Disphenocingulum Explained

Type:	Johnson
Faces:	20 triangles 4 squares
Edges:	38
Vertices:	16
Net:	Johnson solid 90 net.png

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

Properties

The disphenocingulum is named by . 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. The resulting polyhedron has 20 equilateral triangles and 4 squares, making 24 faces.. 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

J _{90} .. It is elementary, meaning that it cannot be separated by a plane into two small regular-faced polyhedra.

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 \sqrt{ 3 })a ^{2} ≈ 12 . 6603 a ^{2} , and its volume is 3 . 7776 a ^{3} .

Cartesian coordinates

Let

a ≈ 0 . 76713 be the second smallest positive root of the polynomialand h = \sqrt{ 2+8 a -8 a ^{2}} and c = \sqrt{ 1- a ^{2}} . Then, the Cartesian coordinates of a disphenocingulum with edge length 2 are given by the union of the orbits of the points$$\backslash left(1,2a,\backslash frac\backslash right),\backslash \; \backslash left(1,0,2c+\backslash frac\backslash right),\backslash \; \backslash left(1+\backslash frac,0,2c-\backslash frac+\backslash frac\backslash right)$$under the action of the group generated by reflections about the xz-plane and the yz-plane