Wallace–Bolyai–Gerwien theorem

In geometry, the Wallace–Bolyai–Gerwien theorem, named after William Wallace, Farkas Bolyai and Paul Gerwien, is a theorem related to dissections of polygons. It answers the question when one polygon can be formed from another by cutting it into a finite number of pieces and recomposing these by translations and rotations. The Wallace-Bolyai-Gerwien theorem states that this can be done if and only if two polygons have the same area. σ ( P , Q ) ≥ d ( P ) d ( Q ) . {displaystyle sigma (P,Q)geq {frac {d(P)}{d(Q)}}.} σ ( P x , Q ) ≤ 2 + ⌈ x 2 − 1 ⌉ , for  x ≥ 1 {displaystyle sigma (P_{x},Q)leq 2+leftlceil {sqrt {x^{2}-1}} ight ceil ,quad { ext{for }}xgeq 1} σ ( P 1 x , Q ) ≤ 2 + ⌈ 1 − x 2 x ⌉ , for  x ≤ 1 {displaystyle sigma left(P_{frac {1}{x}},Q ight)leq 2+leftlceil {frac {sqrt {1-x^{2}}}{x}} ight ceil ,quad { ext{for }}xleq 1} In geometry, the Wallace–Bolyai–Gerwien theorem, named after William Wallace, Farkas Bolyai and Paul Gerwien, is a theorem related to dissections of polygons. It answers the question when one polygon can be formed from another by cutting it into a finite number of pieces and recomposing these by translations and rotations. The Wallace-Bolyai-Gerwien theorem states that this can be done if and only if two polygons have the same area. Farkas Bolyai first formulated the question. Gerwien proved the theorem in 1833, but in fact Wallace had proven the same result already in 1807.