So why are these the only (regular) dice you'll ever see?

The video explains why only five Platonic solids exist, using both geometric and topological proofs. The geometric explanation involves folding regular polygons to form 3D shapes, while the topological proof uses Euler's characteristic to demonstrate the constraints on polyhedra. The video also discusses why other shapes cannot form additional Platonic solids.

How do the angles at a vertex of a polyhedron affect its structure?

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What is Euler's characteristic and how does it relate to convex polyhedra?

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What pairs of values correspond to the five platonic solids?

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