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. By joining both complexes together, the result polyhedron has 18 equilateral triangles and 3 squares, making 21 faces. 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 {\displaystyle J_{89}} . It is an elementary polyhedron, meaning it cannot be separated by a plane into two small regular-faced polyhedra. 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
6 + 9 3 2 a 2 ≈ 10.7942 a 2 , {\displaystyle {\frac {6+9{\sqrt {3}}}{2}}a^{2}\approx 10.7942a^{2},}
and its volume is 2.9129 a 3 {\displaystyle 2.9129a^{3}} .
Cartesian coordinates Let a ≈ 0.21684 {\displaystyle a\approx 0.21684} be the second smallest positive root of the polynomial
26880 x 10 + 35328 x 9 − 25600 x 8 − 39680 x 7 + 6112 x 6
+ 13696 x 5 + 2128 x 4 − 1808 x 3 − 1119 x 2 + 494 x − 47 {\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
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