Sphenocorona

Sphenocorona
Type	Johnson J85 - J86 - J87
Faces	2x2+2x4 triangles 2 squares
Edges	22
Vertices	10
Vertex configuration	4(3^3.4) 2(3^2.4^2) 2x2(3^5)
Symmetry group	C2v
Dual polyhedron	-
Properties	convex
Net

In geometry, the sphenocorona is one of the Johnson solids (J₈₆).

A Johnson solid is one of 92 strictly convex polyhedra that have regular faces but are not uniform (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 sphenocorona is one of the elementary Johnson solids that do not arise from "cut and paste" manipulations of the Platonic and Archimedean solids. However, it contains a cluster of eight triangles that can be aligned with a congruent patch of faces on the regular icosahedron.

Formulae

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

${\displaystyle V=\left({\frac {1}{2}}{\sqrt {1+3{\sqrt {\frac {3}{2}}}+{\sqrt {13+3{\sqrt {6}}}}}}\right)a^{3}\approx 1.51535...a^{3}}$

$A=\left(2+3{\sqrt {3}}\right)a^{2}\approx 7.19615...a^{2}$

See also

• Augmented sphenocorona

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, "Sphenocorona" from Wolfram Alpha. Retrieved July 21, 2010.

External links

• Eric W. Weisstein, Sphenocorona (Johnson solid) at MathWorld.