How a rolling coin draws a heart

2026-05-13

Place a coin on the table. Take a second coin of exactly the same size. Roll the second coin around the first, keeping their edges in contact. Don't let it slip - it has to roll. Now ink a single dot on the rim of the rolling coin and watch what it draws.

It draws a heart.

Press play and let the tan coin do one slow trip around the fixed one. The marker on the rolling rim traces the dashed curve in real time. The shape it leaves behind is a cardioid - literally "heart-shaped". The cusp on the right - the V at the heart's tip - is the instant the marker brushes the fixed coin. The lobe on the left is the marker swinging away.

The hidden trick

Look at the small line on the rolling coin. It's the rolling coin's "12 o'clock" - a fixed mark on the rim, pointing the same way on the coin throughout. Watch how many times that arrow swings around the rolling coin's centre as the coin makes one trip.

Two.

The readout at the bottom counts them. After one trip around the fixed coin, the rolling coin has rotated twice. Most people guess once. Most people are wrong. This is the coin-rotation paradox, and it has tripped up everyone from physics textbook authors to standardised exam writers. (One SAT question in 1982 marked the right answer wrong and had to be thrown out after students complained.)

Where the extra turn comes from

Pretend you are sliding a coin around another coin without rolling - just orbiting, keeping the "12 o'clock" mark always pointing up. That orbit, with no rolling at all, takes the mark through a full 360° rotation in the global frame, simply because the coin has gone all the way around.

Now add rolling on top. The two coins are the same size, so contact arcs are equal - the rolling coin must turn once more to use up its own perimeter against the fixed coin's perimeter. That's another 360°.

One turn for the orbit. One turn for the rolling. Two turns total. The heart shape and the doubled rotation are the same fact in two outfits.

The cardioid family

Change the size of the rolling coin and the heart turns into something else. A rolling coin half as big traces a nephroid - the two-lobed curve you've seen at the bottom of a coffee cup in sunlight. A third the size: a curve with three cusps. A quarter: four. The heart is the special case where the two coins agree on their size.

Roll the coin inside the fixed coin instead and the family changes again. Same size? You get a single point of contact moving back and forth along a diameter - the marker traces a straight line. (Try imagining that without the picture in front of you. Then look at it. The coin is rolling. The marker is moving in a straight line.)

Two identical coins. One rolls around the other. A heart appears. The rolling one spins twice. Both surprises are the same surprise.