The central region in the resulting arrangement of three circles will be a Reuleaux triangle. Next, one draws a second circle, of the same radius, centered at the other marked point and passing through the first marked point.įinally, one draws a third circle, again of the same radius, with its center at one of the two crossing points of the two previous circles, passing through both marked points. The first step is to mark two arbitrary points of the plane (which will eventually become vertices of the triangle), and use the compass to draw a circle centered at one of the marked points, through the other marked point. The same is true more generally of any compass-and-straightedge construction, but the construction for the Reuleaux triangle is particularly simple. The three-circle construction may be performed with a compass alone, not even needing a straightedge. The Reuleaux triangle may be constructed either directly from three circles, or by rounding the sides of an equilateral triangle.
Alternatively, the surface of revolution of the Reuleaux triangle also has constant width. The Reuleaux triangle can also be generalized into three dimensions in multiple ways: the Reuleaux tetrahedron (the intersection of four balls whose centers lie on a regular tetrahedron) does not have constant width, but can be modified by rounding its edges to form the Meissner tetrahedron, which does. Some of these curves have been used as the shapes of coins. The Reuleaux triangle is the first of a sequence of Reuleaux polygons whose boundaries are curves of constant width formed from regular polygons with an odd number of sides. Because of this property of rotating within a square, the Reuleaux triangle is also sometimes known as the Reuleaux rotor. However, although it covers most of the square in this rotation process, it fails to cover a small fraction of the square's area, near its corners. It can perform a complete rotation within a square while at all times touching all four sides of the square, and has the smallest possible area of shapes with this property. It provides the largest constant-width shape avoiding the points of an integer lattice, and is closely related to the shape of the quadrilateral maximizing the ratio of perimeter to diameter. By several numerical measures it is the farthest from being centrally symmetric.
Other applications of the Reuleaux triangle include giving the shape to guitar picks, fire hydrant nuts, pencils, and drill bits for drilling filleted square holes, as well as in graphic design in the shapes of some signs and corporate logos.Īmong constant-width shapes with a given width, the Reuleaux triangle has the minimum area and the sharpest (smallest) possible angle (120°) at its corners. However, these shapes were known before his time, for instance by the designers of Gothic church windows, by Leonardo da Vinci, who used it for a map projection, and by Leonhard Euler in his study of constant-width shapes. They are named after Franz Reuleaux, a 19th-century German engineer who pioneered the study of machines for translating one type of motion into another, and who used Reuleaux triangles in his designs. Reuleaux triangles have also been called spherical triangles, but that term more properly refers to triangles on the curved surface of a sphere. Because all its diameters are the same, the Reuleaux triangle is one answer to the question "Other than a circle, what shape can a manhole cover be made so that it cannot fall down through the hole?" Constant width means that the separation of every two parallel supporting lines is the same, independent of their orientation. It is formed from the intersection of three circular disks, each having its center on the boundary of the other two. All points on a side are equidistant from the opposite vertex.Ī Reuleaux triangle is a curved triangle with constant width, the simplest and best known curve of constant width other than the circle. The boundary of a Reuleaux triangle is a constant width curve based on an equilateral triangle.
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