The Mechanism
The trick is a *Reuleaux triangle* — take an equilateral triangle and replace each straight side with a circular arc centered on the opposite corner. The result is a "curve of constant width": measured across any pair of parallel lines touching it, its width is identical in every direction, exactly like a circle. That's why a Reuleaux triangle rolls smoothly under a flat board without the board bobbing up and down — and why manhole covers, coins (the British 20p and 50p are constant-width curves), and rollers can be made in non-circular shapes that still can't fall through or wobble. Crucially, a Reuleaux triangle can spin inside a square while always touching all four sides; if you slide its center along the right looping path, its corners sweep out a shape that is a square except for four tiny rounded corners. English-American engineer Harry James Watts patented a drill bit based on exactly this in 1917 — a bit that, guided by a floating chuck, bores a hole that is square to within a sliver. The surprise is that "constant width" — the property we assume belongs only to the circle — is shared by infinitely many other shapes, and one of them turns rotary motion into a square.
Why It Matters
It seems impossible because drilling usually means spinning a round tool, and round tools make round holes. The Reuleaux triangle breaks that expectation: it is not circular, but it has constant width, so it can touch parallel sides evenly from every direction. That lets it rotate inside a square in a controlled way. The result is not a perfect mathematical square, but a very close one with tiny rounded corners. The same property also explains why some oddly shaped covers or rollers can still stay stable and not wobble or slip through openings.
Wait — That's Not Quite Right
A common mistake is to think only circles have the same width in every direction. In fact, any curve of constant width does this, and there are infinitely many such shapes. Another wrong idea is that a square hole bit must be square-shaped. Watts's drill bit is based on a Reuleaux triangle and works with guided motion, so the tool and the hole are not the same shape.
Vocabulary
- reuleaux triangle
- constant width
- equilateral triangle
- circular arc
- parallel lines
- rolling
- manhole cover
- floating chuck
- drill bit
- patent
- rotary motion
- square hole
Quick Quiz
5 questions · For classroom or kitchen table
The Experiment
Test Constant Width at Home
Find a coin, a small round lid, and a piece of cardboard. Put the coin on the cardboard and trace around it, then do the same with the lid. Notice how both keep the same width when you turn them, because they are circles.
Now make a paper 'constant-width' shape by drawing a triangle and rounding its sides with three curved arcs, or look up a picture of a Reuleaux triangle and print it. Cut it out and gently roll it under a book or ruler. Watch how it can stay the same height while turning, instead of bobbing up and down much like a circle.
If you want a closer comparison, slide each shape between two pencils or two books placed a small distance apart. The round coin and the Reuleaux shape will both fit snugly in every direction, which helps explain how a non-circle can still behave like a circle in some machines.
coin, small round lid, cardboard, pencil, paper, scissors, tape, two books or two pencils, adult supervision for cutting
Where this came from
- Reuleaux triangle — https://en.wikipedia.org/wiki/Reuleaux_triangle ; Watts square-hole drill, US Patent 1,241,175 (1917); Mathematical Etudes, "Drilling square holes" — https://en.etudes.ru/etudes/drilling-square-holes/
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