Square orthobicupola

Square orthobicupola
Type	Johnson J27 – J28 – J29
Faces	8 triangles 2+8 squares
Edges	32
Vertices	16
Vertex configuration	${\displaystyle 8\times (3^{2}\times 4^{2})+8\times (3\times 4^{3})}$
Symmetry group	${\displaystyle D_{4\mathrm {h} }}$
Properties	convex
Net

In geometry, the square orthobicupola is a Johnson solid constructed by two square cupolas base-to-base.

Construction

The square orthobicupola is started by attaching two square cupolae onto their bases. ^[1] The resulting polyhedron consisted of eight equilateral triangles and ten squares, having eighteen faces in total, as well as thirty-two edges and sixteen vertices. A convex polyhedron in which the faces are all regular polygons is a Johnson solid, and the square orthobicupola is one of them, enumerated as twenty-eighth Johnson solid ${\displaystyle J_{28}}$. ^[2] This construction is similar to the next one, the square gyrobicupola, which is twisted one of the cupolae around 45°. ^[1]

Properties

The square orthobicupola has surface area ${\displaystyle A}$ of a total sum of its area's faces, eight equilateral triangles and two squares. Its volume ${\displaystyle V}$ is twice that of the square cupola's volume. With the edge length ${\displaystyle a}$, they are: ^[2] $${\displaystyle {\begin{aligned}A&=\left(2\cdot {\sqrt {3}}+10\right)a^{2}\approx 13.464a^{2},\\V&=\left(2+{\frac {4{\sqrt {2}}}{3}}\right)a^{3}\approx 3.886a^{3}.\end{aligned}}}$$

The square orthobicupola has an axis of symmetry (a line passing through the center of two cupolas at their top) that rotates around one-, two-, and third-fourth of a full turn, and is reflected over the plane so the appearance remains symmetrical. The solid is also symmetrical by reflection over three mutually orthogonal planes. ^[3]

External links

• Weisstein, Eric W., "Square orthobicupola" ("Johnson solid") at MathWorld .