Triakis octahedron

The triakis octahedron is one of the 13 Catalan solids. It has 24 isosceles triangles as faces, with 6 order-8 and 8 order-3 vertices. It is the dual of the uniform truncated cube.

It can also be obtained as the convex hull of a cube and an octahedron, where the edges of the octahedron are $$\frac{2+\sqrt2}{2} \approx 1.70711$$ times the length of those of the cube. Using an octahedron that is any number more than $$\sqrt{2} \approx 1.41421$$ times the edge length of the cube gives a fully symmetric variant of this polyhedron. The upper limit is $$\frac{3\sqrt2}{2}$$, where the cube's vertices coincide with the face centers of the octahedron.

Each face of this polyhedron is an isosceles triangle with base side length $$\frac{2+\sqrt2}{2} \approx 1.70711$$ times those of the side edges. These triangles have apex angle $$\arccos\left(\frac{1-2\sqrt2}{4}\right) \approx 117.20057^\circ$$ and base angles $$\arccos\left(\frac{2+\sqrt2}{4}\right) \approx 31.39972^\circ$$.

Vertex coordinates
A triakis octahedron with dual edge length 1 has vertex coordinates given by all permutations of:
 * $$\left(\pm\left(1+\sqrt2\right),\,0,\,0\right),$$
 * $$\left(\pm1,\,\pm1,\,\pm1\right).$$