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- Wallace–Bolyai–Gerwien_theorem abstract "In geometry, the Wallace–Bolyai–Gerwien theorem, named after William Wallace, Farkas Bolyai and Paul Gerwien, states that any two simple polygons of equal area are equidecomposable; i.e. one can cut the first into finitely many polygonal pieces and rearrange the pieces to obtain the second polygon."Rearrangement" means that one may apply a translation and a rotation to every polygonal piece. Unlike the generalized solution to Tarski's circle-squaring problem, the axiom of choice is not required for the proof, and the decomposition and reassembly can actually be carried out "physically"; the pieces can, in theory, be cut with scissors from paper and reassembled by hand.The theorem can be understood in two steps. First, every polygon can be cut into triangles: for convex polygons this is immediate, by cutting off each vertex in turn, while for concave polygons this requires more care. Each of these triangles can then be converted to a right triangle, by dropping an altitude (that is, drawing a line perpendicular to the triangle's base and through the top vertex). This is sufficient to easily compute the area, as each right triangle is half a rectangle, or alternatively can be cut halfway up to be reassembled into a rectangle. The second and subtler step is that each right triangle (or equivalently rectangle) can be decomposed into a rectangle with a side of a given (unit) length. Once this is proven, it follows that every polygon can be decomposed into a rectangle with unit width and height equal to its area, which proves the theorem.".
- Wallace–Bolyai–Gerwien_theorem thumbnail Triangledissection.svg?width=300.
- Wallace–Bolyai–Gerwien_theorem wikiPageID "224248".
- Wallace–Bolyai–Gerwien_theorem wikiPageRevisionID "581488474".
- Wallace–Bolyai–Gerwien_theorem subject Category:Euclidean_plane_geometry.
- Wallace–Bolyai–Gerwien_theorem subject Category:Geometric_dissection.
- Wallace–Bolyai–Gerwien_theorem subject Category:Theorems_in_discrete_geometry.
- Wallace–Bolyai–Gerwien_theorem comment "In geometry, the Wallace–Bolyai–Gerwien theorem, named after William Wallace, Farkas Bolyai and Paul Gerwien, states that any two simple polygons of equal area are equidecomposable; i.e. one can cut the first into finitely many polygonal pieces and rearrange the pieces to obtain the second polygon."Rearrangement" means that one may apply a translation and a rotation to every polygonal piece.".
- Wallace–Bolyai–Gerwien_theorem label "Satz von Bolyai-Gerwien".
- Wallace–Bolyai–Gerwien_theorem label "Teorema de Wallace–Bolyai–Gerwien".
- Wallace–Bolyai–Gerwien_theorem label "Teorema di Bolyai-Gerwien".
- Wallace–Bolyai–Gerwien_theorem label "Théorème de Wallace-Bolyai-Gerwien".
- Wallace–Bolyai–Gerwien_theorem label "Wallace–Bolyai–Gerwien theorem".
- Wallace–Bolyai–Gerwien_theorem label "Теорема Бойяи — Гервина".
- Wallace–Bolyai–Gerwien_theorem label "مبرهنة بوياي".
- Wallace–Bolyai–Gerwien_theorem label "ボヤイの定理".
- Wallace–Bolyai–Gerwien_theorem label "華勒斯－波埃伊－格維也納定理".
- Wallace–Bolyai–Gerwien_theorem sameAs Wallace%E2%80%93Bolyai%E2%80%93Gerwien_theorem.
- Wallace–Bolyai–Gerwien_theorem sameAs Satz_von_Bolyai-Gerwien.
- Wallace–Bolyai–Gerwien_theorem sameAs Théorème_de_Wallace-Bolyai-Gerwien.
- Wallace–Bolyai–Gerwien_theorem sameAs Teorema_di_Bolyai-Gerwien.
- Wallace–Bolyai–Gerwien_theorem sameAs ボヤイの定理.
- Wallace–Bolyai–Gerwien_theorem sameAs 보여이-게르빈_정리.
- Wallace–Bolyai–Gerwien_theorem sameAs Teorema_de_Wallace–Bolyai–Gerwien.
- Wallace–Bolyai–Gerwien_theorem sameAs Q834211.
- Wallace–Bolyai–Gerwien_theorem sameAs Q834211.
- Wallace–Bolyai–Gerwien_theorem wasDerivedFrom Wallace–Bolyai–Gerwien_theorem?oldid=581488474.
- Wallace–Bolyai–Gerwien_theorem depiction Triangledissection.svg.
