Dehn invariant (original) (raw)
In der Mathematik ist die Dehn-Invariante eine Invariante von Polyedern. Zwei dreidimensionale Polyeder sind genau dann zerlegungsgleich (d. h. lassen sich in kongruente Stücke zerlegen), wenn Volumen und Dehn-Invariante übereinstimmen.
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| dbo:abstract | In der Mathematik ist die Dehn-Invariante eine Invariante von Polyedern. Zwei dreidimensionale Polyeder sind genau dann zerlegungsgleich (d. h. lassen sich in kongruente Stücke zerlegen), wenn Volumen und Dehn-Invariante übereinstimmen. (de) In geometry, the Dehn invariant is a value used to determine whether one polyhedron can be cut into pieces and reassembled ("dissected") into another, and whether a polyhedron or its dissections can tile space. It is named after Max Dehn, who used it to solve Hilbert's third problem by proving that not all polyhedra with equal volume could be dissected into each other. Two polyhedra have a dissection into polyhedral pieces that can be reassembled into either one, if and only if their volumes and Dehn invariants are equal.A polyhedron can be cut up and reassembled to tile space if and only if its Dehn invariant is zero, so having Dehn invariant zero is a necessary condition for being a space-filling polyhedron. The Dehn invariant of a self-intersection-free flexible polyhedron is invariant as it flexes. The Dehn invariant is zero for the cube but nonzero for the other Platonic solids, implying that the other solids cannot tile space and that they cannot be dissected into a cube. All of the Archimedean solids have Dehn invariants that are rational combinations of the invariants for the Platonic solids. In particular, the truncated octahedron also tiles space and has Dehn invariant zero like the cube. The Dehn invariants of polyhedra are not numbers. Instead, they are elements of an infinite-dimensional tensor space. This space, viewed as an abelian group, is part of an exact sequence involving group homology. Similar invariants can also be defined for some other dissection puzzles, including the problem of dissecting rectilinear polygons into each other by axis-parallel cuts and translations. (en) En geometría, el invariante de Dehn de un poliedro es un valor que se usa para determinar si los poliedros pueden ser congruentemente diseccionados entre sí o si pueden rellenar el espacio. Lleva el nombre de Max Dehn, quien lo usó para resolver el tercer problema de Hilbert, que versa sobre si todos los poliedros con el mismo volumen son congruentemente diseccionables entre sí. La condición hallada por Dehn toma la forma: Un poliedro se puede cortar y volver a ensamblar para teselar el espacio tridimensional si y solo si su invariante de Dehn es cero, por lo que tener el invariante de Dehn cero es una condición necesaria para ser un poliedro que rellena el espacio. El invariante de Dehn de un poliedro flexible libre de auto-intersección es invariante a medida que se flexiona. El invariante de Dehn es cero para un cubo, pero distinto de cero para los otros sólidos platónicos, lo que implica que los otros sólidos no pueden enlosar el espacio y que no se pueden diseccionar en un cubo. Todos los sólidos arquimedianos tienen invariantes de Dehn que son combinaciones racionales de los invariantes de los sólidos platónicos. En particular, el octaedro truncado también rellena el espacio y tiene un invariante de Dehn cero como el cubo. Los invariantes de Dehn de los poliedros son elementos de un espacio vectorial de dimensión infinita. Como grupo abeliano, este espacio es parte de una sucesión exacta que involucra una . También se pueden definir invariantes similares para algunos otros , incluido el problema de diseccionar un polígono rectilíneo entre sí mediante cortes y traslaciones de ejes paralelos. (es) |
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| rdfs:comment | In der Mathematik ist die Dehn-Invariante eine Invariante von Polyedern. Zwei dreidimensionale Polyeder sind genau dann zerlegungsgleich (d. h. lassen sich in kongruente Stücke zerlegen), wenn Volumen und Dehn-Invariante übereinstimmen. (de) In geometry, the Dehn invariant is a value used to determine whether one polyhedron can be cut into pieces and reassembled ("dissected") into another, and whether a polyhedron or its dissections can tile space. It is named after Max Dehn, who used it to solve Hilbert's third problem by proving that not all polyhedra with equal volume could be dissected into each other. (en) En geometría, el invariante de Dehn de un poliedro es un valor que se usa para determinar si los poliedros pueden ser congruentemente diseccionados entre sí o si pueden rellenar el espacio. Lleva el nombre de Max Dehn, quien lo usó para resolver el tercer problema de Hilbert, que versa sobre si todos los poliedros con el mismo volumen son congruentemente diseccionables entre sí. La condición hallada por Dehn toma la forma: (es) |
| rdfs:label | Dehn-Invariante (de) Invariante de Dehn (es) Dehn invariant (en) |
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