Disphenocingulum

Disphenocingulum
File:Disphenocingulum.png
Type	Johnson J89 – J90 – J91
Faces	20 triangles 4 squares
Edges	38
Vertices	16
Vertex configuration	4(32.42) 4(35) 8(34.4)
Symmetry group	D2d
Properties	convex, elementary
Net
File:Johnson solid 90 net.png

File:J90 disphenocingulum.stl

3D model of a disphenocingulum

In geometry, the disphenocingulum is a Johnson solid with 20 equilateral triangles and 4 squares as its faces.

Properties

The disphenocingulum is named by Johnson (1966). The prefix dispheno- refers to two wedgelike complexes, each formed by two adjacent lunes—a figure of two equilateral triangles at the opposite sides of a square. The suffix -cingulum, literally 'belt', refers to a band of 12 triangles joining the two wedges. The resulting polyhedron has 20 equilateral triangles and 4 squares, making 24 faces.. All of the faces are regular, categorizing the disphenocingulum as a Johnson solid—a convex polyhedron in which all of its faces are regular polygon—enumerated as 90th Johnson solid $J_{90}$.. It is an elementary polyhedron, meaning that it cannot be separated by a plane into two small regular-faced polyhedra. The surface area of a disphenocingulum with edge length $a$ can be determined by adding all of its faces, the area of 20 equilateral triangles and 4 squares $(4+5\sqrt{3})a^2\approx 12.6603a^2$, and its volume is $3.7776a^3$.

Cartesian coordinates

Let $a\approx 0.76713$ be the second smallest positive root of the polynomial $$\begin{array}{rl}& 256x^{12}-512x^{11}-1664x^{10}+3712x^9+1552x^8-6592x^7\\ & +1248x^6+4352x^5-2024x^4-944x^3+672x^2-24x-23\end{array}$$ and $h=\sqrt{2+8a-8a^2}$ and $c=\sqrt{1-a^2}$. Then, the Cartesian coordinates of a disphenocingulum with edge length 2 are given by the union of the orbits of the points $$\begin{gathered} (1,2a,\frac{h}{2}), \\ (1,0,2c+\frac{h}{2}), \\ (1+\frac{\sqrt{3-4a^2}}{c},0,2c-\frac{1}{c}+\frac{h}{2}) \end{gathered}$$ under the action of the group generated by reflections about the xz-plane and the yz-plane.

External links

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