Hebesphenomegacorona



In geometry, the hebesphenomegacorona is a Johnson solid with 18 equilateral triangles and 3 squares as its faces.

Properties
The hebesphenomegacorona is named by in which he used the prefix hebespheno- referring to a blunt wedge-like complex formed by three adjacent lunes&mdash;a square with equilateral triangles attached on its opposite sides. The suffix -megacorona refers to a crownlike complex of 12 triangles. By joining both complexes together, the result polyhedron has 18 equilateral triangles and 3 squares, making 21 faces.. All of its faces are regular polygons, categorizing the hebesphenomegacorona as a Johnson solid&mdash;a convex polyhedron in which all of its faces are regular polygons&mdash;enumerated as 89th Johnson solid $$ J_{89} $$. It is elementary, meaning it cannot be separated by a plane into two small regular-faced polyhedra.

The surface area of a hebesphenomegacorona with edge length $$ a $$ can be determined by adding the area of its faces, 18 equilateral triangles and 3 squares $$ \frac{6 + 9\sqrt{3}}{2}a^2 \approx 10.7942a^2, $$ and its volume is $$ 2.9129a^3 $$.

Cartesian coordinates
Let $$ a \approx 0.21684 $$ be the second smallest positive root of the polynomial $$ \begin{align} &26880x^{10} + 35328x^9 - 25600x^8 - 39680x^7 + 6112x^6 \\ &\quad {}+ 13696x^5 + 2128x^4 - 1808x^3 - 1119x^2 + 494x - 47 \end{align}$$ Then, Cartesian coordinates of a hebesphenomegacorona with edge length 2 are given by the union of the orbits of the points $$ \begin{align} &\left(1,1,2\sqrt{1-a^2}\right),\ \left(1+2a,1,0\right),\ \left(0,1+\sqrt{2}\sqrt{\frac{2a-1}{a-1}},-\frac{2a^2+a-1}{\sqrt{1-a^2}}\right),\ \left(1,0,-\sqrt{3-4a^2}\right), \\ &\left(0,\frac{\sqrt{2(3-4a^2)(1-2a)}+\sqrt{1+a}}{2(1-a)\sqrt{1+a}},\frac{(2a-1)\sqrt{3-4a^2}}{2(1-a)}-\frac{\sqrt{2(1-2a)}}{2(1-a)\sqrt{1+a}}\right) \end{align}$$ under the action of the group generated by reflections about the xz-plane and the yz-plane.