrocupolarotunda is one of the Johnson solids. As the name suggests, it can be constructed by elongating a pentagonal gyrocupolarotunda by inserting a decagonal prism between its halves. Rotating either the pentagonal cupola or the pentagonal rotunda through 36 degrees before inserting the prism yields an elongated pentagonal orthocupolarotunda.
In geometry, the elongated pentagonal orthocupolarotunda is one of the Johnson solids. As the name suggests, it can be constructed by elongating a pentagonal orthocupolarotunda by inserting a decagonal prism between its halves. Rotating either the cupola or the rotunda through 36 degrees before inserting the prism yields an elongated pentagonal gyrocupolarotunda.