Tetrakis hexahedron

The tetrakis hexahedron is one of the 13 Catalan solids. It has 24 isosceles triangles as faces, with 6 order-4 and 8 order-6 vertices. It is the dual of the uniform truncated octahedron.

It can also be obtained as the convex hull of a cube and an octahedron, where the edges of the octahedron are $$\frac{3\sqrt2}{4} ≈ 1.06066$$ times the length of those of the cube. Using an octahedron that is any number less than $$\sqrt2 ≈ 1.41421$$ times the edge length of the cube (including if the two have the same edge length) gives a fully symmetric variant of this polyhedron. The lower limit is $$\frac{\sqrt2}{2}$$ times that of the cube, where the octahedron's vertices will coincide with the cube's face centers.

Each face of this polyhedron is an isosceles triangle with base side length $$\frac43 ≈ 1.33333$$ times those of the side edges. These triangles have apex angle $$\arccos\left(\frac19\right) ≈ 83.62063°$$ and base angles $$\arccos\left(\frac23\right) ≈ 48.18969°$$.

Variations
In addition to its fully-symmetric variants, the tetrakis hexahedron has variants that remain isotopic under tetrahedral symmetry. This variant can be called the disdyakis hexahedron and uses scalene triangles for faces.