Sphenocorona

Let k ≈ 0.85273 be the smallest positive root of the quartic polynomial

${\displaystyle 60x^{4}-48x^{3}-100x^{2}+56x+23.}$

Then, Cartesian coordinates of a sphenocorona with edge length 2 are given by the union of the orbits of the points

${\displaystyle \left(0,1,2{\sqrt {1-k^{2}}}\right),\,(2k,1,0),\left(0,1+{\frac {\sqrt {3-4k^{2}}}{\sqrt {1-k^{2}}}},{\frac {1-2k^{2}}{\sqrt {1-k^{2}}}}\right),\,\left(1,0,-{\sqrt {2+4k-4k^{2}}}\right)}$

under the action of the group generated by reflections about the xz-plane and the yz-plane.[3]

One may then calculate the surface area of a sphenocorona of edge length a as

${\displaystyle A=\left(2+3{\sqrt {3}}\right)a^{2}\approx 7.19615a^{2},}$[4]

and its volume as

${\displaystyle \left({\frac {1}{2}}{\sqrt {1+3{\sqrt {\frac {3}{2}}}+{\sqrt {13+3{\sqrt {6}}}}}}\right)a^{3}\approx 1.51535a^{3}.}$[5]

The sphenocorona is also the vertex figure of the isogonal n-gonal double antiprismoid where n is an odd number greater than one, including the grand antiprism with pairs of trapezoid rather than square faces.

• Augmented sphenocorona

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