Sphenomegacorona

Let k ≈ 0.59463 be the smallest positive root of the polynomial

${\displaystyle 1680x^{16}-4800x^{15}-3712x^{14}+17216x^{13}+1568x^{12}-24576x^{11}+2464x^{10}+17248x^{9}}$

${\displaystyle {}-3384x^{8}-5584x^{7}+2000x^{6}+240x^{5}-776x^{4}+304x^{3}+200x^{2}-56x-23.}$

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

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

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

under the action of the group generated by reflections about the xz-plane and the yz-plane.[2]

We may then calculate the surface area of a sphenomegacorona of edge length a as

${\displaystyle A=(2+4{\sqrt {3}})a^{2}\approx 8.92820a^{2},}$[3]

and its volume as

${\displaystyle V=\xi a^{3}\approx 1.94811a^{3},}$

where the decimal expansion of ξ is given by A334114.[4]

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