Sphenomegacorona

The sphenomegacorona, or wamco, is one of the 92 Johnson solids (J88). It consists of 2+2+4+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 "megacorona", denoting a large crown-like structure composed of 12 triangles, as opposed to the smaller "corona" of 8 triangles found in the sphenocorona.

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


 * $$\begin{align}&1680 x^{16}- 4800 x^{15} - 3712 x^{14} + 17216 x^{13}+ 1568 x^{12} - 24576 x^{11} + 2464 x^{10} + 17248 x^9\\{}&-3384 x^8 - 5584 x^7 + 2000 x^6+ 240 x^5- 776 x^4+ 304 x^3 + 200 x^2 - 56 x -23.\end{align}$$

Then, coordinates for the vertices of a sphenomegacorona with edge length 1 are given by the points:


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

Measures
From the coordinates of the sphenomegacorona, one may calculate its volume for unit edge length as approximately 1.94811. The exact value is the greatest real root of the polynomial


 * $$\begin{align}&521578814501447328359509917696x^{32} - 985204427391622731345740955648x^{30}\\

{} &- 16645447351681991898880656015360x^{28} + 79710816694053483249372512649216x^{26}\\ {} &- 152195045391070538203422101864448x^{24} + 156280253448056209478031589244928x^{22}\\ {} &- 96188116617075838858708654227456x^{20} + 30636368373570166303441645731840x^{18}\\ {} &+ 5828527077458909552923002273792x^{16} - 8060049780765551057159394951168x^{14}\\ {} &+ 1018074792115156107372011716608x^{12} + 35220131544370794950945931264x^{10}\\ {} &+ 327511698517355918956755959808x^8 - 116978732884218191486738706432x^6\\ {} &+ 10231563774949176791703149568x^4 - 366323949299263261553952192x^2\\ {} &+ 3071435678740442112675625.\end{align}$$

The dihedral angles may also be calculated in terms of the constant k given in § Vertex coordinates:


 * $$\text{4–4: }2 \text{asin}(k) \approx 72.97300^\circ,$$
 * $$\text{3–3: }2 \text{asin}\left(\frac{2k}{\sqrt{3}}\right) \approx 86.72683^\circ,$$
 * $$\text{3–3: }2 \text{asin}\left(\sqrt{\frac{1+2k}{3}}\right) \approx 117.35557^\circ,$$
 * $$\text{3–3: }2 \text{asin}\left(\sqrt{\frac{-3+4k^2}{3\left(-1+k^2\right)}}\right) \approx 129.44457^\circ,$$
 * $$\text{4–3: acos}\left(\frac{3-\left(\sqrt{4-4k^2}+\sqrt{2+4k-4k^2}\right)^2}{2\sqrt{3}}\right) \approx 137.24008^\circ,$$
 * $$\text{3–3: }2 \text{asin}\left(\frac{\sqrt{2+2k^4+\sqrt{3-7k^2+4k^2}-2k^2\left(2+\sqrt{3-7k^2+4k^2}\right)}}{\sqrt{3}\left(1-k^2\right)}\right) \approx 143.73833^\circ,$$
 * $$\text{4–3: acos}\left(-\sqrt{\frac{-3+4k^2}{3\left(-1+k^2\right)}}\right) \approx 154.72228^\circ,$$
 * $$\text{3–3: }2 \text{asin}\left(\frac{\sqrt{3-4k^2}(2k^2-1)}{2(k^2-1)\sqrt{3(1-k^2)}}+\frac{1}{2\sqrt{3}}\right) \approx 161.48285^\circ,$$
 * $$\text{3–3: }2 \text{asin}\left(\frac{1}{2\sqrt{3}}\sqrt{1+\left(\sqrt{2+k-k^2}+\frac{2-4k^2}{\sqrt{1-k^2}}\right)^2+\left(1+\sqrt{\frac{-3+4k^2}{-1+k^2}}\right)^2}\right) \approx 171.64574^\circ.$$