Sphenocorona Explained

Type:	Johnson
Faces:	12 triangles 2 squares
Edges:	22
Vertices:	10
Dual:	-
Properties:	convex
Net:	Johnson solid 86 net.png

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

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:$$A=\backslash left(2+3\backslash sqrt\backslash right)a^2\backslash approx7.19615a^2,$$and its volume as:$$\backslash left(\backslash frac\backslash sqrt\backslash right)a^3\backslash approx1.51535a^3.$$

Cartesian coordinates

Let

k ≈ 0 . 85273 be the smallest positive root of the quartic polynomial 60 x ^{4} - 48 x ^{3} - 100 x ^{2} + 56 x + 23 . Then, Cartesian coordinates of a sphenocorona with edge length 2 are given by the union of the orbits of the points$$\backslash left(0,1,2\backslash sqrt\backslash right),\backslash ,(2k,1,0),\backslash left(0,1+\backslash frac,\backslash frac\backslash right),\backslash ,\backslash left(1,0,-\backslash sqrt\backslash right)$$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.

See also

• Augmented sphenocorona