A Reuleaux triangle is a shape formed from an equilateral triangle by drawing arcs with a radius equal to the length of each side, centred at the opposite vertices. This creates a rounded, curved triangle with three identical curved edges. The shape may be familiar to you as the centre of a three-circle Venn diagram.
Despite its name, it’s not a true triangle. The most interesting property of the Reuleaux triangle is that it has a constant width, meaning that the distance between any two parallel lines that touch its boundary is the same, regardless of orientation. This is a rare trait shared only by circles and other specialised shapes.
And they exist in 3D form as well. The intersection of three spheres forms what’s called a Reuleaux tetrahedron. Sandwich some of these shapes between two flat surfaces and you can slide the top surface back and forward smoothly. It’s odd to watch things that aren’t balls, functioning like balls.
The shape is 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.
However, these shapes were known before his time, for instance by the designers of Gothic church windows, and by Leonardo da Vinci, who used it for a map projection.
Other applications of the Reuleaux triangle include giving the shape to guitar picks, fire hydrant nuts that can only be opened by firefighters who have the special wrench, coins used in vending machines that require the equal width to recognise the denomination, and specialised drill bits for drilling square holes (HERE is a video that shows it) – as well as in graphic design as the shape of some signs and corporate logos.
Also, because its width is constant, a manhole cover shaped like a Reuleaux triangle cannot fall down through the hole. [Ed: So circle, what we’re trying to say is that you’re not so special after all.]
Finally, the Wankel engine is a type of internal combustion engine using an eccentric rotary design to convert pressure into rotating motion. The Wankel engine's rotor, which creates the turning motion, is similar in shape to a Reuleaux triangle, but with the sides having less curvature.
Story Idea: Nils Benson
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References
wikipedia.org/wiki/Reuleaux_triangle
makezine.com/article/craft/fine-art/follow-this-process-to-create-the-perfect-gothic-arch-or-reuleaux-triangle
indiatimes.com/trending/human-interest/triangle-cycle-wheels-new-design-by-square-wheel-inventor
Video: New Reuleaux Triangular Magic by Mathologer on YouTube
Images
1. Reuleaux triangle
2. Construction of a Reuleaux triangle
3. Franz Reuleaux (1829–1905), German mechanical engineer
4. Reuleaux triangle shaped window of the Church of Our Lady, Bruges in Belgium. Photo credit: LEMeZza
5. Leonardo da Vinci's world map in eight Reuleaux-triangle quadrants
6. Coins of constant width. Credit: Mathologer
7. Reuleaux triangular guitar pick. Credit: musicradar.com
8. Hydrant with Reuleaux triangle nut in Philadelphia. Photo: Bryan McClelland
9. Construction of a Reuleaux tetrahedron
10. Book: How Round Is Your Circle?: Where Engineering and Mathematics Meet, John Bryant and Chris Sangwin, 2011
11. Video: Who said wheels have to be round? The Q on YouTube
12. Wankel motor in Deutschen Museum Bonn, 2005. Photo: Ralf Pfeifer
