Tetrakis hexahedron

The tetrakis hexahedron, or tekah, 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 $$\frac{2\sqrt2}{3}$$ times the length of those of the cube. Using an octahedron that is any number less than $$\sqrt2$$ times the edge length of the cube (including if the two have the same edge length) gives a fully symmetric variant of this polyhedron.

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