Gyroelongated pentagonal birotunda

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.

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

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 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

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

and the volume is

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

See also

• Birotunda

External links

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