This is the 24 cell, a polyhedron in four dimensional space.
The 24 cell is a regular 4-polytope. It can be realized in four dimensional space as the 24 points (a,b,0,0),(a,0,b,0),(a,0,0,b),(0,a,b,0),(0,a,0,b),(0,0,a,b) where a,b are either -1 or 1. The polytope is selfdual. It has also 24 chambers, which is the reason for the name 24 cell.
This is part of a thesis project of Liz Slavkovsky