Hebesphenomegacorona

89th Johnson solid (21 faces)

Hebesphenomegacorona

Type	Johnson J₈₈ – J₈₉ – J₉₀
Faces	3x2+3x4 triangles 1+2 squares
Edges	33
Vertices	14
Vertex configuration	4(3².4²) 2+2x2(3⁵) 4(3⁴.4)
Symmetry group	C_2v
Properties	convex
Net

In geometry, the hebesphenomegacorona is a Johnson solid with 18 equilateral triangles and 3 squares as its faces.

Properties

The hebesphenomegacorona is named by Johnson (1966) in which he used the prefix hebespheno- referring to a blunt wedge-like complex formed by three adjacent lunes—a square with equilateral triangles attached on its opposite sides. The suffix -megacorona refers to a crownlike complex of 12 triangles.^[1] By joining both complexes together, the result polyhedron has 18 equilateral triangles and 3 squares, making 21 faces.^[2]. All of its faces are regular polygons, categorizing the hebesphenomegacorona as a Johnson solid—a convex polyhedron in which all of its faces are regular polygons—enumerated as 89th Johnson solid $J_{89}$ .^[3] It is elementary, meaning it does not arise from "cut-and-paste" manipulations of both Platonic and Archimedean solids.^[4]

The surface area of a hebesphenomegacorona with edge length $a$ can be determined by adding the area of its faces, 18 equilateral triangles and 3 squares

{\frac {6+9{\sqrt {3}}}{2}}a^{2}\approx 10.7942a^{2},

and its volume is

2.9129a^{3}

.^[2]

Cartesian coordinates

Let $a\approx 0.21684$ be the second smallest positive root of the polynomial

{\begin{aligned}&26880x^{10}+35328x^{9}-25600x^{8}-39680x^{7}+6112x^{6}\\&\quad {}+13696x^{5}+2128x^{4}-1808x^{3}-1119x^{2}+494x-47\end{aligned}}

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

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

under the action of the group generated by reflections about the xz-plane and the yz-plane.^[5]

References

^ 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.
^ ^a ^b 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.
^ Francis, D. (August 2013). "Johnson solids & their acronyms". Word Ways. 46 (3): 177.
^ Cromwell, P. R. (1997). Polyhedra. Cambridge University Press. p. 87. ISBN 978-0-521-66405-9.
^ 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.