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Impossible pictures

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

A question to start

Look at this triangle, made of cubes.

Each corner looks perfectly normal. The cubes look solid, and you could imagine picking them up. But try to build it with real blocks, and you will find that you cannot. Nobody can. It is an impossible object: a picture of a thing that could never exist.

What is wrong with it? Look for a while before you read on. (A hint: look at the orange tops of the cubes.)

The idea

The impossible triangle

The shape above is called the Penrose triangle, or the impossible triangle. A Swedish artist, Oscar Reutersvärd, drew one made of cubes in 1934. In 1958, a British scientist, Lionel Penrose, and his son Roger Penrose, a mathematician, drew it with smooth beams and wrote about it, and that is how it got its name.

Each corner is fine

Cover two corners of the triangle with your hands, so that you can see only one. What you see is an ordinary corner: two straight rows of cubes meeting, like a corner of a picture frame. Nothing is wrong. Now move your hands and look at a different corner: also fine. And the third corner: fine too.

So every part of the picture makes sense. The problem only shows up when you look at the whole thing at once.

Follow the cubes

Let us take a walk round the triangle and keep track of how high we are.

  • Start at the top left cube.
  • Walk along the top side, down to the right corner. Look at the cubes: you can see their orange tops, all in a neat row. They are lying flat, side by side, like blocks on a floor. No climbing.
  • Walk along the lower side, from the right corner to the bottom left. Again, the cubes show their orange tops in a row. Flat again. No climbing.
  • Now walk up the left side. These cubes are stacked one on top of another, like a tower: you cannot see their tops, because each one is covered by the cube above it. You climb up the tower, 4 cubes high.
  • And at the top of the tower, you arrive at ... the top left cube, where you started!

So you walked flat, walked flat again, climbed up 4 cubes, and ended up where you started, without ever coming down. That cannot happen in a real place. If you climb up and never go down, you must end up higher than you started. In the picture, you end up at the same cube. That is why the triangle is impossible.

How can a picture lie?

A picture on paper is flat. It shows left and right, up and down, but it cannot really show near and far. Our eyes have to guess what is near and what is far, using clues like which cube covers which.

The trick of the impossible triangle is that the cube at the top of the tower is really (if it were real) far away from the top left cube, but on the paper the two are drawn in exactly the same place. From one special spot, a near thing and a far thing can look like they touch. Your eyes see them touching and believe it.

Artists have built real sculptures that look like the impossible triangle from one special spot. Walk round to the side, and you see the trick: there is a big gap, or one beam is bent. The picture only works from that one spot.

The never-ending staircase

The Penroses drew a second impossible picture: a staircase that goes round a square and always goes up, yet comes back to where it started. You could climb it forever and never get any higher. It is called the Penrose stairs.

Here is a map of it, seen from above. Each square is one step. Each step is one step higher than the step before it, going round clockwise: 1, 2, 3, ...

1 2 3 4 5 6 7 8 9 10 11 12 each step is one higher than the step before after step 12 comes step 1 again

Walk round: step 1, step 2, up and up, all the way to step 12. Each time you go up. And after step 12 ... you are back on step 1.

Why the staircase cannot be built

Let us count. Each step is one higher than the step before. Going round from step 1 to step 12 is 11 steps up, and from step 12 back onto step 1 is one more step up. That is 12 steps up for one trip round.

So when you get back to step 1, you should be 12 steps higher than when you left it. But it is the same step! A step cannot be 12 steps higher than itself. So no real staircase can be built like this.

In a drawing, the artist hides the problem: he makes the stairs on one side look a little shorter, and uses the trick of near and far, so your eyes cannot tell. Every part looks right, and the whole thing is impossible, just like the triangle.

The Dutch artist M. C. Escher loved these pictures. In 1960 he made a print called Ascending and Descending, with people walking round and round a never-ending staircase on the roof of a building: some going up forever, some going down forever.

Loops that should not close

The staircase and the triangle are both loops, like the circle in Going round and round. A real loop is fine: you go round a race track and come back to the start, at the same height. What makes these loops impossible is that they keep going up and still come back. A real loop that goes up somewhere must come down somewhere else.

This is a bit like the stories in Stories that never end: a story can loop forever because words can do anything. Stairs are made of stone, and stone has to obey the rule "what goes up must come down before you get back".

Worked examples

1. A smaller staircase. A drawing shows 8 steps round a square, each one step higher than the one before, and the 8th leads back to the 1st. How much higher should step 1 be after one trip round?

Going round is 8 steps up (7 from step 1 to step 8, and 1 more back onto step 1). So step 1 would have to be 8 steps higher than itself. Impossible.

2. Fixing the staircase. How could you change the 12-step staircase so that it can be built? Make some steps go down. For example: 6 steps up, then 6 steps down. Then after one trip round you are \(6 - 6 = 0\) steps higher, back at the start. That is just an ordinary hill path.

3. Covering corners. Cover the right corner of the cube triangle with your hand. What do you see? Two straight rows of cubes: one flat row along the top going off to the right, and one tower on the left, joined at the top left. That could be built from real blocks. Only the whole triangle is impossible.

4. Counting the cubes. In the triangle picture, count all the cubes you can see. Count each side, and be careful not to count a corner cube twice. In our picture there are 12 cubes: 5 along the top side (including both corners), 4 more down the lower side (including the bottom left corner), and 3 more in the tower in between.

Try it

  1. What is an impossible object?
  2. In the cube triangle, cover one corner. Is what you can still see possible or impossible?
  3. A drawing shows a staircase of 20 steps round a square, all going up, back to the start. How many steps higher than itself would the first step have to be?
  4. A real path goes up 5 steps, then down 3 steps, then down 2 steps. Is it back at its starting height? Could it be a real loop?
  5. Why can a flat drawing show something impossible?
  6. Change the 12-step staircase so that it is a real staircase, going up first and then down, back to the start.
  7. Escher's people on the staircase walk round and round. Do the people going "up" ever get to the top?
Answers
  1. A picture of a thing that looks solid and real, but could never be built in the real world.
  2. Possible. Each part of the triangle makes sense on its own. Only the whole thing is impossible.
  3. 20 steps higher. Each of the 20 steps going round is one step up.
  4. Up 5, then down 3 and down 2: down \(3 + 2 = 5\) in all. Up 5 and down 5, so it is back at the start height. Yes, it could be a real loop.
  5. A drawing is flat: it cannot really show near and far. A near thing and a far thing can be drawn in the same spot, and our eyes believe they touch.
  6. For example: steps 1 to 7 go up (6 steps up), and steps 7 to 12 and back to 1 go down (6 steps down). \(6 - 6 = 0\), so you get back to the start at the same height. Many other answers are right too, as long as the ups and downs are the same.
  7. No. There is no top. The staircase loops back to where it started, so they climb forever and never get higher. That is what makes it impossible.

Common mistake

Checking only the parts. Each corner of the triangle, and each flight of the staircase, looks right. To find the problem, follow the whole loop and keep track of how high you are.

Thinking a clever builder could make it. The triangle and the staircase are not just hard to build. They break a rule that every real thing obeys: if you only ever go up, you cannot come back to where you started.

Where it leads