Gyroelongated pentagonal birotunda

Gyroelongated pentagonal birotunda
Type	Johnson J47 – J48 – J49
Faces	4x10 triangles 2+10 pentagons
Edges	90
Vertices	40
Vertex configuration	2x10(3.5.3.5) 2.10(34.5)
Symmetry group	D5
Dual polyhedron	-
Properties	convex, chiral
Net

In geometry, the gyroelongated pentagonal birotunda is one of the Johnson solids (J₄₈). As the name suggests, it can be constructed by gyroelongating a pentagonal birotunda (either J₃₄ or the icosidodecahedron) 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. ^[1]

The gyroelongated pentagonal birotunda 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 pentagonal 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 pentagonal face above it and to the right. The two chiral forms of J₄₈ are not considered different Johnson solids.

Area and Volume

With edge length a, the surface area is

${\displaystyle A=\left(10{\sqrt {3}}+3{\sqrt {25+10{\sqrt {5}}}}\right)a^{2}\approx 37.966236883...a^{2},}$

and the volume is

${\displaystyle V=\left({\frac {45}{6}}+{\frac {17}{6}}{\sqrt {5}}+{\frac {5}{6}}{\sqrt {2{\sqrt {650+290{\sqrt {5}}}}-2{\sqrt {5}}-2}}\right)a^{3}\approx 20.584813812...a^{3}.}$

See also

• Birotunda

External links

• Weisstein, Eric W. "Johnson Solid". MathWorld .
  • Weisstein, Eric W. "Gyroelongated pentagonal birotunda". 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 .