Disphenocingulum
Type	Johnson J89 – J90 – J91
Faces	4+2x8 triangles 4 squares
Edges	38
Vertices	16
Vertex configuration	4(32.42) 4(35) 8(34.4)
Symmetry group	D2d
Dual polyhedron	Order-5 truncated snub disphenoid
Properties	convex
Net

3D model of a disphenocingulum

In geometry, the disphenocingulum or pentakis elongated gyrobifastigium is one of the Johnson solids (J₉₀). It is one of the elementary Johnson solids that do not arise from "cut and paste" manipulations of the Platonic and Archimedean solids.

A Johnson solid is one of 92 strictly convex polyhedra that is composed of regular polygon faces but are not uniform polyhedra (that is, they are not Platonic solids, Archimedean solids, prisms, or antiprisms). They were named by Norman Johnson, who first listed these polyhedra in 1966.^[1]

Cartesian coordinates

Let a ≈ 0.76713 be the second smallest positive root of the polynomial

${\displaystyle {\begin{aligned}&256x^{12}-512x^{11}-1664x^{10}+3712x^{9}+1552x^{8}-6592x^{7}\\&\quad {}+1248x^{6}+4352x^{5}-2024x^{4}-944x^{3}+672x^{2}-24x-23\end{aligned}}}$

and ${\displaystyle h={\sqrt {2+8a-8a^{2}}}}$ and ${\displaystyle c={\sqrt {1-a^{2}}}}$.

Then, Cartesian coordinates of a disphenocingulum with edge length 2 are given by the union of the orbits of the points

${\displaystyle \left(1,2a,{\frac {h}{2}}\right),\ \left(1,0,2c+{\frac {h}{2}}\right),\ \left(1+{\frac {\sqrt {3-4a^{2}}}{c}},0,2c-{\frac {1}{c}}+{\frac {h}{2}}\right)}$

under the action of the group generated by reflections about the xz-plane and the yz-plane.^[2]

References

External links

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