Hebesphenomegacorona

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

Hebesphenomegacorona
Type	Johnson J88 – J89 – J90
Faces	3x2+3x4 triangles 1+2 squares
Edges	33
Vertices	14
Vertex configuration	4(32.42) 2+2x2(35) 4(34.4)
Symmetry group	C2v
Properties	convex, elementary
Net

3D model of a hebesphenomegacorona

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.^[1] By joining both complexes together, the result polyhedron has 18 equilateral triangles and 3 squares, making 21 faces.^[2]. 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 ${\displaystyle J_{89}}$.^[3] It is an elementary polyhedron, meaning it cannot be separated by a plane into two small regular-faced polyhedra.^[4]

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

Cartesian coordinates

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