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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
The first mathematician to discover the existence of curves of constant width, and to observe that the Reuleaux triangle has constant width, may have been Leonhard Euler. [5] In a paper that he presented in 1771 and published in 1781 entitled De curvis triangularibus , Euler studied curvilinear triangles as well as the curves of constant width ...
In geometry, a curve of constant width is a simple closed curve in the plane whose width (the distance between parallel supporting lines) is the same in all directions. The shape bounded by a curve of constant width is a body of constant width or an orbiform, the name given to these shapes by Leonhard Euler. [1]
Bonnesen and Fenchel [4] conjectured that Meissner tetrahedra are the minimum-volume three-dimensional shapes of constant width, a conjecture which is still open. [5] In 2011 Anciaux and Guilfoyle [6] proved that the minimizer must consist of pieces of spheres and tubes over curves, which, being true for the Meissner tetrahedra, supports the conjecture.
In particular, the unit sphere has surface area , while the surface of revolution of a Reuleaux triangle with the same constant width has surface area . [ 5 ] Instead, Barbier's theorem generalizes to bodies of constant brightness , three-dimensional convex sets for which every two-dimensional projection has the same area.
All curves of constant width have the same perimeter, the same value πw as the circumference of a circle with that width (this is Barbier's theorem). Therefore, every surface of constant width is also a surface of constant girth: its girth in all directions is the same number πw. Hermann Minkowski proved, conversely, that every convex surface ...
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Surface of constant width; This page was last edited on 9 November 2020, at 07:49 (UTC). Text is available under the Creative Commons Attribution ...