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Rolling on Wheels That Aren’t Round

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What shape has a constant width, no matter how it’s oriented? While most of you would say a circle (and you’d be right), that’s not the only answer. Other shapes, like the Reuleaux triangle, also fit the mold. At the base, the triangle is shaped like an equilateral triangle, but each side curves on a circular arc.

In the video “Shapes and Solids of Constant Width,” Numberphile’s Steve Mould discusses and demonstrates the use of these shapes in more depth. In an attempt to see what could replace the wheel, he even tries to balance on a wooden board supported by 3-D shapes of constant width that aren’t circles.

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  1. 1. gmcdavid 9:14 pm 12/6/2013

    The Reuleaux triangle was used by the science fiction writer Poul Anderson in his 1963 story _The Three Cornered Wheel_. See

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  2. 2. asozasis 12:49 am 12/7/2013

    I believe the meissner tetrahedron could be turned on a lathe. It would need to be done in four stages, with each stage using a perpendicular axis of rotation through each vertex to the centre of the opposing face. The face through which the axis of rotation passes could then be machined to a spherical section.

    The machined object would be balanced for the first and final stages, but not for the two intermediate stages, though. Question to lathe enthusiasts: is that a deal breaker, or is there a way around the imbalance? Counterweights, maybe?

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