Construction
With a compass, sweep an arc sufficient to enclose the desired figure. With radius unchanged, sweep a sufficient arc centred at a point on the first arc to intersect that arc. With the same radius and the centre at that intersection sweep a third arc to intersect the other arcs. The result is a curve of constant width.
Equivalently, given an equilateral triangle T of side length s, take the boundary of the intersection of the disks with radius s centered at the vertices of T.
By the Blaschke–Lebesgue theorem, the Reuleaux triangle has the least area of any curve of given constant width. This area is, where s is the constant width. The existence of Reuleaux polygons shows that diameter measurements alone cannot verify that an object has a circular cross-section.
The area of Reuleaux triangle is smaller than that of the disk of the same width (i.e. diameter); the area of such a disk is .
Read more about this topic: Reuleaux Triangle
Famous quotes containing the word construction:
“The construction of life is at present in the power of facts far more than convictions.”
—Walter Benjamin (1892–1940)
“When the leaders choose to make themselves bidders at an auction of popularity, their talents, in the construction of the state, will be of no service. They will become flatterers instead of legislators; the instruments, not the guides, of the people.”
—Edmund Burke (1729–1797)
“No construction stiff working overtime takes more stress and straining than we did just to stay high.”
—Gus Van Sant, U.S. screenwriter and director, and Dan Yost. Bob Hughes (Matt Dillon)