Reuleaux Triangle in 3D
A Reuleaux triangle has constant width: rolled between two flat plates, it behaves like a circle even though it is not round. This exploration constructs one, proves the constant-width property, and extends the idea to 3D solids of constant width.
Where this idea comes from
Start here — this is the source that inspired this exploration.
Introduction
A shape has constant width if the distance between two parallel supporting lines touching it is the same no matter which direction you measure. Circles are the obvious example, but the Reuleaux triangle, built from three circular arcs centred on the corners of an equilateral triangle, has the same property despite not being round. In 2023, mathematicians found new constant-width shapes in higher dimensions using a construction related to the Reuleaux triangle, resolving a question that had been open for decades. This exploration starts with the 2D Reuleaux triangle, proves its constant-width property, and extends the construction to 3D.
Guiding Questions
- What does it mean for a shape to have constant width, and how can you check it for the Reuleaux triangle?
- How do you construct a Reuleaux triangle with compass and ruler, and what is its area compared with a circle of the same width?
- Why do these shapes roll smoothly between two flat plates but make terrible wheels?
- Could you extend the idea to 3D solids of constant width and calculate or estimate their volume?