Augmented sphenocorona

The augmented sphenocorona, or auwaco, is one of the 92 Johnson solids (J87). It consists of 1+1+1+1+2+2+2+2+2+2 triangles and 1 square. It can be constructed by attaching a square pyramid to one of the square faces of the sphenocorona.

Coordinates
Coordinates for the vertices of an sphenocorona with unit edge length are given by: where k ≈ 0.85273 is the smallest positive root of the quartic polynomial
 * $$(0,\pm1/2,\sqrt{1-k^2}),$$
 * $$(\pm k, \pm1/2, 0),$$
 * $$\left(0,\pm\left(\frac12+\frac{\sqrt{3-4k^2}}{2\sqrt{1-k^2}}\right), \frac{1-2k^2}{2\sqrt{1-k^2}}\right),$$
 * $$\left(\pm\frac12,0,-\sqrt{\frac12+k-k^2}\right),$$
 * $$\left(\frac{k+\sqrt{2-2k^2}}2,0,\frac{k+\sqrt{2-2k^2}}2\right).$$
 * $$60x^4−48x^3−100x^2+56x+23.$$ This root can also be given as $$k = \frac{6+\sqrt6+2\sqrt{213-57\sqrt6}}{30}$$