The Disphenocingulum

The disphenocingulum is the 90th Johnson solid (J90). Its surface consists of 20 equilateral triangles and 4 squares.


The disphenocingulum is one of the special Johnson solids at the end of Norman Johnson's list that are not directly derived from the uniform polyhedra by cut-and-paste operations. As Norman Johnson explains, a lune is a square with two opposite edges attached to equilateral triangles, and a spheno (Latin for wedge) complex is two lunes joined together to form a wedge-like structure. The prefix di- means two, and cingulum (Latin for belt) refers to a belt of 12 triangles. Thus, di-spheno-cingulum refers to taking two spheno complexes and joining them to either side of the belt of 12 triangles. It so happens that if the spheno complexes are rotated 90° with respect to each other, the result can be closed up into a polyhedron with regular faces.


Here are some views of the disphenocingulum from various angles:

Projection Description

Top view.

Front view.

45° side view.


The Cartesian coordinates of the disphenocingulum, centered on the origin with edge length 2, are:

where B is the root of the following polynomial between 1.5 and 1.6:

B12 − 4B11 − 26B10 + 116B9 + 97B8 − 824B7 + 312B6 + 2176B5 − 2024B4 − 1888B3 + 2688B2 − 192B − 368 = 0


C=√((1+2B−B2) / 2)
A=C + √(4−B2)
E=(A2−B2−C2) / (2√(4−B2))
D=1 + √(4−(A−E)2)

Numerically, A, B, C, D, and E have the approximate values:

Last updated 25 Mar 2019.

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