A surprising overlap

Every so often, my muggle side and mathematical side conflict, and this clip from @marksettle shows one of them.

My muggle side says "wait, what, how can that be?" My mathematician says "aha! neat! Arc lengths!"

The two curved sides of the track are - presumably - arcs of concentric circles with radius $r$. The smaller arc has length $r\theta$, and the longer length $(r+w)\theta)$, where $w$ is the width of the track and $\theta$ the common angle at the centre. The overlap is the difference between them, $w\theta$.

We can estimate $\theta$, fairly roughly: I can imagine two of the pieces of track making a quarter-turn, or possibly three; that puts the angle somewhere between $\frac{\pi}{4}$ and $\frac{\pi}{6}$. The overlap, then, is somewhere between half and three-quarters (roughly) of the width.

Looking at (rather than measuring) the picture, that looks like it may be a slight underestimate: this could be because the two arcs aren't flush against each other - the inner one is a tighter circle. Finding the difference that makes... well, that's a problem for another day.

The surprising overlap is known as the Jastrow illusion.


Colin is a Weymouth maths tutor, author of several Maths For Dummies books and A-level maths guides. He started Flying Colours Maths in 2008. He lives with an espresso pot and nothing to prove.


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I teach in my home in Abbotsbury Road, Weymouth.

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