Disphenocingulum

Disphenocingulum
Type	Johnson J89 – J90 – J91
Faces	20 triangles 4 squares
Edges	38
Vertices	16
Vertex configuration	4(32.42) 4(35) 8(34.4)
Symmetry group	D2d
Properties	convex, elementary
Net

3D model of a disphenocingulum

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

Properties

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 ${\displaystyle J_{90}}$. ^[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 ${\displaystyle a}$ can be determined by adding all of its faces, the area of 20 equilateral triangles and 4 squares ${\displaystyle (4+5{\sqrt {3}})a^{2}\approx 12.6603a^{2}}$, and its volume is ${\displaystyle 3.7776a^{3}}$. ^[2]

Cartesian coordinates

Let ${\displaystyle a\approx 0.76713}$ be the second smallest positive root of the polynomial $${\displaystyle {\begin{aligned}&256x^{12}-512x^{11}-1664x^{10}+3712x^{9}+1552x^{8}-6592x^{7}\\&\quad {}+1248x^{6}+4352x^{5}-2024x^{4}-944x^{3}+672x^{2}-24x-23\end{aligned}}}$$ and ${\displaystyle h={\sqrt {2+8a-8a^{2}}}}$ and ${\displaystyle c={\sqrt {1-a^{2}}}}$. Then, the Cartesian coordinates of a disphenocingulum with edge length 2 are given by the union of the orbits of the points $${\displaystyle \left(1,2a,{\frac {h}{2}}\right),\ \left(1,0,2c+{\frac {h}{2}}\right),\ \left(1+{\frac {\sqrt {3-4a^{2}}}{c}},0,2c-{\frac {1}{c}}+{\frac {h}{2}}\right)}$$ under the action of the group generated by reflections about the xz-plane and the yz-plane. ^[5]

References

1. Johnson, N. W. (1966). "Convex polyhedra with regular faces". Canadian Journal of Mathematics . 18: 169–200. doi: 10.4153/cjm-1966-021-8 . MR   0185507. S2CID   122006114. Zbl   0132.14603.
2. Berman, M. (1971). "Regular-faced convex polyhedra". Journal of the Franklin Institute. 291 (5): 329–352. doi:10.1016/0016-0032(71)90071-8. MR   0290245.
3. Francis, D. (August 2013). "Johnson solids & their acronyms". Word Ways. 46 (3): 177.
4. Cromwell, P. R. (1997). Polyhedra. Cambridge University Press. p. 86–87, 89. ISBN   978-0-521-66405-9.
5. Timofeenko, A. V. (2009). "The non-Platonic and non-Archimedean noncomposite polyhedra". Journal of Mathematical Science. 162 (5): 717. doi:10.1007/s10958-009-9655-0. S2CID   120114341.

External links

• Weisstein, Eric W., "Disphenocingulum" ("Johnson solid") at MathWorld .