Abstract
The Reuleaux triangle is a figure with the remarkable property of having constant width, a typical property of the circle. It takes its name from Franz Reuleaux, a 19th century German engineer, who studied its properties, in particular the ones related to applications to mechanics. However, this figure was previously known: actually, we find it in the shape of the windows and in the ornaments of some Gothic architecture. Furthermore, Leonardo da Vinci, to represent the terrestrial globe, used eight Reuleaux triangles, each one corresponding to an octant of the spherical surface. Even the mathematician Euler encountered this figure in his study of geometric forms with constant width.
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Notes
- 1.
Given a plane, convex and closed figure, the distance between two parallel straight lines, each of them having at least one intersection point with the border of the figure but none with the interior of the figure, is called width relative to the direction of the straight lines. If the width is the same for every direction, the figure is said to be of constant width.
- 2.
See Bragastini [2].
- 3.
Franz Reuleaux was born on 30 September 1829 in Eschweiler (Germany). To complete his training, he worked from 1844 to 1846 in a foundry and then in a machinery assembly office. Later he enrolled at the Karlsruhe Polytechnic, completing his studies in two years; finally, he attended the Faculty of Philosophy in Berlin. After graduating he taught courses about machine constructions in Bonn. From 1856 to 1864 he was professor of machine design at the Zurich Federal Polytechnic, where he developed many of his ideas on kinematics. From 1864 he was professor at the Gewerbe Akademie in Berlin, later becoming its president. He attended numerous international fairs as a head of delegation. He died on May 20, 1905 in Charlottenburg. More than an inventor, Reuleaux can be defined as a “scientific engineer” and a machine theorist; he is considered the father of modern kinematics (the latter word, coined by Ampère). He criticized German militarism; in fact, after seeing a cannon built by Krupp, he said: “here is a murderer”.
Reuleaux had a certain reputation for his studies; proof of this is the fact that Wittgenstein wanted to enroll himself at the school where Reuleaux had taught in 1906.
His work is vast. His written works include: Konstruktionslehre für den Maschinenbau (1854–62); Theoretische Kinematik (1875); Kurzgefasste Gechichte der Dampfma-schine (1891); Die praktischen Beziehungen der Kinematik zur Geometrie und Mechanik (1900); he also directed the Buch der Erfindungen, Gewerbe und Industrien (vol. 8, Leipzig 1883–1889), translated into Italian with the title Le grandi scoperte e le loro applicazioni (vol. 13, Turin 1886–96). Reuleaux had created in Berlin a collection of over 800 models of mechanisms, many of which by means of his triangle, which were widely used in Europe before the Second World War. Most of them were lost in the destructions of 1941–45 war. The Reuleaux Collection of Kinematic Mechanisms, located at Cornell University, contains a series of 219 models, which are probably the last remaining.
- 4.
To prove this, just use Cauchy’s formula \(L = 1/2\int \nolimits_{0}^{2\pi } B(\theta )d\theta\) where \(B(\theta )\) is the length of the projection of C along a straight line with direction corresponding to the angle \(\theta\).
- 5.
Actually, not all scholars agree on the paternity of this map and even on the type of projection used.
- 6.
See Paris Manuscript A, 15v.
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Conti, G., Paoletti, R. (2020). Reuleaux Triangle in Architecture and Applications. In: Magnaghi-Delfino, P., Mele, G., Norando, T. (eds) Faces of Geometry. From Agnesi to Mirzakhani. Lecture Notes in Networks and Systems, vol 88. Springer, Cham. https://doi.org/10.1007/978-3-030-29796-1_7
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