Sphenomegacorona

Sphenomegacorona
Type	Johnson J87 – J88 – J89
Faces	16 triangles 2 squares
Edges	28
Vertices	12
Vertex configuration	2(34) 2(32.42) 2x2(35) 4(34.4)
Symmetry group	C2v
Dual polyhedron	-
Properties	convex
Net

3D model of a sphenomegacorona

In geometry, the sphenomegacorona is one of the Johnson solids (J₈₈). It is one of the elementary Johnson solids that do not arise from "cut and paste" manipulations of the 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]

Johnson uses the prefix spheno- to refer to a wedge-like complex formed by two adjacent lunes, a lune being a square with equilateral triangles attached on opposite sides. Likewise, the suffix -megacorona refers to a crownlike complex of 12 triangles, contrasted with the smaller triangular complex that makes the sphenocorona. Joining both complexes together results in the sphenomegacorona. ^[1]

Cartesian coordinates

Let k ≈ 0.59463 be the smallest positive root of the polynomial

$${\displaystyle 1680x^{16}-4800x^{15}-3712x^{14}+17216x^{13}+1568x^{12}-24576x^{11}+2464x^{10}+17248x^{9}-3384x^{8}-5584x^{7}+2000x^{6}+240x^{5}-776x^{4}+304x^{3}+200x^{2}-56x-23.}$$

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

$${\displaystyle {\begin{aligned}&\left(0,1,2{\sqrt {1-k^{2}}}\right),\,(2k,1,0),\,\left(0,{\frac {\sqrt {3-4k^{2}}}{\sqrt {1-k^{2}}}}+1,{\frac {1-2k^{2}}{\sqrt {1-k^{2}}}}\right),\\&\left(1,0,-{\sqrt {2+4k-4k^{2}}}\right),\,\left(0,{\frac {{\sqrt {3-4k^{2}}}\left(2k^{2}-1\right)}{\left(k^{2}-1\right){\sqrt {1-k^{2}}}}}+1,{\frac {2k^{4}-1}{\left(1-k^{2}\right)^{\frac {3}{2}}}}\right)\end{aligned}}}$$

under the action of the group generated by reflections about the xz-plane and the yz-plane. ^[2]

The surface area of a sphenomegacorona with edge length a can be calculated as:

$${\displaystyle A=\left(2+4{\sqrt {3}}\right)a^{2}\approx 8.92820a^{2},}$$

and its volume as

$${\displaystyle V=\xi a^{3}\approx 1.94811a^{3},}$$

where the decimal expansion of ξ is given by A334114. ^{[3] [4]}

References

1. Johnson, Norman 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. Timofeenko, A. V. (2009), "The non-Platonic and non-Archimedean noncomposite polyhedra", Journal of Mathematical Science, 162 (5): 720, doi:10.1007/s10958-009-9655-0, S2CID   120114341
3. Berman, Martin (1971), "Regular-faced convex polyhedra", Journal of the Franklin Institute, 291 (5): 329–352, doi:10.1016/0016-0032(71)90071-8, MR   0290245
4. OEIS Foundation Inc. (2020), The On-Line Encyclopedia of Integer Sequences, A334114.

External links

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