A wheel that is not round
Tonight's idea
A circle is the same width whichever way you measure it. That is why a plank on round rollers rides level. But a circle is not the only shape that can do this.
Draw a triangle with three equal sides of length \(w\). With a compass point on one corner, draw the arc joining the other two. Repeat from each corner. The bulging shape is a Reuleaux triangle (say "ruh-low").
Stand it between two parallel lines. One touches a corner, the other the arc opposite. Every point of that arc is exactly \(w\) from the corner, because the corner is the arc's centre. So the gap is \(w\) however the shape is turned: a shape of constant width. It rolls between the lines, and it turns inside a square hole of side \(w\), always touching all four sides.
Some coins use the same idea with seven curved sides. The British 20p and 50p are like this, so a machine measures the same width whichever way round the coin arrives.
Watch one: how far round is it?
Take \(w = 10\) cm. Each corner of an equal-sided triangle is 60°, and 60° is one sixth of a full turn. So each arc is one sixth of a circle of radius 10.
A whole circle of radius 10 has circumference \(2 \times \pi \times 10 \approx 62.8\) cm. One arc is \(62.8 \div 6 \approx 10.5\) cm. Three arcs make about 31.4 cm.
That is \(\pi \times 10\): exactly the circumference of a circle 10 cm wide. Same width, same distance round.
Perimeter of a Reuleaux triangle of width \(w\): \(3 \times \dfrac{2 \pi w}{6} = \pi w\).
Your turn
- A Reuleaux triangle is 6 cm wide. (a) How long is one of its arcs? (b) How far is it round the whole shape? (c) A different one has perimeter 22 cm. How wide is it, to the nearest centimetre?
- A square lid can fall down its own hole if you turn it on edge along the hole's diagonal. Why can a round lid never fall down a round hole slightly smaller than itself? Is a Reuleaux lid safe too?
- A curved five-sided shape of constant width \(w\) has five equal arcs. Each arc has radius \(w\) and covers 36° of a circle. (a) What fraction of a full circle is one arc? (b) Show that the distance round this shape is \(\pi w\) as well.
- The centre of a Reuleaux triangle 10 cm wide is about 5.8 cm from each corner. (a) How far is the centre from the middle of each arc? (b) Rolling on the ground, how high is the centre when a corner is straight below it? When the middle of an arc is on the ground? (c) Why is this a bad bicycle wheel but a good roller under a plank?
Stretch: seven sides
A coin has seven equal curved sides and constant width 27 mm. Each side is an arc of radius 27 mm, and the distance round the coin is \(\pi \times 27\) mm. What angle does each arc cover? Check your method on the triangle (60°) and the five-sided shape (36°).
Answers
- (a) One sixth of \(2 \times \pi \times 6\), which is \(2\pi \approx 6.3\) cm. (b) \(\pi \times 6 \approx 18.8\) cm. (c) \(22 \div \pi \approx 7.0\), so 7 cm.
- A square is wider corner to corner than side to side, so the lid, turned on edge, slips through the long diagonal of the hole. A circle is the same width in every direction, and the hole is narrower than that in every direction, so there is no way through. The Reuleaux lid is safe for the same reason: its width never drops below \(w\).
- (a) \(36/360 = 1/10\). (b) One arc is a tenth of \(2\pi w\). Five arcs make \(5 \times \dfrac{2\pi w}{10} = \pi w\).
- (a) From a corner, through the centre, to the middle of the opposite arc is 10 cm. So the centre is \(10 - 5.8 = 4.2\) cm from the arc. (b) 5.8 cm, then 4.2 cm. (c) An axle through the centre would bob up and down by about 1.6 cm three times in every turn. A plank does not care where the centre is. It rests on the top of the roller, and the top is always 10 cm above the ground.
Stretch. A full circle of radius 27 has circumference \(2 \times \pi \times 27\). The coin's perimeter is half of that, so the seven arcs together cover half a turn, 180°. Each one covers \(180 \div 7 \approx 25.7°\). Check: \(180 \div 3 = 60\) and \(180 \div 5 = 36\).