Sphenomegacorona

In geometry, the sphenomegacorona is one of the Johnson solids (J_88). It is one of the elementary Johnson solids that do not arise from "cut and paste" manipulations of the Platonic and Archimedean solids. Johnson uses the prefix spheno- to refer to a wedge-like complex formed by two adjacent lunes, a lune being a square with equilateral triangles attached on opposite sides. Likewise, the suffix -megacorona refers to a crownlike complex of 12 triangles, contrasted with the smaller triangular complex that makes the sphenocorona. Joining both complexes together results in the sphenomegacorona. Let k ≈ 0.59463 be the smallest positive root of the polynomial Then, Cartesian coordinates of a sphenomegacorona with edge length 2 are given by the union of the orbits of the points under the action of the group generated by reflections about the xz-plane and the yz-plane. We may then calculate the surface area of a sphenomegacorona of edge length a as and its volume as where the decimal expansion of ξ is given by .