MYSTERIES OF THE EQUILATERAL TRIANGLE - HIKARI Ltd

56 Mathematical Properties
(a) (b)
Figure 2.37: Curves of Constant Width: (a) Sharp Reuleaux Triangle. (b)
Rounded Reuleaux Triangle.
Like the circle, it is a curve of constant breadth, the breadth (width) being
equal to the length of the triangle side [291]. Whereas the circle encloses
the largest area amongst constant-width curves with a fixed width, w, the
Reuleaux triangle encloses the smallest area amongst such curves (Blaschke-
Lebesgue Theorem) [39]. (The enclosed area is equal to (π− √ 3)w 2 /2 [125]; by
Barbier’s theorem [39], the perimeter equals πw) Moreover, a constant-width
curve cannot be more pointed than 120 ◦ , with the Reuleaux triangle being the
only one with a corner of 120 ◦ [249]. Like all curves of constant-width, the
Reuleaux triangle is a rotor for a square, i.e. it can be rotated so as to maintain
contact with the sides of the square, but the center of rotation is not fixed [241].
In the case of the Reuleaux triangle, the square with rounded corners that is
out swept out has area that is approximately equal to .9877 times the area of
the square. (See Application 25.) The corners of the Reuleaux triangle can
be smoothed by extending each side of the equilateral triangle a fixed distance
at each end and then constructing six circular arcs centered at the vertices
as shown in Figure 2.37(b). The constant-width of the resulting smoothed
Reuleaux triangle is equal to the sum of the two radii so employed [125].
Property 57 (Least-Area Rotor). The least-area rotor for an equilateral
triangle is formed from two 60 ◦ circular arcs with radius equal to the altitude
of the triangle [291] (which coincides with the length of the rotor [322]) (Figure
2.38). (The incircle is the greatest-area rotor [39].)
As it rotates, its corners trace the entire boundary of the triangle without
rounding of corners [125], although it must slip as it rolls [291].