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 (Johnson 1966) in which he used the prefix hebespheno- referring to a blunt wedge-like complex formed by three adjacent lunes—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—a convex polyhedron in which all of its faces are regular polygons—enumerated as 89th Johnson solid $J_{89}$. It is an elementary polyhedron, 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 $$\begin{array}{rl}& 26880x^{10}+35328x^9-25600x^8-39680x^7+6112x^6\\ & +13696x^5+2128x^4-1808x^3-1119x^2+494x-47\end{array}$$ Then, Cartesian coordinates of a hebesphenomegacorona with edge length 2 are given by the union of the orbits of the points $$\begin{array}{rl}& (1,1,2\sqrt{1-a^2}),(1+2a,1,0),(0,1+\sqrt{2}\sqrt{\frac{2a-1}{a-1}},-\frac{2a^2+a-1}{\sqrt{1-a^2}}),(1,0,-\sqrt{3-4a^2}),\\ & (0,\frac{\sqrt{2(3-4a^2)(1-2a)}+\sqrt{1+a}}{2(1-a)\sqrt{1+a}},\frac{(2a-1)\sqrt{3-4a^2}}{2(1-a)}-\frac{\sqrt{2(1-2a)}}{2(1-a)\sqrt{1+a}})\end{array}$$ under the action of the group generated by reflections about the xz-plane and the yz-plane.

External links

• Eric W. Weisstein, Hebesphenomegacorona (Johnson solid) at MathWorld.