Sphenomegacorona

In geometry, the sphenomegacorona is one of the Johnson solids (J88). It is one of the elementary Johnson solids that do not arise from "cut and paste" manipulations of the Platonic and Archimedean solids.

Johnson uses the prefix spheno- to refer to a wedge-like complex formed by two adjacent lunes, a lune being a square with equilateral triangles attached on opposite sides. Likewise, the suffix -megacorona refers to a crownlike complex of 12 triangles, contrasted with the smaller triangular complex that makes the sphenocorona. Joining both complexes together results in the sphenomegacorona.

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:


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