Gyroelongated triangular cupola

Gyroelongated triangular cupola
Type	Johnson J21 - J22 - J23
Faces	1+3x3+6 triangles 3 squares 1 hexagon
Edges	33
Vertices	15
Vertex configuration	3(3.4.3.4) 2.3(33.6) 6(34.4)
Symmetry group	C3v
Dual polyhedron	-
Properties	convex
Net

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.

The gyroelongated triangular cupola can also be seen as a gyroelongated triangular bicupola (J₄₄) with one triangular cupola removed. Like all cupolae, the base polygon has twice as many sides as the top (in this case, the bottom polygon is a hexagon because the top is a triangle).

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]

Formulae

The following formulae for volume and surface area can be used if all faces are regular, with edge length a: ^[2]

${\displaystyle V=\left({\frac {1}{3}}{\sqrt {{\frac {61}{2}}+18{\sqrt {3}}+30{\sqrt {1+{\sqrt {3}}}}}}\right)a^{3}\approx 3.51605...a^{3}}$

${\displaystyle A=\left(3+{\frac {11{\sqrt {3}}}{2}}\right)a^{2}\approx 12.5263...a^{2}}$

Dual polyhedron

The dual of the gyroelongated triangular cupola has 15 faces: 6 kites, 3 rhombi, and 6 pentagons.

Dual gyroelongated triangular cupola	Net of dual

References

1. Johnson, Norman W. (1966), "Convex polyhedra with regular faces", Canadian Journal of Mathematics , 18: 169–200, doi:10.4153/cjm-1966-021-8, MR   0185507, Zbl   0132.14603 .
2. Stephen Wolfram, "Gyroelongated triangular cupola" from Wolfram Alpha. Retrieved July 22, 2010.

External links

• Weisstein, Eric W., "Gyroelongated triangular cupola" ("Johnson solid") at MathWorld .