Skip to content

Mirrors facing each other

Infinity and strange math · level 1, lesson 4 of 6

A question to start

In some lifts, shops and hair salons, there are two mirrors on opposite walls, facing each other. Stand between them and look into one. What do you see?

You see yourself, of course. But behind you, you see the back of your head in the other mirror. And behind that, yourself again, a little smaller. And behind that, the back of your head again, even smaller. A long, long tunnel of you, going back and back.

How many of you are in there? Ten? A hundred? Does the tunnel ever end?

The idea

What a mirror does

A mirror is a very smooth, shiny surface. When light hits it, the light bounces back. That is how you see yourself: light leaves your face, hits the mirror, bounces back, and goes into your eyes. Your eyes think the light came from a person behind the mirror, so you see a copy of yourself there. We call that copy a reflection.

Two mirrors make copies of copies

Now put a second mirror on the wall behind you, facing the first one. Light that bounces off the first mirror goes back across the room and hits the second mirror. And it bounces again! Then it crosses the room again, hits the first mirror again, and bounces again. Back and forth, back and forth.

Each time the light bounces, your eyes see one more copy. Here is a picture from above, looking down at the room. The light is the zigzag line.

the light bounces back and forth between the mirrors mirror mirror

And here is roughly what you see when you look into one of the mirrors: a picture inside a picture inside a picture.

Why the copies get smaller

Each new copy comes from light that has crossed the room one more time. So each copy looks further away than the one before. Things that are far away look small: a bus far down the street looks tiny, even though it is really big. That is why the copies in the mirror tunnel get smaller and smaller.

The first copy of you looks as far away as the other mirror is from you. The next copy looks twice as far. The next, three times as far. And so on. The further away, the smaller it looks.

In a perfect mirror, the copies never end

Now imagine perfect mirrors: perfectly flat, perfectly shiny, bouncing back every bit of light. Then the light would bounce back and forth forever, and every bounce would make another copy.

So in perfect mirrors, the tunnel would never end. After every copy there would be another copy, smaller and further away. It is like the counting numbers from lesson 1: after every copy, one more. And it is like the cookie from lesson 2: the copies get smaller and smaller but never disappear.

Real mirrors are not perfect

When you really stand between two mirrors, you see lots of copies, but not forever. Why not?

  1. Each bounce loses a little light. A mirror sends back most of the light that hits it, but not all. After many bounces, there is so little light left that the far copies look dark. With ordinary glass mirrors, the far copies often also look a little green, because the glass soaks up a tiny bit of the other colours on every trip.
  2. The copies get too small to see. After a while a copy is smaller than your eyes can make out.
  3. You are in the way! Your own head blocks the light coming down the middle of the tunnel. To see deeper, you have to tilt or lean, and then the tunnel curves away to the side.

So the real tunnel fades away. But the idea of the tunnel, with perfect mirrors, goes on forever. This is a pattern you will see again and again in this topic: real things stop, but the math idea behind them can go on without end.

Pictures inside pictures

The mirror tunnel is a picture that contains itself: inside the big picture is a smaller copy of the same big picture, and inside that, a smaller copy again. Look for this in other places:

  • A video call where someone points their camera at their own screen.
  • A cereal box with a picture of a child holding the same cereal box.
  • Nesting dolls: a doll that opens up, with a smaller doll inside, and a smaller doll inside that one.

Nesting dolls are a good test. Do they go on forever? No: when you open them all, the last little doll is solid. It does not open. Real nesting dolls stop. But you can imagine a set that never stops, where every doll has a smaller doll inside.

Worked examples

1. Count the copies. In the picture of the mirror tunnel above, how many frames can you count, including the big one? Count from the outside in: 10 frames. In a perfect mirror there would be more and more, too small to draw.

2. Each copy half as wide. Mia draws a picture inside a picture. The big picture is 32 centimetres wide. Each copy inside is half as wide as the one around it. How wide are the copies?

Halve each time: 32, then 16, then 8, then 4, then 2, then 1. So the copies are 16, 8, 4, 2 and 1 centimetres wide. The next one would be half a centimetre wide, which is very hard to draw with a pencil. Mia's pencil stops her, not the idea.

3. Nesting dolls. A set of nesting dolls has 6 dolls. You open the biggest one. How many dolls are still inside it? There are 6 dolls in all, and you are holding the biggest. So 5 dolls are inside it. The set stops after 6. It is big, but not infinite.

4. Mirrors and counting. Leo stands between two mirrors. He counts 3 copies of himself, then 4, then sees more: 5, 6, 7 ... Could he keep counting forever? With perfect mirrors, yes, there is always one more. With real mirrors, no: the copies get too dark and too small, and Leo has to stop.

Try it

  1. Why do you see yourself in a mirror?
  2. Why do the copies in a mirror tunnel get smaller and smaller?
  3. A picture is 16 centimetres wide. Each copy inside it is half as wide. How wide is the first copy? The second copy?
  4. In the picture in question 3, which copy is 1 centimetre wide? (Count: the first copy, the second copy, ...)
  5. Name two reasons why you cannot see forever into real mirrors.
  6. Nesting dolls: do they go on forever? Explain.
  7. Draw a picture with a copy of itself inside, and a copy inside that. How many copies can you draw before your pencil is too big?
Answers
  1. Light from your face hits the mirror and bounces back into your eyes. Your eyes see it as a copy of you behind the mirror.
  2. Each copy looks further away than the one before, because its light has crossed the room more times. Things that are far away look small.
  3. The first copy is 8 centimetres wide (half of 16). The second copy is 4 centimetres wide (half of 8).
  4. 16, then 8 (first copy), 4 (second copy), 2 (third copy), 1 (fourth copy). The fourth copy is 1 centimetre wide.
  5. Any two of these: each bounce loses a little light, so far copies get dark; the copies get too small to see; your own head gets in the way.
  6. No. The smallest doll is solid and does not open. Real nesting dolls always stop after a number of dolls. You can only imagine a set that never ends.
  7. Everyone's drawing is different. Most people manage 4 or 5 copies before the copies are smaller than a pencil point. The idea could go on forever; the pencil cannot.

Common mistake

Thinking the tunnel in a real mirror really goes on forever. It looks like it, but real mirrors lose a little light at every bounce, and the copies get too small and dark to see. "Forever" lives in the idea of perfect mirrors.

Thinking the copies are the same size. They are copies of the same thing, but they look smaller and smaller, because each one is further away.

Where it leads