Sphenocorona
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In geometry, the sphenocorona is a Johnson solid with 12 equilateral triangles and 2 squares as its faces.
PropertiesEdit
The sphenocorona was named by Template:Harvtxt 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.Template:R By joining both complexes together, the resulting polyhedron has 12 equilateral triangles and 2 squares, making 14 faces.Template:R 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>.Template:R It is an elementary polyhedron, meaning it cannot be separated by a plane into two small regular-faced polyhedra.Template:R
The surface area of a sphenocorona with edge length <math> a </math> can be calculated as:Template:R <math display="block"> A=\left(2+3\sqrt{3}\right)a^2\approx7.19615a^2,</math> and its volume as:Template:R <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 coordinatesEdit
Let <math> k \approx 0.85273 </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.Template:R
VariationsEdit
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.
