Sphenocorona

The sphenocorona was named by Johnson (1966) in which he used the prefix spheno- referring to a wedge-like complex formed by two adjacent lunes—a square with equilateral triangles attached on its opposite sides. The suffix -corona refers to a crownlike complex of 8 equilateral triangles.^[1] By joining both complexes together, the resulting polyhedron has 12 equilateral triangles and 2 squares, making 14 faces.^[2] A convex polyhedron in which all faces are regular polygons is called a Johnson solid. The sphenocorona is among them, enumerated as the 86th Johnson solid ${\displaystyle J_{86}}$.^[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 sphenocorona with edge length ${\displaystyle a}$ can be calculated as^[2]$${\displaystyle A=\left(2+3{\sqrt {3}}\right)a^{2}\approx 7.19615a^{2},}$$and its volume as^[2]$${\displaystyle \left({\frac {1}{2}}{\sqrt {1+3{\sqrt {\frac {3}{2}}}+{\sqrt {13+3{\sqrt {6}}}}}}\right)a^{3}\approx 1.51535a^{3}.}$$

Let

${\displaystyle {\begin{aligned}k&={\frac {6+{\sqrt {6}}+2{\sqrt {213-57{\sqrt {6}}}}}{30}}\\&\approx 0.85273\end{aligned}}}$

be the smallest positive root of the quartic polynomial ${\displaystyle 60x^{4}-48x^{3}-100x^{2}+56x+23}$. Then, Cartesian coordinates of a sphenocorona with edge length 2 are given by the union of the orbits of the points$${\displaystyle \left(0,1,2{\sqrt {1-k^{2}}}\right),\,(2k,1,0),\left(0,1+{\frac {\sqrt {3-4k^{2}}}{\sqrt {1-k^{2}}}},{\frac {1-2k^{2}}{\sqrt {1-k^{2}}}}\right),\,\left(1,0,-{\sqrt {2+4k-4k^{2}}}\right)}$$under the action of the group generated by reflections about the xz-plane and the yz-plane.^[5]

The sphenocorona is also the vertex figure of the isogonal n-gonal double antiprismoid where n is an odd number greater than one, including the grand antiprism with pairs of trapezoid rather than square faces.