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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
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]
[60] [61] Placing the antennae on a curve of constant width causes the observatory to have the same spatial resolution in all directions, and provides a circular observation beam. As the most asymmetric curve of constant width, the Reuleaux triangle leads to the most uniform coverage of the plane for the Fourier transform of the signal from the ...
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 ...
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.
The constant width of these shapes allows their use as coins that can be used in coin-operated machines. For instance, the United Kingdom has made 20-pence and 50-pence coins in the shape of a regular Reuleaux heptagon. [5] The Canadian loonie dollar coin uses another regular Reuleaux polygon with 11 sides. [6]
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.
A diagram illustrating great-circle distance (drawn in red) between two points on a sphere, P and Q. Two antipodal points, u and v are also shown. The great-circle distance, orthodromic distance, or spherical distance is the distance between two points on a sphere, measured along the great-circle arc between them. This arc is the shortest path ...