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- Surface_of_constant_width abstract "In geometry, a surface of constant width is a convex form whose width, measured by the distance between two opposite parallelplanes touching its boundary, is the same regardless of the direction of those two parallel planes. One defines the width of the surface in a given direction to be the perpendicular distance between the parallels perpendicular to that direction. Thus, a surface of constant width is the three-dimensional analogue of a curve of constant width, a two-dimensional shape with a constant distance between pairs of parallel tangent lines.More generally, any compact convex body D has one pair of parallel supporting planes in a given direction. A supporting plane is a plane that intersects the boundary of D but not the interior of D. One defines the width of the body as before. If the width of D is the same in all directions, then one says that the body is of constant width and calls its boundary a surface of constant width, and the body itself is referred to as a spheroform.A sphere, a surface of constant radius and thus diameter, is a surface of constant width. Contrary to common belief the Reuleaux tetrahedron is not a surface of constant width. However, there are two different ways of smoothing subsets of the edges of the Reuleaux tetrahedron to form Meissner tetrahedra, surfaces of constant width which were conjectured by Bonnesen & Fenchel (1934) to have the minimum volume among all shapes with the same constant width; this conjecture remains unsolved. Among all surfaces of revolution with the same constant width, the one with minimum volume is the shape swept out by a Reuleaux triangle rotating about one of its axes of symmetry (Campi, Colesanti & Gronchi 1996); conversely, the one with maximum volume is the sphere.Every parallel projection of a surface of constant width is a curve of constant width. By Barbier's theorem, it follows that every surface of constant width is also a surface of constant girth, where the girth of a shape is the perimeter of one of its parallel projections. Conversely, Hermann Minkowski proved that every surface of constant girth is also a surface of constant width (Hilbert & Cohn-Vossen 1952).".
- Surface_of_constant_width wikiPageExternalLink Spheroforms.html.
- Surface_of_constant_width wikiPageExternalLink books?id=7WY5AAAAQBAJ&pg=PA216.
- Surface_of_constant_width wikiPageExternalLink solids.html.
- Surface_of_constant_width wikiPageExternalLink spheroforms.pdf.
- Surface_of_constant_width wikiPageID "12403587".
- Surface_of_constant_width wikiPageLength "4553".
- Surface_of_constant_width wikiPageOutDegree "20".
- Surface_of_constant_width wikiPageRevisionID "660852703".
- Surface_of_constant_width wikiPageWikiLink Barbiers_theorem.
- Surface_of_constant_width wikiPageWikiLink Brady_Haran.
- Surface_of_constant_width wikiPageWikiLink Category:Euclidean_solid_geometry.
- Surface_of_constant_width wikiPageWikiLink Category:Geometric_shapes.
- Surface_of_constant_width wikiPageWikiLink Compact_space.
- Surface_of_constant_width wikiPageWikiLink Convex_set.
- Surface_of_constant_width wikiPageWikiLink Curve_of_constant_width.
- Surface_of_constant_width wikiPageWikiLink Geometry.
- Surface_of_constant_width wikiPageWikiLink Girth_(geometry).
- Surface_of_constant_width wikiPageWikiLink Hermann_Minkowski.
- Surface_of_constant_width wikiPageWikiLink Parallel_projection.
- Surface_of_constant_width wikiPageWikiLink Perpendicular.
- Surface_of_constant_width wikiPageWikiLink Reuleaux_tetrahedron.
- Surface_of_constant_width wikiPageWikiLink Reuleaux_triangle.
- Surface_of_constant_width wikiPageWikiLink Sphere.
- Surface_of_constant_width wikiPageWikiLink Surface_of_revolution.
- Surface_of_constant_width wikiPageWikiLink Tangent_space.
- Surface_of_constant_width wikiPageWikiLinkText "Surface of constant width".
- Surface_of_constant_width wikiPageWikiLinkText "bodies of constant width".
- Surface_of_constant_width wikiPageWikiLinkText "surface of constant width".
- Surface_of_constant_width wikiPageWikiLinkText "surfaces of constant width".
- Surface_of_constant_width wikiPageUsesTemplate Template:Citation.
- Surface_of_constant_width wikiPageUsesTemplate Template:Cite_web.
- Surface_of_constant_width wikiPageUsesTemplate Template:Geometry-stub.
- Surface_of_constant_width wikiPageUsesTemplate Template:Harv.
- Surface_of_constant_width wikiPageUsesTemplate Template:Harvtxt.
- Surface_of_constant_width wikiPageUsesTemplate Template:Unsolved.
- Surface_of_constant_width subject Category:Euclidean_solid_geometry.
- Surface_of_constant_width subject Category:Geometric_shapes.
- Surface_of_constant_width hypernym Form.
- Surface_of_constant_width comment "In geometry, a surface of constant width is a convex form whose width, measured by the distance between two opposite parallelplanes touching its boundary, is the same regardless of the direction of those two parallel planes. One defines the width of the surface in a given direction to be the perpendicular distance between the parallels perpendicular to that direction.".
- Surface_of_constant_width label "Surface of constant width".
- Surface_of_constant_width sameAs Q1972992.
- Surface_of_constant_width sameAs Cuerpo_de_espesor_constante.
- Surface_of_constant_width sameAs Solide_dxc3xa9paisseur_constante.
- Surface_of_constant_width sameAs m.03by1k4.
- Surface_of_constant_width sameAs Тело_постоянной_ширины.
- Surface_of_constant_width sameAs Q1972992.
- Surface_of_constant_width wasDerivedFrom Surface_of_constant_width?oldid=660852703.
- Surface_of_constant_width isPrimaryTopicOf Surface_of_constant_width.