90th Johnson solid (22 faces)
3D model of a disphenocingulum In geometry , the disphenocingulum or pentakis elongated gyrobifastigium is one of the Johnson solids (J 90 Platonic and Archimedean solids.
A Johnson solid is one of 92 strictly convex polyhedra that is composed of regular polygon faces but are not uniform polyhedra (that is, they are not Platonic solids , Archimedean solids , prisms , or antiprisms ). They were named by Norman Johnson , who first listed these polyhedra in 1966.[1]
Cartesian coordinates
Let a ≈ 0.76713 be the second smallest positive root of the polynomial
256
x
12
−
512
x
11
−
1664
x
10
+
3712
x
9
+
1552
x
8
−
6592
x
7
+
1248
x
6
+
4352
x
5
−
2024
x
4
−
944
x
3
+
672
x
2
−
24
x
−
23
{\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
h
=
2
+
8
a
−
8
a
2
{\displaystyle h={\sqrt {2+8a-8a^{2))))
c
=
1
−
a
2
{\displaystyle c={\sqrt {1-a^{2))))
Then, Cartesian coordinates of a disphenocingulum with edge length 2 are given by the union of the orbits of the points
(
1
,
2
a
,
h
2
)
,
(
1
,
0
,
2
c
+
h
2
)
,
(
1
+
3
−
4
a
2
c
,
0
,
2
c
−
1
c
+
h
2
)
{\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.[2]
