Disphenocingulum

The disphenocingulum is one of the 92 Johnson solids (J90). It consists of 4+8+8 triangles and 4 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 - "di" signifying two such complexes) and "cingulum" denoting a crown-like belt of 12 triangles.

Coordinates
Coordinates for a disphenocingulum with unit edge length are given by together with all sign changes of the first two coordinates, and the points that result from swapping the first two coordinates and negating the third, where k is the second smallest real root of the polynomial
 * $$\left(\frac12, 0, u + \frac v4\right),$$
 * $$\left(\frac12 + \frac{w}{2u}, 0, u - \frac1{2u} + \frac v4\right),$$
 * $$\left(\frac12, k, \frac v4\right),$$
 * $$256x^{12} - 512x^{11} - 1664x^{10} + 3712x^9 + 1552x^8 - 6592x^7$$
 * $${} + 1248x^6 + 4352x^5 - 2024x^4 - 944x^3 + 672x^2 - 24x - 23,$$

and u, v, w are given by
 * $$u=\sqrt{1-k^2},\ v=\sqrt{2+8k-8k^2},\ w=\sqrt{3-4k^2}.$$

From these coordinates, its volume can be calculated as ξ ≈ 3.77765, where ξ is the largest real root of
 * $$1213025622610333925376 x^{24}+54451372392730545094656 x^{22}$$
 * $${}-796837093078664749252608 x^{20}-4133410366404688544268288 x^{18}$$
 * $${}+20902529024429842816303104 x^{16}-133907540390420673677230080 x^{14}$$
 * $${}+246234688242991598853881856 x^{12}-63327534106871321714442240 x^{10}$$
 * $${}+14389309497459555704164608 x^8+48042947402464500749392128 x^6$$
 * $${}-5891096640600351061013664 x^4-3212114716816853362953264 x^2$$
 * $${}+479556973248657693884401.$$