Gyroelongated square cupola

Gyroelongated square cupola
Type	Johnson J22 - J23 - J24
Faces	3x4+8 triangles 1+4 squares 1 octagon
Edges	44
Vertices	20
Vertex configuration	4(3.43) 2.4(33.8) 8(34.4)
Symmetry group	C4v
Dual polyhedron	-
Properties	convex
Net

An unfolded gyroelongated square cupola

An unfolded gyroelongated square cupola, faces colored by symmetry

In geometry, the gyroelongated square cupola is one of the Johnson solids (J₂₃). As the name suggests, it can be constructed by gyroelongating a square cupola (J₄) by attaching an octagonal antiprism to its base. It can also be seen as a gyroelongated square bicupola (J₄₅) with one square bicupola removed.

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]

Area and Volume

The surface area is,

${\displaystyle A=\left(7+2{\sqrt {2}}+5{\sqrt {3}}\right)a^{2}\approx 18.4886811...a^{2}.}$

The volume is the sum of the volume of a square cupola and the volume of an octagonal prism,

${\displaystyle V=\left(1+{\frac {2}{3}}{\sqrt {2}}+{\frac {2}{3}}{\sqrt {4+2{\sqrt {2}}+2{\sqrt {146+103{\sqrt {2}}}}}}\right)a^{3}\approx 6.2107658...a^{3}.}$

Dual polyhedron

The dual of the gyroelongated square cupola has 20 faces: 8 kites, 4 rhombi, and 8 pentagons.

Dual gyroelongated square cupola	Net of dual

External links

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

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 .