Sphenocorona



In geometry, the sphenocorona is a Johnson solid with 12 equilateral triangles and 2 squares as its faces.

Properties
The sphenocorona was named by in which he used the prefix spheno- referring to a wedge-like complex formed by two adjacent lunes&mdash;a square with equilateral triangles attached on its opposite sides. The suffix -corona refers to a crownlike complex of 8 equilateral triangles. By joining both complexes together, the resulting polyhedron has 12 equilateral triangles and 2 squares, making 14 faces. 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 $$ J_{86} $$. It is elementary, meaning it cannot be separated by a plane into two small regular-faced polyhedra.

The surface area of a sphenocorona with edge length $$ a $$ can be calculated as: $$ A=\left(2+3\sqrt{3}\right)a^2\approx7.19615a^2,$$ and its volume as: $$\left(\frac{1}{2}\sqrt{1 + 3 \sqrt{\frac{3}{2}} + \sqrt{13 + 3 \sqrt{6}}}\right)a^3\approx1.51535a^3.$$

Cartesian coordinates
Let $$ k \approx 0.85273 $$ be the smallest positive root of the quartic polynomial $$ 60x^4 - 48x^3 - 100x^2 + 56x + 23 $$. Then, Cartesian coordinates of a sphenocorona with edge length 2 are given by the union of the orbits of the points $$ \left(0,1,2\sqrt{1-k^2}\right),\,(2k,1,0),\left(0,1+\frac{\sqrt{3-4k^2}}{\sqrt{1-k^2}},\frac{1-2k^2}{\sqrt{1-k^2}}\right),\,\left(1,0,-\sqrt{2+4k-4k^2}\right)$$ under the action of the group generated by reflections about the xz-plane and the yz-plane.

Variations
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.
 * Grand antiprism verf.png