Arbitrary Curves of Constant Width
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Both the circle and the Reuleaux triangle are examples of curves of constant width. Such curves, if fitted into a square, can rotate in constant contact with all four sides. Any triangle can serve as a template for a curve of constant width by putting three pairs of arcs of circles around it, centered at each of the three vertices, as shown by this Demonstration.[more]
Barbier's theorem  proves that a curve with constant width 1 has a perimeter of π.[less]
Contributed by: Ed Pegg Jr (January 2013)
Open content licensed under CC BY-NC-SA
 Wikipedia, Barbier's theorem.
"Arbitrary Curves of Constant Width"
Wolfram Demonstrations Project
Published: January 15 2013