This is an octahedron.

If you augment each face of an octahedron with more octahedra, you end up with this.

One can then augment each triangular face of this with yet more octahedra.

Here’s the next iteration:

This could, of course, go on forever, but one more step in the series is all you will see here. I don’t want to get caught in an infinite loop.

Performing various manipulations of polyhedra is easy with *Stella 4d: Polyhedron Navigator*, which I used to make all five of these rotating images. If you’d like to try this program for yourself, just check out http://www.software3d.com/Stella.php.

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## About RobertLovesPi

I go by RobertLovesPi on-line, and am interested in many things. The majority of these things are geometrical. Welcome to my little slice of the Internet.
The viewpoints and opinions expressed on this website are my own. They should not be confused with the views of my employer, nor any other organization, nor institution, of any kind.

If you stop augmenting the octahedron at the 2nd step, you can observe the small dihedral angle between adjacent red faces. You cannot do anything with this angle in 3 dimensional space. However, if you go to 4 dimensional space (you figure will be on 3D hyperplane) than you can fold the 8 octahedra in new dimension so that the red faces will touch. After that you can add more octahedra. As result, you get a nice 4 dimensional 24-cell – a regular polytope which has no analog in 3D space – https://en.wikipedia.org/wiki/24-cell.

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