Elongated pentagonal gyrobirotunda

From HandWiki
Short description: 43rd Johnson solid
Elongated pentagonal gyrobirotunda
Elongated pentagonal gyrobirotunda.png
TypeJohnson
J42J43J44
Faces10+10 triangles
10 squares
2+10 pentagons
Edges80
Vertices40
Vertex configuration20(3.42.5)
2.10(3.5.3.5)
Symmetry groupD5d
Dual polyhedron-
Propertiesconvex
Net
Johnson solid 43 net.png

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]

Formulae

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

[math]\displaystyle{ V=\frac{1}{6}\left(45+17\sqrt{5}+15\sqrt{5+2\sqrt{5}}\right)a^3 \approx 21.5297 a^3 }[/math]
[math]\displaystyle{ A=\left(10+\sqrt{30\left(10+3\sqrt{5}+\sqrt{75+30\sqrt{5}}\right)}\right)a^2 \approx 39.306 a^2 }[/math]

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

External links