Schläfli orthoscheme (original) (raw)
In geometry, a Schläfli orthoscheme is a type of simplex. The orthoscheme is the generalization of the right triangle to simplex figures of any number of dimensions. Orthoschemes are defined by a sequence of edges that are mutually orthogonal. They were introduced by Ludwig Schläfli, who called them orthoschemes and studied their volume in Euclidean, hyperbolic, and spherical geometries. H. S. M. Coxeter later named them after Schläfli. As right triangles provide the basis for trigonometry, orthoschemes form the basis of a trigonometry of n dimensions, as developed by Schoute who called it polygonometry. J.-P. Sydler and Børge Jessen studied orthoschemes extensively in connection with Hilbert's third problem.
| Property | Value |
|---|---|
| dbo:abstract | In geometry, a Schläfli orthoscheme is a type of simplex. The orthoscheme is the generalization of the right triangle to simplex figures of any number of dimensions. Orthoschemes are defined by a sequence of edges that are mutually orthogonal. They were introduced by Ludwig Schläfli, who called them orthoschemes and studied their volume in Euclidean, hyperbolic, and spherical geometries. H. S. M. Coxeter later named them after Schläfli. As right triangles provide the basis for trigonometry, orthoschemes form the basis of a trigonometry of n dimensions, as developed by Schoute who called it polygonometry. J.-P. Sydler and Børge Jessen studied orthoschemes extensively in connection with Hilbert's third problem. Orthoschemes, also called path-simplices in the applied mathematics literature, are a special case of a more general class of simplices studied by Fiedler, and later rediscovered by Coxeter. These simplices are the convex hulls of trees in which all edges are mutually perpendicular. In an orthoscheme, the underlying tree is a path. In three dimensions, an orthoscheme is also called a birectangular tetrahedron (because its path makes two right angles at vertices each having two right angles) or a quadrirectangular tetrahedron (because it contains four right angles). (en) |
| dbo:thumbnail | wiki-commons:Special:FilePath/Triangulated_cube.svg?width=300 |
| dbo:wikiPageID | 20740340 (xsd:integer) |
| dbo:wikiPageLength | 10018 (xsd:nonNegativeInteger) |
| dbo:wikiPageRevisionID | 1116578228 (xsd:integer) |
| dbo:wikiPageWikiLink | dbr:Quadrirectangular_tetrahedron dbr:Midpoint dbr:Birectangular_tetrahedron dbr:Applied_mathematics dbr:Path_graph dbr:Volume dbr:Coxeter dbr:Orthogonal dbr:Pieter_Hendrik_Schoute dbr:Geometry dbr:Miroslav_Fiedler dbr:Convex_hull dbr:Convex_polytope dbr:Order_polytope dbr:Ludwig_Schläfli dbr:Simplex dbr:Børge_Jessen dbr:Tree_(graph_theory) dbr:Trigonometry dbr:Dissection_problem dbr:Coxeter-Dynkin_diagram dbr:3-orthoscheme dbr:4-orthoscheme dbr:5-cell dbr:Acute_angle dbr:Euclidean_space dbr:Goursat_tetrahedron dbr:Hilbert's_third_problem dbr:Hill_tetrahedron dbr:Regular_polytope dbr:Right_angle dbr:Coxeter_group dbr:Tetrahedron dbr:Hyperbolic_geometry dbr:Hypercube dbr:Hyperrectangle dbc:Polytopes dbr:Chiral dbr:Edge_(geometry) dbr:Jean-Pierre_Sydler dbr:Triangle dbr:Dihedral_angle dbr:Dilogarithm dbr:Polygonometry dbr:Circumscribed_sphere dbr:Hugo_Hadwiger dbr:H._S._M._Coxeter dbr:Face_(geometry) dbr:Facet_(geometry) dbr:Lobachevsky_function dbr:Spherical_geometry dbr:Right_triangle dbr:Fundamental_region dbr:Symmetry_(mathematics) dbr:File:Triangulated_cube.svg |
| dbp:wikiPageUsesTemplate | dbt:Blockquote dbt:Main dbt:R dbt:Reflist dbt:Short_description |
| dcterms:subject | dbc:Polytopes |
| rdfs:comment | In geometry, a Schläfli orthoscheme is a type of simplex. The orthoscheme is the generalization of the right triangle to simplex figures of any number of dimensions. Orthoschemes are defined by a sequence of edges that are mutually orthogonal. They were introduced by Ludwig Schläfli, who called them orthoschemes and studied their volume in Euclidean, hyperbolic, and spherical geometries. H. S. M. Coxeter later named them after Schläfli. As right triangles provide the basis for trigonometry, orthoschemes form the basis of a trigonometry of n dimensions, as developed by Schoute who called it polygonometry. J.-P. Sydler and Børge Jessen studied orthoschemes extensively in connection with Hilbert's third problem. (en) |
| rdfs:label | Schläfli orthoscheme (en) |
| owl:sameAs | freebase:Schläfli orthoscheme wikidata:Schläfli orthoscheme dbpedia-he:Schläfli orthoscheme https://global.dbpedia.org/id/4uwzX |
| prov:wasDerivedFrom | wikipedia-en:Schläfli_orthoscheme?oldid=1116578228&ns=0 |
| foaf:depiction | wiki-commons:Special:FilePath/Triangulated_cube.svg |
| foaf:isPrimaryTopicOf | wikipedia-en:Schläfli_orthoscheme |
| is dbo:wikiPageRedirects of | dbr:Schlafli_orthoscheme dbr:Orthoscheme |
| is dbo:wikiPageWikiLink of | dbr:Coxeter–Dynkin_diagram dbr:Generalized_trigonometry dbr:Order_polytope dbr:Ludwig_Schläfli dbr:Simplex dbr:Tetragonal_disphenoid_honeycomb dbr:5-cell dbr:Euler_brick dbr:Hill_tetrahedron dbr:Tetrahedron dbr:Heronian_tetrahedron dbr:Dissection_into_orthoschemes dbr:Simplex_noise dbr:Schlafli_orthoscheme dbr:Orthoscheme |
| is foaf:primaryTopic of | wikipedia-en:Schläfli_orthoscheme |
