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 cutandpaste 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 wedgelike
structure. The prefix di means two
, and cingulum
(Latin for belt) refers to a belt of 12 triangles. Thus,
disphenocingulum 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:
256A^{24} − 1024A^{22} − 5376A^{20} − 4864A^{18} + 17248A^{16} + 780864A^{14} + 649904A^{12} − 8229072A^{10} − 6987567A^{8} + 12237856A^{6} − 4124152A^{4} + 2524544A^{2} + 234256 = 0 
2.1 ≤ A ≤ 2.3 
B^{12} − 4B^{11} − 26B^{10} + 116B^{9} + 97B^{8} − 824B^{7} + 312B^{6} + 2176B^{5} − 2024B^{4} − 1888B^{3} + 2688B^{2} − 192B − 368 = 0 
1.5 ≤ B ≤ 1.6 
256C^{24} + 6144C^{22} + 39168C^{20} + 34048C^{18} − 271776C^{16} − 237888C^{14} + 999920C^{12} + 35184C^{10} − 1647087C^{8} + 1403420C^{6} − 353354C^{4} − 12820C^{2} + 5041 = 0 
0.9 ≤ C ≤ 0.93 
−15D^{12} + 108D^{11} + 82D^{10} − 2404D^{9} + 5191D^{8} + 5560D^{7} − 24804D^{6} + 440D^{5} + 43007D^{4} − 3940D^{3} − 32302D^{2} − 5524D + 3337 = 0 
2 ≤ D ≤ 3 
−57600E^{24} − 1956864E^{22} − 24341248E^{20} − 134413568E^{18} − 295149408E^{16} + 49235392E^{14} + 1010394448E^{12} + 1017962960E^{10} + 754898127E^{8} − 94099264E^{6} − 216014664E^{4} + 5746176E^{2} − 35344 = 0 
0.5 ≤ E ≤ 1 