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 (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. 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 an elementary polyhedron, 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$$A=(2+3\sqrt{3})a^2\approx 7.19615a^2,$$and its volume as$$\left(\frac{1}{2}\sqrt{1+3\sqrt{\frac{3}{2}}+\sqrt{13+3\sqrt{6}}}\right)a^3\approx 1.51535a^3.$$

Cartesian coordinates

Let

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

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$$(0,1,2\sqrt{1-k^2}),(2k,1,0),(0,1+\frac{\sqrt{3-4k^2}}{\sqrt{1-k^2}},\frac{1-2k^2}{\sqrt{1-k^2}}),(1,0,-\sqrt{2+4k-4k^2})$$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.

160px

See also

• Augmented sphenocorona

External links

• Eric W. Weisstein, Sphenocorona (Johnson solid) at MathWorld.