Cutting Shapes to Make Equations

Source: Quanta, Oct 2019

If you have two flat paper shapes and a pair of scissors, can you cut up one shape and rearrange it as the other? If you can, the shapes are “scissors congruent.”

But, mathematicians wonder, can you tell if two shapes share this relationship even without using scissors? In other words, are there characteristics of each shape you could measure ahead of time to determine whether they’re scissors congruent?

For two-dimensional shapes, the answer is easy: Just determine their areas. If they’re the same, the shapes are scissors congruent.

But for higher-dimensional shapes — like a three-dimensional ball, or an 11-dimensional doughnut that’s impossible to picture — the question of cutting one up and reassembling it as another is a lot harder. Despite centuries of effort, mathematicians have been unable to identify characteristics that dictate scissors congruence for most higher-dimensional shapes.

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