Gyroelongated pentagonal cupolarotunda

Gyroelongated pentagonal cupolarotunda
Type	Johnson J46 – J47 – J48
Faces	7x5 triangles 5 squares 2+5 pentagons
Edges	80
Vertices	35
Vertex configuration	5(3.4.5.4) 2.5(3.5.3.5) 2.5(34.4) 2.5(34.5)
Symmetry group	C5
Dual polyhedron	-
Properties	convex, chiral
Net

In geometry, the gyroelongated pentagonal cupolarotunda is one of the Johnson solids (J₄₇). As the name suggests, it can be constructed by gyroelongating a pentagonal cupolarotunda (J₃₂ or J₃₃) by inserting a decagonal antiprism between its two halves.

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 gyroelongated pentagonal cupolarotunda 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 pentagonal 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 left. In the figure of opposite chirality (the mirror image of the illustrated figure), each bottom pentagon would be connected to a square face above it and to the right. The two chiral forms of J₄₇ are not considered different Johnson solids.

Area and Volume

With edge length a, the surface area is

$A={\frac {1}{4}}\left(20+35{\sqrt {3}}+7{\sqrt {25+10{\sqrt {5}}}}\right)a^{2}\approx 32.198786370...a^{2},$

and the volume is

$V=\left({\frac {55}{12}}+{\frac {25}{12}}{\sqrt {5}}+{\frac {5}{6}}{\sqrt {2{\sqrt {650+290{\sqrt {5}}}}-2{\sqrt {5}}-2}}\right)a^{3}\approx 15.991096162...a^{3}.$

External links

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