Triangular orthobicupola

Triangular orthobicupola
Type	Johnson J26 – J27 – J28
Faces	2+6 triangles 6 squares
Edges	24
Vertices	12
Vertex configuration	6(32.42) 6(3.4.3.4)
Symmetry group	D3h
Dual polyhedron	Trapezo-rhombic dodecahedron
Properties	convex
Net

In geometry, the triangular orthobicupola is one of the Johnson solids (J₂₇). As the name suggests, it can be constructed by attaching two triangular cupolas (J₃) along their bases. It has an equal number of squares and triangles at each vertex; however, it is not vertex-transitive. It is also called an anticuboctahedron, twisted cuboctahedron or disheptahedron. It is also a canonical polyhedron.

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.

The triangular orthobicupola is the first in an infinite set of orthobicupolae.

Construction

The triangular orthobicupola can be constructed by attaching two triangular cupolas onto their bases. Similar to the cuboctahedron, which would be known as the triangular gyrobicupola, the difference is that the two triangular cupolas that make up the triangular orthobicupola are joined so that pairs of matching sides abut (hence, "ortho"); the cuboctahedron is joined so that triangles abut squares and vice versa. Given a triangular orthobicupola, a 60-degree rotation of one cupola before the joining yields a cuboctahedron. Hence, another name for the triangular orthobicupola is the anticuboctahedron. Because the triangular orthobicupola has the property of convexity and its faces are regular polygons—eight equilateral triangles and six squares—it is categorized as a Johnson solid. It is enumerated as the twenty-seventh Johnson solid $J_{27}$

Properties

The surface area $A$ and the volume $V$ of a triangular orthobicupola are the same as those with cuboctahedron. Its surface area can be obtained by summing all of its polygonal faces, and its volume is by slicing it off into two triangular cupolas and adding their volume. With edge length $a$, they are: $${\begin{aligned}A&=2\left(3+{\sqrt {3}}\right)a^{2}\approx 9.464a^{2},\\V&={\frac {5{\sqrt {2}}}{3}}a^{3}\approx 2.357a^{3}.\end{aligned}}$$

The dual polyhedron of a triangular orthobicupola is the trapezo-rhombic dodecahedron. It has 6 rhombic and 6 trapezoidal faces, and is similar to the rhombic dodecahedron.

External links

• Weisstein, Eric W., "Johnson solid" ("Triangular orthobicupola") at MathWorld.