The cocoa is in the can – the math is on it

Mathematicians gave the name 'Droste effect' to a recursive picture, one that repeats itself forever.

I love math, don't you? Math is all around us, and I like to find it when it's hidden.

Today I'm sipping my hot chocolate and thinking about mirrors. Did you ever go shopping for clothes with Mom when she made you stand in front of three mirrors together? From that angle you can see the back of yourself.

Sometimes, when conditions are just right, you can see yourself multiple times. Try this: Take a mirror and put it next to another mirror. A bathroom is a great place to do this. Set something small between the two mirrors – a stack of five pennies perhaps. Put the mirrors almost at a corner.

Can you see your pennies more than once? How many times can you see them? Now try moving one of the mirrors. Move them a little closer together.

Ah ... more math. The shape that the mirrors make when they meet along their edge is called an angle.

When we move the mirrors closer to each other, we are making a smaller angle. When we move them away from each other, it is considered a larger angle. This is geometry.

See how many times you can see the pennies in the mirror. Do you see more copies of the pennies when the mirrors are closer together or farther apart?

In mathematics we call that "recursion" – when we see something again and again and again. Anything that repeats itself is called recursion.

Sometimes we see this in pictures. I had a book of stories one time. On the cover it showed a woman reading a book. And on the cover of that book was a picture of that very same book – with the woman reading that book again. Wow! How long could that go on?

In mathematics we say it could go on to infinity – forever. (In real life the picture would get so small you couldn't see it anymore – but I like to think mathematically.)

Mathematicians have a name for that effect. They call it the "Droste effect." It's named after a brand of cocoa made in the Netherlands.

The Droste cocoa tin shows a picture of a woman wearing a large white hat. She's carrying a tray with a cup and a cocoa tin on it. What kind of cocoa do you think she is carrying? You guessed it! The cocoa she's carrying is a Droste cocoa can, and it has a picture of the woman on it. And in that picture she also is carrying a tray with a Droste cocoa can on it. And so on and so on.

That's why mathematicians gave the name "Droste effect" to a recursive picture such as that.

So as I take another sip of hot cocoa, I'll think about other times I've seen that kind of picture – one that goes on forever. Maybe you can, too.

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