The Disphenocingulum


The disphenocingulum is the 90th Johnson solid (J90). Its surface consists of 24 faces (20 equilateral triangles and 4 squares), 38 edges, and 16 vertices.

The
disphenocingulum

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.

Projections

Here are some views of the disphenocingulum from various angles:

Projection Description

Top view.

Front view.

45° side view.

Coordinates

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

and:

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 18 Jun 2019.

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