# Circles rolling on circles

### by Yutaka Nishiyama

How many revolutions will the smaller coin make when rolling around the bigger one?

Imagine a circle with radius 1 cm rolling completely along the circumference of a circle with radius 4 cm. How many rotations did the smaller circle make?

The circumference of a circle with radius is , so the circumference of a circle with radius would be . Since

I figured the answer must be four revolutions. So imagine my surprise when I saw that the answer was given to be five!

I read the explanation of why this was indeed the correct answer, and although the reasoning seemed sound, it took some time before I could truly convince myself that my solution was in error. It’s an interesting problem, so I presented it to a number of people, most of whom immediately answered "four" as I did, and, like me, were difficult to convince otherwise; only a very few could intuitively see "five" as the correct answer.

Here’s a better way to think of this problem: rather than rolling along the larger circle, start by imagining the smaller circle as rolling on a line the same length as the circumference of the larger circle. In this case it is straightforward to think of the line as being units long, and so the smaller circle clearly must have made rotations. Next, think of the circle as sliding along the line, without rolling, so that the point on the coin at which it touches the line remains the same. Now consider the difference between sliding along a straight line and doing the same along the circumference of a circle; if you slide units along a straight line, you arrive at your destination just as you started, without having ever changed orientation. But if you do the same along the circumference of a circle you will have made a full rotation at the time you return to your starting point. When *rolling* along the same circumference, therefore, you will have made the four rolling rotations plus the one sliding revolution, for a total of five!

Put differently: when the small circle rolls along the circumference of the larger circle, two kinds of movement simultaneously occur, revolution and rotation. The four movements one initially considers are the four revolutions, perhaps because these are readily seen. Rotation, on the other hand, is more difficult to grasp.

It’s difficult to make progress on this kind of problem just by thinking about it, so it is important to verify the situation through experimentation. For example, you should try modelling this problem using two coins; if the problem followed the predictions of most people, then when using two same-sized coins the moving one would rotate times, but as you will see it does so twice. For example, you might predict that rolling from the top of the fixed coin to its bottom would result in the rolling coin finishing upside-down, but in fact it will have unexpectedly performed a complete rotation by this point. I highly recommend trying this yourself.

If you find it difficult to grasp how things work on a circle, you might also want to imagine what would happen on a square. When a circle rolling along the outer periphery of a square encounters the first corner, it will have to rotate an "extra" 90° to continue along the next side. This will happen again at each corner, and since 90° x 4 = 360°, this accounts for an additional full revolution.

Each of the above explanations describes the circle's movement as a decomposition into rotation and revolution, but in reality no such decomposition is taking place. Just as a human's heart and lungs work simultaneously, rotation and revolution take place together. Separating revolution from rotation is helpful for understanding, but doing so does not provide a fundamental solution. Some say that the makeup of our brains does not allow for multitasking, but learning to simultaneously comprehend such phenomena would be of great value.

*A similar problem appeared in Aha! Gotcha: Paradoxes to Puzzle and Delight by Martin Gardner and also in Scientific American in 1868. If you can think of an alternative proof or explanation for this problem, please post a comment or email us!*

### About the author

Yutaka Nishiyama is a professor at Osaka University of Economics, Japan. After studying mathematics at the University of Kyoto he went on to work for IBM Japan for 14 years. He is interested in the mathematics that occurs in daily life, and has written ten books about the subject. The most recent one is *The Mysterious Number 6174: One of 30 mathematical topics in daily life*, published by Gendai Sugakusha in July 2013 (ISBN978-4-7687-6174-8). You can visit his website here.

## Comments

## Orbital motion

The article reminds me of a surprising fact I learnt recently about the Moon's orbit round the Earth. It keeps the same face towards us all of the time (except for a relatively small effect known as libration) because it does one complete rotation for every revolution. I believe this synchronous orbit is explained as a gravitational effect called tidal locking.

This motion seems to be similar to what's happening when, as described in the article, you slide but don't roll a coin A round the edge of another B so that the same point on A's edge is always in contact with B. Coin A then does one full rotation, but not two.

It's almost as if gravity answers the reverse of the question posed in the article by explaining why we get one less rotation than we should.

Chris G

## Circles Rolling on Circles

In the case of coins/circles of equal radius, the moving coin rotates once with respect to the static coin/circle but twice with respect to the observer.

The important point is that were it different, a driven planet gear of radius ‘r’, would rotate more than once when powered by a drive-cog of radius ‘r’. This would involve a creation of energy.

In the case of a coin/circle of radius ‘r’ moving along a line of length 2πr, the moving coin rotates once with respect to the static coin/circle and also once with respect to the observer.

In the animation at http://www.geogebratube.org/student/m107691 the black arrow is “with respect to the observer” and the red arrow is “with respect to the static circle.”

The mathematical formula Xπr / Yπr = X/Y remains true in all cases: anything else is an illusion.

## David Van Leeuwen's animation

4 or 5? David Van Leeuwen's animation by GeoGebra is a good explanation.

http://www.geogebratube.org/student/m107691

Yutaka Nishiyama

## alternative solution

The CENTRE of the small coin needs to travel around a circle of radius 4+1=5cm. So the CENTRE needs to travel 10pi, as though it is traveling along a straight line of length 10pi while revolving. This gives the solution.

## different rotations

The centre of rotation of a point on the smaller circle is the centre of the smaller circle. Therefore for a full rotation the smaller circle will travel a distance of its circumference around the larger circle.

But the centre of rotation of the centre of the smaller circle is the centre of the larger circle. Hence it rotates around the circle by the larger radius as suggested.

## Ok I get that it's more

Ok I get that it's more complex than it first appears, but what about circles, with circles within them rotating. Do the maths, my head hurts.

http://math.ucr.edu/home/baez/mathematical/tusi_couple_animation.gif

## circles rolling on circles

Could one also consider that the edge of the circle are always curved away from the circle rolling on it, and because of this the circle must roll extra to meet this edge verses a straight line?

## Food for thought

I prefer to think of this problem as a coin rotating around a spot; i.e. the 'big' circle has a radius of zero.

It's easy to see that the 'small' circle will rotate once, even though it's travelled a circumference on 0cm.

Then, if the 'big' circle has any additional circumference (in this case 4cm), the 'small' circle will rotate that once, then travel the additional circumference.

I hope this makes sense!

## Arrived at the same reasoning.

I arrived at your reasoning independently before looking at the comments.

## Another similar way of explaining...

Think of the circle at the end of the 8xpixr line… It rotated four times. Now bend the line to make it a circle. The small circle has to rotate one more time to follow the line.