What is this triangle?
If an enormously heavy object has to be moved from one spot to another, it may not be practical to move it on wheels. Instead the object is placed on a flat platform that in turn rests on cylindrical rollers (Figure 1). As the platform is pushed forward, the rollers left behind are picked up and put down in front. An object moved this way over a flat horizontal surface does not bob up and down as it rolls along. The reason is that cylindrical rollers have a circular cross section, and a circle is closed curve "with constant width." What does that mean? If a closed convex curve is placed between two parallel lines and the lines are moved together until they touch the curve, the distance between the parallel lines is the curve's "width" in one direction. Because a circle has the same width in all directions, it can be rotated between two parallel lines without altering the distance between the lines.
Is a circle the only curve with constant width? Actually
there are infinitely many such curves. The simplest noncircular such
curve is named the Reuleaux triangle. Mathematicians knew it earlier
(some references go back to Leonard
Euler in the 18th century), but some curved triangles can be
seen in Leonardo da Vinci's 15th-century Codex Madrid. In addition the
13th-century Notre Dame cathedral in Bruges, Belgium, has several
windows in the clear shape of a Reuleaux triangle (Figure 2). But Franz
Reuleaux was the first to demonstrate its constant-width properties
and the first to use the triangle in mechanisms. See models B01, B02, B03, B04, L01, L02, L03, L04, L05, and L06. A
modern application of the Reuleaux triangle can be seen in the Wankel
engine (Figure 3).
How to construct a Reuleaux triangle
To construct a Reuleaux triangle begin with an equilateral triangle of side s, and then replace each side by a circular arc with the other two original sides as radii (Figure 4).
The corners of a Reuleaux triangle are the sharpest possible on a curve with constant width. Extending each side of an equilateral triangle a uniform distance at each end can round these corners. The resulting curve has a width, in all directions, that is the sum of the same two radii (Figure 5).
Other symmetrical curves with constant width result if you start with a regular pentagon (or any regular polygon with an odd number of sides) and follow similar procedures. This construction is used in the design of some British coins (Figure 6).
This result can be generalized to the case when the polygon is not regular, but each of its vertices is the endpoint of two diagonals of the same length h (and the lengths of other diagonals are less than h). Of course, the polygon has to have an odd number of vertices.
Here is another really surprising method of constructing curves with constant width:
Draw as many straight lines as you like, but all mutually intersecting. Each arc of the curve will be drawn with the compass point at the intersection of the two lines that bound the arc. Start with any arc, then proceed around the curve, connecting each arc to the preceding one. If you do it carefully, the curve will close and will have a constant width. (You can try to prove it! It is not difficult at all.) The curves drawn in this way may have arcs of as many different circles as you wish. Here is one example (Figure 7), but you will really enjoy making your own! After you have made one, you can make more copies of it and check that your wheels really roll!
Another interesting result about curves of constant width is that the inscribed and circumscribed circles of an arbitrary figure of constant width h are concentric and the sum of their radii is equal to h.
Reuleaux Triangle Links
Problems about Reuleaux Triangles and other Curves with Constant Width
The three dimensional analog of a curve with constant width is the solid with constant width. A sphere is not the only such solid that will rotate within a cube, at all times touching all six sides of the cube; all solids of constant width share this property. Rotating a Reuleaux triangle around one of its axes of symmetry generates the simplest example of a nonspherical solid of this type. There are an infinite number of other such examples. The solids with constant width that have the smallest volumes are derived from the regular tetrahedron in somewhat the same way that the Reuleaux triangle is derived from the equilateral triangle: Spherical caps are first placed on each face of the tetrahedron, and then three of tghe edges must be slightly altered. These altered edges may either form a triangle or radiate from one corner. Since all curves with the same constant width have the same perimeter, it might be supposed that all solids width the same constant width have the same surface area. It was proved by Hermann Minkowski that all the shadows of solids with constant width are curves of the same constant width (when the projecting rays are parallel and the shadow falls on a plane perpendicular to the rays). All such shadows have equal perimeters. Michael Goldberg (1957, 1960, 1962) has introduced the term "rotor" for any convex figure that can be rotated inside a polygon or polyhedron while at all times touching every side or face. The Reuleaux triangle is the rotor of least area in a square. The least area rotor for the equilateral triangle is a biangle (a lens shaped figure) formed from two 60-degree arcs of the circle with radius equal to the triangle’s altitude. As the biangle rotates its corners trace the entire boundary of the triangle, with no rounding of corners. See B01.
Closely related to the theory of rotors is a famous problem named the Kakeya Needle Problem, which was first posed in 1917 by the Japanese mathematician Soichi Kakeya: What is the plane figure of least area, in which a line segment of length 1 can be rotated 360 degrees? The rotation obviously can be made inside a circle of unit diameter, but that is far from the smallest area. Ten years after the Kakeya problem was posed, the Russian mathematician Abram Besicovitch showed that there is no minimum area as an answer to Kakeya’s needle problem.
Reuleaux Tetrahedron Links
Kakeya Needle Problem Links
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