Hebesphenomegacorona

The hebesphenomegacorona is one of the 92 Johnson solids (J89). It consists of 18 triangles and 3 squares.

It is one of several polyhedra near the end of the list of Johnson solids with no obvious relation to any of the uniform polyhedra. The name is derived from "hebespheno" (meaning a wedge-like arrangement of three "lunes", where each lune consists of a square attached to two triangles) and "megacorona" denoting a crown-like structure composed of 12 triangles (as opposed to the smaller "corona" of 8 triangles found in the sphenocorona).

It has a weak relation to the icosahedron. If the middle square is contracted to an edge such that the neighboring squares become triangles and the neighboring triangles touch, the result is an icosahedron.

Coordinates
Coordinates for a hebesphenomegacorona with unit edge length are given by where k ≈ 0.21684 is the second to smallest positive real root of
 * $$\left(\pm\frac12, \pm\frac12, u\right),$$
 * $$\left(\pm\left(\frac12+k\right), \pm\frac12, u\right),$$
 * $$\left(\pm\frac12, 0,-\frac w2\right),$$
 * $$\left(0,\pm\left(\frac 12+\frac v{2u}\right), \frac{v^2}{4u}\right),$$
 * $$\left(0,\pm\frac{wv+k+1}{4u^2}, \frac{(2k-1)w}{4(1-k)}-\frac{v}{4u^2}\right),$$
 * $$26880 x^{10}+35328 x^9-25600 x^8-39680 x^7+6112 x^6$$
 * $${}+13696 x^5+2128 x^4-1808 x^3-1119 x^2+494 x-47,$$

and with u, v, w given by:
 * $$u=\sqrt{1-k^2},\ v=\sqrt{2-2k-4k^2},\ w=\sqrt{3-4k^2}.$$

From these coordinates, the volume can be calculated by ξ ≈ 2.91291, where ξ is given as the greatest real root of
 * $$2693461945329 + 132615435213216 x^2 - 22211277300912896 x^4$$
 * $${} - 8337259437908852736 x^6 + 366229890219212144640 x^8$$
 * $${} - 3065290664181478981632 x^{10} + 8973584611317745975296 x^{12}$$
 * $${} - 8432333285523990773760 x^{14} + 3596480447590271287296 x^{16}$$
 * $${} - 722445512980071186432 x^{18} + 47330370277129322496 x^{20}$$