Gyroelongated pentagonal bicupola

Gyroelongated pentagonal bicupola
Type	Johnson J45 – J46 – J47
Faces	3.10 triangles 10 squares 2 pentagons
Edges	70
Vertices	30
Vertex configuration	10(3.4.5.4) 2.10(34.4)
Symmetry group	D5
Dual polyhedron	-
Properties	convex, chiral
Net

In geometry, the gyroelongated pentagonal bicupola is one of the Johnson solids (J₄₆). As the name suggests, it can be constructed by gyroelongating a pentagonal bicupola (J₃₀ or J₃₁) by inserting a decagonal antiprism between its congruent halves.

The gyroelongated pentagonal bicupola is one of five Johnson solids which are chiral, meaning that they have a "left-handed" and a "right-handed" form. In the illustration to the right, each square face on the bottom half of the figure is connected by a path of two triangular faces to a square face above it and to the right. In the figure of opposite chirality (the mirror image of the illustrated figure), each bottom square would be connected to a square face above it and to the left. The two chiral forms of J₄₆ are not considered different Johnson solids.

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.

Area and Volume

With edge length a, the surface area is

$A={\frac {1}{2}}\left(20+15{\sqrt {3}}+{\sqrt {25+10{\sqrt {5}}}}\right)a^{2}\approx 26.431335858...a^{2},$

and the volume is

$V=\left({\frac {5}{3}}+{\frac {4}{3}}{\sqrt {5}}+{\frac {5}{6}}{\sqrt {2{\sqrt {650+290{\sqrt {5}}}}-2{\sqrt {5}}-2}}\right)a^{3}\approx 11.397378512...a^{3}.$

External links

• Weisstein, Eric W., "Gyroelongated pentagonal bicupola" ("Johnson solid") at MathWorld.