The Cube-octahedron is the most important of the 13 Archimedean Solids due to its being the 3 dimensional Vector Equilibrium.
The 4th dimensional vector equilibrium is one of the most remarkable shapes in any dimension. It is one of the 6 Platonic Solids, the '24 cell', and I call it Metatron's Compass.
This form is one of my studies in creating a 3D Metatron's Cube. If each of the 24 vertexes is connected to all the others, and each one is the center of a sphere that touches its neighbors, then you have the exact analog of Metatron's Cube in 3D