# 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.

## 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:

• (±1, 0, A)
• (±1, ±B, C)
• (±D, 0, E)
• (0, ±D, −E)
• (±B, ±1, −C)
• (0, ±1, −A)

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

• A = 2.208875884159868
• B = 1.534262227966894
• C = 0.925895207830690
• D = 2.252966294157922
• E = 0.650005951899941

The exact values of A, B, C, D, and E may be obtained by extracting the roots of the following polynomials within the indicated ranges:

 256A24 − 1024A22 − 5376A20 − 4864A18 + 17248A16 + 780864A14 + 649904A12 − 8229072A10 − 6987567A8 + 12237856A6 − 4124152A4 + 2524544A2 + 234256 = 0 2.1 ≤ A ≤ 2.3 B12 − 4B11 − 26B10 + 116B9 + 97B8 − 824B7 + 312B6 + 2176B5 − 2024B4 − 1888B3 + 2688B2 − 192B − 368 = 0 1.5 ≤ B ≤ 1.6 256C24 + 6144C22 + 39168C20 + 34048C18 − 271776C16 − 237888C14 + 999920C12 + 35184C10 − 1647087C8 + 1403420C6 − 353354C4 − 12820C2 + 5041 = 0 0.9 ≤ C ≤ 0.93 −15D12 + 108D11 + 82D10 − 2404D9 + 5191D8 + 5560D7 − 24804D6 + 440D5 + 43007D4 − 3940D3 − 32302D2 − 5524D + 3337 = 0 2 ≤ D ≤ 3 −57600E24 − 1956864E22 − 24341248E20 − 134413568E18 − 295149408E16 + 49235392E14 + 1010394448E12 + 1017962960E10 + 754898127E8 − 94099264E6 − 216014664E4 + 5746176E2 − 35344 = 0 0.5 ≤ E ≤ 1

Last updated 03 Sep 2018.