Sphenomegacorona

In geometry, the sphenomegacorona is a Johnson solid with 16 equilateral triangles and 2 squares as its faces.

Sphenomegacorona
Type	Johnson J87 – J88 – J89
Faces	16 triangles 2 squares
Edges	28
Vertices	12
Vertex configuration	2(34) 2(32.42) 2x2(35) 4(34.4)
Symmetry group	C2v
Dual polyhedron	-
Properties	convex, elementary
Net

3D model of a sphenomegacorona

Properties

The sphenomegacorona was named by Johnson (1966) in which he used the prefix spheno- referring to a wedge-like complex formed by two adjacent lunes—a square with equilateral triangles attached on its opposite sides. The suffix -megacorona refers to a crownlike complex of 12 triangles, contrasted with the smaller triangular complex that makes the sphenocorona.^[1] By joining both complexes, the resulting polyhedron has 16 equilateral triangles and 2 squares, making 18 faces.^[2] All of its faces are regular polygons, categorizing the sphenomegacorona as a Johnson solid—a convex polyhedron in which all of the faces are regular polygons—enumerated as the 88th Johnson solid ${\displaystyle J_{88}}$.^[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 sphenomegacorona ${\displaystyle A}$ is the total of polygonal faces' area—16 equilateral triangles and 2 squares. The volume of a sphenomegacorona is obtained by finding the root of a polynomial, and its decimal expansion—denoted as ${\displaystyle \xi }$—is given by A334114. With edge length ${\displaystyle a}$, its surface area and volume can be formulated as:^[2][5] $${\displaystyle {\begin{aligned}A&=\left(2+4{\sqrt {3}}\right)a^{2}&\approx 8.928a^{2},\\V&=\xi a^{3}&\approx 1.948a^{3}.\end{aligned}}}$$

Cartesian coordinates

Let ${\displaystyle k\approx 0.59463}$ be the smallest positive root of the polynomial $${\displaystyle {\begin{aligned}&1680x^{16}-4800x^{15}-3712x^{14}+17216x^{13}\\+&1568x^{12}-24576x^{11}+2464x^{10}+17248x^{9}\\-&3384x^{8}-5584x^{7}+2000x^{6}+240x^{5}\\-&776x^{4}+304x^{3}+200x^{2}-56x-23\end{aligned}}}$$ Then, Cartesian coordinates of a sphenomegacorona with edge length 2 are given by the union of the orbits of the points $${\displaystyle {\begin{aligned}&\left(0,1,2{\sqrt {1-k^{2}}}\right),\,(2k,1,0),\,\left(0,{\frac {\sqrt {3-4k^{2}}}{\sqrt {1-k^{2}}}}+1,{\frac {1-2k^{2}}{\sqrt {1-k^{2}}}}\right),\\&\left(1,0,-{\sqrt {2+4k-4k^{2}}}\right),\,\left(0,{\frac {{\sqrt {3-4k^{2}}}\left(2k^{2}-1\right)}{\left(k^{2}-1\right){\sqrt {1-k^{2}}}}}+1,{\frac {2k^{4}-1}{\left(1-k^{2}\right)^{\frac {3}{2}}}}\right)\end{aligned}}}$$ under the action of the group generated by reflections about the xz-plane and the yz-plane.^[6]