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Problem:You Suppose you care about the real world and objects you can hold in your hands. Show that any flexible polyhedron maintains a constant volume while it is flexed. This was known as the Bellows Conjecture. Solution: With a little commutative algebra, you can prove that 12*volume is an algebraic integer in $\mathbb Q$ adjoin the lengths of the sides. Any continuous function from $\mathbb R$ to a countable set is constant. In fact, the volume is a root of a single polynomial. |
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Problem: You care about the real world and objects you can hold in your hands. Show that any flexible polyhedron maintains a constant volume while it is flexed. This was known as the Bellows Conjecture. Solution: With a little commutative algebra, you can prove that 12*volume is an algebraic integer in $\mathbb Q$ adjoin the lengths of the sides. Any continuous function from $\mathbb R$ to a countable set is constant. In fact, the volume is a root of a single polynomial. |
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