thumb|3D model of a 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—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 <math> J_{86} </math>. 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 <math> a </math> can be calculated as<math display="block"> A=\left(2+3\sqrt{3}\right)a^2\approx7.19615a^2,</math>and its volume as<math display="block">\left(\frac{1}{2}\sqrt{1 + 3 \sqrt{\frac{3}{2 + \sqrt{13 + 3 \sqrt{6}\right)a^3\approx1.51535a^3.</math>
Cartesian coordinates
Let
:<math> \begin{align} k &= \frac{6 + \sqrt{6} + 2\sqrt{213-57\sqrt{6}{30} \\ &\approx 0.85273 \end{align} </math>
be the smallest positive root of the quartic polynomial <math> 60x^4 - 48x^3 - 100x^2 + 56x + 23 </math>. Then, Cartesian coordinates of a sphenocorona with edge length 2 are given by the union of the orbits of the points<math display="block"> \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)</math>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.
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See also
- Augmented sphenocorona