Not a circle, but rolls like one — and can drill square holes!
What shape, other than a circle, has constant width? Most people guess "none" — but they're wrong!
The Reuleaux triangle is the same width in every direction, rolls smoothly between parallel plates, and can even drill nearly-square holes. It's why British 50p coins work in vending machines despite having 7 sides!
The shape stays in contact with both rails at all times!
Watch the Reuleaux triangle trace a nearly-square path!
The corners are slightly rounded, but the sides are perfectly straight!
Uncovered area: only ~2% of the square!
Width = Diameter everywhere
The "obvious" constant width shape
Width = Side length everywhere
The "surprising" constant width shape!
Each arc has radius s, centered at the opposite vertex. When you measure between any two parallel tangent lines, one line touches an arc while the other touches the opposite vertex — which is exactly distance s away (the arc's radius)!
These coins are Reuleaux heptagons (7 curved sides) — not regular heptagons! Their constant width means vending machines can measure their diameter at any angle and get the same value.
The 50p was introduced in 1969, making it the world's first seven-sided coin. The design was created by Hugh Conway, who also worked on the Concorde!
Patented by Harry Watts in 1914. A Reuleaux triangle drill creates holes with straight sides and only slightly rounded corners (~2% uncovered).
The rotor in a Wankel rotary engine is a Reuleaux triangle. Used in Mazda RX-7 and RX-8 sports cars.
Like circles, Reuleaux shapes can't fall through their own holes. They're used where non-circular aesthetics are desired.
Some pencils have Reuleaux triangle cross-sections. They roll less than round pencils and encourage proper grip.
The shape provides three sharp points for articulation while maintaining comfortable grip from any angle.
Medieval architects used Reuleaux triangles in church windows long before Reuleaux studied them mathematically!
The Reuleaux triangle isn't the only curve of constant width!
Among all curves of constant width w, the Reuleaux triangle has the smallest area. The circle has the largest. This was proven in 1915 and shows the Reuleaux triangle is "maximally non-circular" while maintaining constant width!