 Cross polytope

The convex hull of the positive and negative coordinate unit vectors. Its boundary triangulates a sphere, and each face contains at most one member of each antipodal pair of coordinate unit vectors. This makes its barycentric subdivision useful for triangulating Real projective space.

Examples: square, octahedron.

Take $(0,0,0,\dots ,0)$ and flip one of 0's to $\pm 1$. Therefore has $2\times D$ vertices.

Each edge E is linked to every other edge, except it's opposite -E.