Sphenomegacorona

Sphenomegacorona
File:Sphenomegacorona.png
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
File:Johnson solid 88 net.png

File:J88 sphenomegacorona.stl

3D model of a sphenomegacorona

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

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. By joining both complexes, the resulting polyhedron has 16 equilateral triangles and 2 squares, making 18 faces. 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 $J_{88}$. It is an elementary polyhedron, meaning it cannot be separated by a plane into two small regular-faced polyhedra. The surface area of a sphenomegacorona $A$ 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 $\xi$—is given by A334114. With edge length $a$, its surface area and volume can be formulated as: $$\begin{array}{rlr}A& =(2+4\sqrt{3})a^2& \approx 8.928a^2,\\ V& =\xi a^3& \approx 1.948a^3.\end{array}$$

Cartesian coordinates

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

External links

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