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|The FastGeometry™ logo contains a
shape known as Reuleaux's triangle. This is a constant width
shape with the following interesting property:
A better-known 2D-shape of constant width is the circle, with its diameter being also its width.
A lesser known constant width shape is Reuleaux's triangle. The simplest form of this interesting shape, shown above, can be constructed by starting with an equilateral triangle and then drawing arcs with radii equal to the sides, centered at the vertices and just outside the opposite edge, intersecting and terminating at the two ends of the edge, i.e. the other two vertices. The resulting triangle can be contained inside a square.
Another interpretation of Reuleaux's triangle, at least the basic form above, is the intersection of 3 circles, which is highlighted below.
An alternative, more general form of Reuleaux's triangle is to construct two arcs centered at each vertex, with the new arc typically located opposite the triangle with respect to the vertex. The result, shown below, is a more rounded shape with a constant width equal to the sum of the two radii.
Reuleaux's triangle has not found many engineering applications, most likely because as it does not have a fixed centre as it rotates within a band or square equivalent to its width. One application is the Wankel rotary engine found in the now discontinued Mazda RX7 sports car, wherein the rotor is in the shape of a Reuleaux's triangle. It can be argued, however, this design does not directly exploit the constant width property.
The triangle is not the only polygon which can be modified into a constant-width shape. This is true of all regular convex polygons with an odd number of sides. The following figure illustrates Reuleaux's pentagon. In this case, the radii of the arcs, or rather the width of the shape, are equal to the distance from any given vertex to the farthest of the other vertices. Once again, a more rounded version can be made by using two arcs centered at each vertex instead of one.