Elongated pentagonal gyrobirotunda

Elongated pentagonal gyrobirotunda
Type	Johnson J42 – J43 – J44
Faces	10+10 triangles 10 squares 2+10 pentagons
Edges	80
Vertices	40
Vertex configuration	20(3.42.5) 2.10(3.5.3.5)
Symmetry group	D5d
Dual polyhedron	-
Properties	convex
Net

In geometry, the elongated pentagonal gyrobirotunda or elongated icosidodecahedron is one of the Johnson solids (J₄₃). As the name suggests, it can be constructed by elongating a "pentagonal gyrobirotunda," or icosidodecahedron (one of the Archimedean solids), by inserting a decagonal prism between its congruent halves. Rotating one of the pentagonal rotundae (J₆) through 36 degrees before inserting the prism yields an elongated pentagonal orthobirotunda (J₄₂).

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]

3D model of an elongated pentagonal gyrobirotunda

Formulae

The following formulae for volume and surface area can be used if all faces are regular, with edge length a: ^[2]

${\displaystyle V={\frac {1}{6}}\left(45+17{\sqrt {5}}+15{\sqrt {5+2{\sqrt {5}}}}\right)a^{3}\approx 21.5297a^{3}}$

${\displaystyle A=\left(10+{\sqrt {30\left(10+3{\sqrt {5}}+{\sqrt {75+30{\sqrt {5}}}}\right)}}\right)a^{2}\approx 39.306a^{2}}$

References

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 .
2. Stephen Wolfram, "Elongated pentagonal gyrobirotunda" from Wolfram Alpha. Retrieved July 26, 2010.

External links

• Weisstein, Eric W., "Elongated pentagonal gyrobirotunda" ("Johnson solid") at MathWorld .