24-cell
Yes, footballnutt got the right object. Those are the names for it plus it's also called the 24-cell because it has 24 octahedral cells. 3d space doesn't have enough room to fit together all of those 24 without distorting them, hence the projections used in the pictures.
Looking at a projection of the 4d object into 3d space, like in the first picture is looking at the 3d shadow of the 4d object. It's like looking at the 2d shadow of a 3d object. The other picture with red squares is a projection into 2d space.
Those red squares are the 24 points connected by 96 edges, which make up 96 triangular faces that are put together into 24 octahedral cells.
The 24-cell is the special polytope that is self-dual while not being a polygon or a
simplex. There is no others like that in any other dimension of Euclidean space. A polytope that is
dual to another one has vertices that correspond to the faces of the other one and vice versa.
For example the cube and octahedron are duals of each other. The 6 points of the octahedron correspond to the 6 faces of the cube and the 8 corners of the cube correspond to the 8 triangles of the octahedron. The tetrahedron is the dual of itself, being the simplex in 3d space, as well as the regular triangle, which is the 2d simplex, plus all other regular polygons. The 24-cell is self-dual, yet isn't a polygon or simplex, making it special.
I used the same program used to make some of those pictures to make the picture of the bunch of 24 red octahedra I posted. Fold that in 4d and you get the 24-cell.
Here is the tiling of the 24-cell across 4d space I also referred to. Many copies of the 24-cell put together can fill 4d space like many copies of the cube can fill 3d space.
4d space is said to be the richest one because there's so many forms with 64 convex uniform ones outside of infinite series, as opposed to the 18 we have besides prisms in 3d. Including nonconvex uniform ones brings us to 75 (plus one special one) in 3d and well over 1000 in 4d. Then for some reason, spaces of five and higher dimensions have only 3 convex regular polytopes each, rather than 5 and 6 for 3d and 4d space.
And that's just flat Euclidean space. There's more in curved space, both positively and negatively curved.
Whew I tried to explain all of the clues I used.
Those special properties, especially the self-dual one, was why I picked the 24-cell. I thought the 4d analogy of the cube would be too easy. This was inspired by jenni-m's posting of the dodecahedron picture, which I recognized instantly.
For all of your trouble, I give you a link to a calendar printed onto a dodecahedron, one month on each side. Enjoy!
12 sided calendar
You are next footballnutt!