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

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

89

{\displaystyle J_{89}}

.

[3] It is an elementary polyhedron, meaning it cannot be separated by a plane into two small regular-faced polyhedra.

[4] The surface area of a hebesphenomegacorona with edge length

a

{\displaystyle a}

can be determined by adding the area of its faces, 18 equilateral triangles and 3 squares

2

2

and its volume is

a

be the second smallest positive root of the polynomial

{\displaystyle {\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

{\displaystyle {\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] This polyhedron-related article is a stub.

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