Sphenomegacorona



In geometry, the sphenomegacorona is a Johnson solid with 16 equilateral triangles and 2 squares as its faces.

Properties
The sphenomegacorona was named by in which he used the prefix spheno- referring to a wedge-like complex formed by two adjacent lunes&mdash;a square with equilateral triangles attached on its opposite sides. The suffix -megacorona refers to a crownlike complex of 12 triangles, contrasted with the smaller triangular complex that makes the sphenocorona. By joining both complexes, the resulting polyhedron has 16 equilateral triangles and 2 squares, making 18 faces. All of its faces are regular polygons, categorizing the sphenomegacorona as a Johnson solid&mdash;a convex polyhedron in which all of the faces are regular polygons&mdash;enumerated as the 88th Johnson solid $$ J_{88} $$. It is elementary, meaning it cannot be separated by a plane into two small regular-faced polyhedra.

The surface area of a sphenomegacorona $$ A $$ is the total of polygonal faces' area&mdash;16 equilateral triangles and 2 squares. The volume of a sphenomegacorona is obtained by finding the root of a polynomial, and its decimal expansion&mdash;denoted as $$ \xi $$&mdash;is given by. With edge length $$ a $$, its surface area and volume can be formulated as: $$ \begin{align} A &= \left(2+4\sqrt{3}\right)a^2 &\approx 8.928a^2, \\ V &= \xi a^3 &\approx 1.948a^3. \end{align} $$

Cartesian coordinates
Let $$ k \approx 0.59463 $$ be the smallest positive root of the polynomial $$ 1680 x^{16}- 4800 x^{15} - 3712 x^{14} + 17216 x^{13}+ 1568 x^{12} - 24576 x^{11} + 2464 x^{10} + 17248 x^9 -3384 x^8 - 5584 x^7 + 2000 x^6+ 240 x^5- 776 x^4+ 304 x^3 + 200 x^2 - 56 x -23. $$ Then, Cartesian coordinates of a sphenomegacorona with edge length 2 are given by the union of the orbits of the points $$\begin{align} &\left(0,1,2\sqrt{1-k^2}\right),\,(2k,1,0),\,\left(0,\frac{\sqrt{3-4k^2}}{\sqrt{1-k^2}}+1,\frac{1-2k^2}{\sqrt{1-k^2}}\right), \\ &\left(1,0,-\sqrt{2+4k-4k^2}\right),\,\left(0,\frac{\sqrt{3-4k^2}\left(2k^2-1\right)}{\left(k^2-1\right)\sqrt{1-k^2}}+1,\frac{2k^4-1}{\left(1-k^2\right)^\frac32}\right) \end{align}$$ under the action of the group generated by reflections about the xz-plane and the yz-plane.