Hi!

Having recently gotten a missing piece of insight into 4D visualization I'm revisiting the regular polytopes trying to observe them mentally. I'm fine with the 5-cell, 8-cell and 16-cell, working on the 24-cell.

Anyway, after the 16-cell I was considering the following figure:

- Start with a cube in 3 space. (Say at ±1 on x, y, z and w=0).

- Now add two vertices placed on each side of the w axis, at the center of the cube in xyz.

- Connect the vertices of the cube to form pyramids towards each side of the w axis. So we have square pyramids coming in and out the w axis, meeting in these two "apices".

Now if the vertices on each side of the cube are distant from each other of the same length than the diagonals of the cube faces, these back to back pyramids actually make up regular octahedra (right?).

I'm not sure that description was the best way to convey the figure. Hopefully someone else can see it.

When I observe it mentally I count the following:

- 10 vertices.

- 28 edges.

- 6 cells (all regular octahedra).

I feel this figure to be somewhat special because it's made entirely of regular polyhedra while still being rather simple.

The vertex figure of the vertices at each end are cubes.

The vertex figure of the other vertices are the figure formed by two tetrahedra back to back.

I've reviewed the cubic pyramid and the octahedral pyramid but they only ever go into one direction.

Did I mixup something while imagining this solid?

Is there a family for polytopes made of regular polyhedra but that don't have interchangeable vertices?

Thanks