Geometric Dissection - Leong, Yee Hang (2020)
At the beginning of this project, the dissection of some polygons were studied and analysed. One of them is the solution of Haberdasher’s problem which is a four-pieces dissection from an equilateral triangle to a square given by Henry Dudeney. His original construction idea is applied to construct the dissection from a square to an equilateral triangle. After that, equidecomposability of polygons and polyhedra are discussed. Wallace-Bolyai-Gerwien Theorem states that any polygons with same area are equidecomposable. Two proofs for this theorem are given. A stronger result tells that equidecomposable polygons have a common hinged dissection. Hilbert’s Third Problem asks whether two polyhedra of equal volume are equidecomposable. Max Dehn gave an negative answer to this problem. A recent alternative solution based on Bricard’s condition is studied.