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The extended family I've been sharing digs with... See alt-text for code
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Platonic Solids and Logic
When I first learned there are just 5 platonic solids I was surprised and a bit confused. Why 5? 0 or 1 or $\infty$ all seem like reasonable possibilities, but 5 is just so ... arbitrary. And how on earth can you ever be sure there aren't more that just haven't been found yet? The first proof I saw went like this: If 6 equilateral triangles share a vertex the total angle subtended at the vertex is $60^{\circ} \times 6 = 360^{\circ}$ meaning it's flat (they're in a plane). So platonic solids with equilateral triangles must have 3, 4, or 5 trangles sharing each vertex. That's 3 platonic solids! If 4 squares share a vertex the total angle subtended at the vertex is $90^{\circ} \times 4 = 360^{\circ}$, and likewise that means it's in a plane. So there can only one platonic solid made from squares, namely one with 3 squares sharing each vertex. One to go! Since $108^{\circ} \times 3 < 360^{\circ} < 108^{\circ} \times 4$ the onl...