In geometry, the disphenocingulum is a Johnson solid with 20 equilateral triangles and 4 squares as its faces.

The disphenocingulum is named by Johnson (1966).

The prefix dispheno- refers to two wedgelike complexes, each formed by two adjacent lunes—a figure of two equilateral triangles at the opposite sides of a square.

The suffix -cingulum, literally 'belt', refers to a band of 12 triangles joining the two wedges.

[1] The resulting polyhedron has 20 equilateral triangles and 4 squares, making 24 faces.[2].

All of the faces are regular, categorizing the disphenocingulum as a Johnson solid—a convex polyhedron in which all of its faces are regular polygon—enumerated as 90th Johnson solid

.

[3].

It is an elementary polyhedron, meaning that it cannot be separated by a plane into two small regular-faced polyhedra.

[4] The surface area of a disphenocingulum with edge length

a

can be determined by adding all of its faces, the area of 20 equilateral triangles and 4 squares

( 4 + 5

2

, and its volume is

be the second smallest positive root of the polynomial

x

11

{\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}}}

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

under the action of the group generated by reflections about the xz-plane and the yz-plane.