This is a truncated 4-dimensional cube. You can take an ordinary 3-dimensional cube, cut off its corners, and get a uniform polyhedron with $2 \times 3 = 6$ octagonal faces and $2^3 = 8$ triangular faces. It’s called the truncated cube. Similarly, you can take a 4-dimensional cube, cut off its corners, and get a 4d uniform polytope with $2 \times 4 = 8$ truncated cubes as facets and $2^4 = 16$ tetrahedral facets! It’s called the **truncated 4-cube**.

This particular truncated 4-cube was drawn in a curved style by Jos Leys. You can see more of his 4d polytopes here:

• Jos Leys, 4d Polychora.

A **polychoron** is just another name for a 4-dimensional polytope. The truncated 4-cube is also called the **truncated tesseract**, and you can learn more about it here:

• Truncated tesseract, Wikipedia.

The Coxeter diagram of the truncated 4d cube is

**●—4—●—3—o—3—o**

where the black dots are often drawn as dots with rings around them, and the white ones are often drawn as dots without rings. The unmarked diagram

**o—4—o—3—o—3—o**

describes the symmetry group of the 4-cube, including both rotations and reflections. This group, called a Coxeter group, has four generators $s_1, \dots, s_4$ obeying relations that are encoded in the diagram:

$$ (s_1 s_2)^4 = 1 $$

$$ (s_2 s_3)^3 = 1 $$

$$ (s_3 s_4)^3 = 1 $$

together with relations

$$s_i^2 = 1$$

and

$$ s_i s_j = s_j s_i \; \textrm{ if } \; |i – j| > 1 $$

Marking the Coxeter diagram lets us describe many uniform polytopes with the same symmetry group as the 4-cube. You can think of the 4 dots as corresponding to the vertices, edges, 2d faces and 3d facets of the cube. Blackening the vertex and edge dots:

**●—4—●—3—o—3—o**

is a way to indicate that the truncated 4-cube has a vertex for each **vertex-edge flag**: that is, each pair consisting of a vertex and an edge of the 4-cube, where the vertex lies on the edge.

All this generalizes from 4 dimensions to higher (or lower) dimensions. The **truncated $n$-cube** has $2n$ truncated $(n-1)$-cubes and $2^n$ $(n-1)$-simplices as faces, and it is described by a Coxeter diagram just like the one above, but with $n$ dots. For example, the truncated 5-cube has this diagram:

**●—4—●—3—o—3—o—3—o**

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