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 .[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 can be determined by adding the area of its faces, 18 equilateral triangles and 3 squares
and its volume is .[2]
Cartesian coordinates
Let be the second smallest positive root of the polynomial
Then, Cartesian coordinates of a hebesphenomegacorona with edge length 2 are given by the union of the orbits of the points
under the action of the group generated by reflections about the xz-plane and the yz-plane.[5]
^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. S2CID120114341.