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Mathematics

Regular Polytopes in Higher Dimensions and the Euler Characteristic

Quick fact

There are exactly five Platonic solids in 3D, but in four dimensions there are six regular polytopes; however, in every dimension higher than 4, there are only three—the simplex, hypercube, and cross-polytope—and the Euler characteristic plays a key role in proving this.

Why this is interesting

You know a cube has 6 faces, 12 edges, and 8 vertices. But what if I told you that counting these simple parts reveals why only five perfectly regular solids exist, and why the same logic rules out many 'regular' shapes in higher dimensions?