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Picture in a picture

Infinity and strange math · level 2, lesson 5 of 6

A question to start

Look at this picture. It shows a sun, a hill, and a framed picture hanging on the right.

Now look closely at the framed picture. What is in it? A sun, a hill, and ... a framed picture. And in that framed picture? A sun, a hill, and a framed picture again.

How many pictures are hiding in there? If the picture were perfect, would they ever stop?

The idea

A picture that contains itself

The picture above has a copy of itself inside it. The copy is smaller, but it shows exactly the same thing: the sun, the hill and the frame. And because the copy shows the frame, the copy has a copy inside it too. And that copy has a copy, and so on.

This is called a picture in a picture. In level 1 you saw the same thing in mirrors facing each other, and heard it in a story that tells itself. Now you will learn how it works, and how to draw one yourself.

The cocoa tin

More than a hundred years ago, in the early 1900s, a Dutch company called Droste sold cocoa in tins. The picture on the tin showed a nurse carrying a tray. On the tray was a cup of cocoa and a tin of Droste cocoa. And on that tin was the same picture: the nurse, the tray, the tin ... In the Netherlands, a picture that contains a smaller copy of itself came to be called the Droste effect, and the name has spread around the world.

You can find the Droste effect in lots of places once you look:

  • A cereal box with a picture of a child holding the same cereal box.
  • A video call where someone points their camera at their own screen: a screen in a screen in a screen.
  • A "You are here" map in a park. The map shows the whole park, and the park has the map stand in it. So a perfect map would show a tiny map stand, with a tiny map on it, showing the park, with a tinier map stand ...
  • A famous print by the Dutch artist M. C. Escher, called Print Gallery (1956): a young man in a gallery looks at a picture of a town, and the town has the gallery in it, with the young man inside.

Why the copies get smaller

Each copy has to fit inside the spot where it hangs. The spot is smaller than the whole picture, so the copy is smaller than the whole picture.

In our drawing, each copy is half as wide and half as tall as the picture around it. Look at the widths:

Picture How wide (in our drawing)
the big picture 300
copy 1 150
copy 2 75
copy 3 37 and a half
copy 4 about 19
copy 5 about 9

Each copy is half the size of the one before. That is the halving from Halving a cookie again! And just like the cookie, halving never reaches zero. So in a perfect picture, there would always be one more copy, smaller and smaller, forever.

Where do the copies go?

Look at where the copies sit. Each one is in the bottom right of the one before. So the copies march off towards one special spot, a little up and to the left of the bottom right corner, getting tinier and tinier. They squeeze closer and closer to that spot but never fill it up: there is always room for one more.

In the mirror tunnel, the copies all sit in the middle, one inside the next. In the cocoa tin, the copies sit off to one side. Both are pictures in pictures: the copies just march towards a different spot.

Real pictures stop

On paper, the copies stop. Your pencil line has a thickness, and after five or six copies, a copy is smaller than the tip of your pencil. A printer stops at the tiniest dot it can print. A screen stops at its tiniest dot of light.

But the idea, the rule "inside the picture, draw the whole picture again, smaller", never stops. The rule can be followed again and again forever. Real pictures stop; the rule does not.

Drawing your own

Here is how to draw your own picture in a picture.

  1. Choose a picture with a frame in it. For example: a TV in a living room, a poster on a bedroom wall, a book cover with a child reading the book, a window, a phone screen.
  2. Draw the big picture, with the frame empty. Use a ruler for the frame, and make the frame the same shape as the whole page, just smaller. (If your page is wide and short, the frame should be wide and short.)
  3. Inside the frame, draw everything again, smaller: the furniture, the people, and the empty frame in the same place.
  4. Repeat. Inside the new frame, draw everything again, even smaller.
  5. Stop when the frame is too tiny to draw inside. Colour the last tiny frame in, or put a dot in it, to show "and so on, forever".

A tip: keep the drawing simple. A sun, a tree and a frame are much easier to copy five times than a busy room.

Worked examples

1. Half each time. A picture is 16 centimetres wide. Each copy inside is half as wide as the one around it. How wide are the copies, and which copy is the first one narrower than 1 centimetre?

Halve each time: copy 1 is 8 cm, copy 2 is 4 cm, copy 3 is 2 cm, copy 4 is 1 cm, copy 5 is half a centimetre. So copy 5 is the first one narrower than 1 centimetre. Copies 6, 7, 8, ... would be a quarter, an eighth, a sixteenth of a centimetre: too small for a pencil, but still there in the idea.

2. A third each time. A poster is 27 centimetres wide. Each copy is one third as wide. The copies are 9, 3 and 1 centimetres wide: share 27 into 3 equal parts and each part is 9; share 9 into 3 and get 3; share 3 into 3 and get 1. Shrinking by thirds is faster than shrinking by halves, so the copies get tiny sooner.

3. Counting the pictures. In the sun-and-hill drawing at the top of this page, count the frames, including the big one. There are 7 frames. The seventh is so small that you have to squint to see it. In a perfect picture there would be infinitely many.

4. The map in the park. A park map shows the whole park, 400 metres wide, on a board 1 metre wide. So on the map, everything is 400 times smaller. The map stand on the map is 400 times smaller than the real map: 1 metre shared into 400 parts, which is just two and a half millimetres. And on that tiny map is an even tinier one, 400 times smaller again. These copies shrink so fast that the second one is already far too small to see.

Try it

  1. What is the Droste effect?
  2. A picture is 32 centimetres wide. Each copy is half as wide. How wide are the first three copies?
  3. In question 2, which copy is 1 centimetre wide?
  4. Why do the copies in a picture in a picture get smaller?
  5. Why can you not draw infinitely many copies on real paper?
  6. Name two places where you have seen (or could see) a picture in a picture.
  7. Draw your own picture in a picture, with at least three copies. Where do your copies march off to?
Answers
  1. A picture that has a smaller copy of itself inside it, which has a smaller copy inside it, and so on. It is named after the Dutch cocoa tin that showed a picture of the same tin.
  2. Half of 32 is 16, half of 16 is 8, half of 8 is 4. So copies 1, 2 and 3 are 16, 8 and 4 centimetres wide.
  3. Copy 4 is 2 cm, copy 5 is 1 cm. So copy 5.
  4. Each copy has to fit inside the frame of the picture around it, and that frame is smaller than the whole picture.
  5. After a few copies, a copy is smaller than the tip of your pencil (or the smallest dot a printer can print). The rule could go on forever, but real paper and pencils cannot.
  6. Many answers are right: a cereal box, a camera filming its own screen, two mirrors facing each other, a "You are here" map, a book cover showing someone reading that book, the cocoa tin.
  7. Everyone's drawing is different. The copies march off towards the spot where your frame is: if your frame is in the top left, the copies go towards the top left.

Common mistake

Making the frame a different shape from the page. If the page is wide and the frame is tall and thin, the copy will not fit nicely, and the picture looks squashed. Make the frame the same shape as the whole picture, just smaller.

Thinking the copies stop because the idea stops. They stop on paper because of the pencil. The rule "draw the whole picture inside the frame" can always be done once more.

Where it leads