Sphenocorona

The sphenocorona, or waco, is one of the 92 Johnson solids (J86). It consists of 2+2+4+4 triangles and 2 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 "spheno" (meaning a wedge-like arrangement of two "lunes", where each lune consists of a square attached to two triangles) and "corona" denoting a crown-like structure composed of 8 triangles.

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

Related polyhedra
A square pyramid can be attached to one of the sphenocorona's square faces to form the augmented sphenocorona.

The sphenocorona has a weak relation to the elongated pentagonal bipyramid: if one of the squares of the elongated pentagonal bipyramid is contracted to an edge, bringing two triangles together and converting two squares to triangles, the result is a sphenocorona.

In vertex figures
The sphenocorona appears as the vertex figure of the nonuniform triangular double antiprismoid. This vertex figure has an edge length of 1, and has no corealmic realization, because the Johnson sphenocorona has no circumscribed sphere.

Variants made by changing the edge connecting the two isosceles trapezoids also appear as the vertex figure of the generally nonuniform double antiprismoids, uniform only for the pentagon-based grand antiprism, which has an aforementioned edge length of (1+$\sqrt{2}$)/2. This vertex figure is also equivalent to a regular icosahedron with two adjacent vertices removed.