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BGW - Math402 Bolyai—Gerwien—Wallace Theorem Both in...

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Unformatted text preview: Math402 Bolyai—Gerwien—Wallace Theorem Both in Euclidean and in Hyperbolic Geometry we can assign a positive real number to each polygon so that these numbers add under non—overlapping unions of polygons. In Euclidean geometry the number is the area, in Hyperbolic geometry it is the defect. The following works in both situations. Definition. Two polygons are scissor—equivalent (R1 ~ R2) if one can be cut into finitely—many polygonal pieces and these pieces can be rearranged into the other polygon. Here are two examples of scissor—equivalences: Triangle — Square Square ~ 21—gon (4 pieces) I (14 pieces) i /A\ / \\ / \\ / \ \\ .\ /\ Theorem. Scissor-equivalence is an equivalence relation. In particular it is transitive. Since area/ defect is additive, equivalent polygons have equal areas. The converse is a non—trivial theorem. Bolyai—Gerwien—Wallace Theorem. If two polygons have the same area/ defect then they are scissor-equivalent. Proofs. For a pair of hyperbolic triangles this is essentially Theorem 7.20 from the book (the same proof works in Euclidean Geometry). BGW Theorem for a pair of general polygons has been proved in class by induction on the number of triangles in a triangular decomposition of the two polygons. In Euclidean Geometry there is a nicer proof via rectangles essentially done in Problem Set 9. All the proofs are constructive, 226. each describes an algorithm for cutting polygons into pieces and rearranging them (although these algorithms are not very effective — the number of pieces is much greater than actually required). 3d-analog of BGW Theorem is false: it is impossible to saw a regular tetrahedron with volume one into pieces and rearrange these pieces into the unit cube (Dehn’s negative answer to the Third Hilbert’s problem). So in 3d volume is not the only invariant of sawing; there is another number additive under sawing of polyhedra—Dehn’s invariant. ...
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