The Koch snowflake, and how its sides and perimeter grow at each step

0:00Here is a straight line, one unit long, and here is a rule that will be applied to it. Cut the line into three equal pieces. On the middle piece, build an equilateral triangle standing outward. Then rub out the middle piece, the one the triangle is now standing on. What is left has a peak in it. Call it a bump. The line has gone. The bump has taken its place.

0:28That is the whole rule, and the rest of this is that rule done again and again, on every straight piece you can see. It is named after von Koch, the Swedish mathematician who first described it. Now count what the bump is made of, because this is where it goes wrong. You cut the line into three, so three pieces. But the middle one has been rubbed out. That leaves two.

0:56And the triangle you raised has two sides of its own, and both of them are part of the boundary now. Two, plus two, is four. Three pieces, minus the one removed, plus the two raised. Four segments where there was one. And every one of the four is a third as long as the line you started with. So the bump measures four thirds of what it replaced. Longer already, at one segment.

1:24Four, not three. Miscount this and every number after it is wrong. Now begin with an equilateral triangle whose sides are one unit, and do it to all three sides at once. Each side becomes a bump, so three peaks stand out from the three sides. What you get is a six pointed star. And here is the second place it goes wrong. That star is not a hexagon. Count its edges. Twelve.

1:56Three sides became twelve, because each one became four. Three of those twelve corners are corners you started with. They have not moved. Do it again, to all twelve, and you have forty eight. Again, and it is one hundred and ninety two. Then seven hundred and sixty eight. Those counts are not a pattern somebody spotted in a list. They come from one side. One side becomes four. That is true of every side, and each side is dealt with on its own, so nothing interferes with anything else.

2:34So whatever the count is, the next count is four times it. Three, twelve, forty eight, one hundred and ninety two. Which means that after n steps the count is three, times four to the n. The three is the triangle you began with. The four to the n is the rule, applied n times. The other half of the story is how long those sides are. Every new segment is one of the thirds, so every new side is a third of the side it came from.

3:08Start at one unit. After one step, a third. After two, a ninth. After three, a twenty seventh. After n steps, the side length is a third to the n. So one number is being multiplied by four while the other is being divided by three, at every single step. More sides, shorter sides. Those pull in opposite directions. The whole question is which of them wins. The perimeter is the count times the length, and that is allowed because every side is the same length as every other one.

3:47Three times one is three. Twelve times a third is four. Forty eight times a ninth is sixteen thirds, about five point three three. One hundred and ninety two times a twenty seventh is sixty four ninths, about seven point one one. Then two hundred and fifty six twenty sevenths, about nine point four eight. In general, three, times four to the n, times a third to the n. Which is three, times four thirds, to the n.

4:20Four on top, three underneath. The four is the sides. The three is the shrinking. There is a second way to reach that four thirds, and it counts nothing at all. Look at one side again. It loses its middle third. It gains two new segments, and each of those is a third as well. So it loses one third and gains two. On balance it has gained a third of itself.

4:48Every side does that, so the whole boundary gains a third of itself. And something plus a third of itself is four thirds of it. Two completely different readings. Count the pieces, or edit the boundary. The same four thirds falls out of both. That agreement is worth more than either route on its own. But say that sentence carelessly and you get a very different picture. It gains a third each step. A third of what?

5:20Hear it as a third of a unit, and the perimeter goes three, three and a third, three and two thirds, four, four and a third. Slow. Nothing much happening. It would take forever to get anywhere. The two readings agree at exactly one step, and after that they come apart and never meet again. By the fifth step the careless reading says four and two thirds. The true one says twelve point six four.

5:49A third of itself is multiplication. A third of a unit is addition. Everything that follows depends on which of those two you are doing. Four thirds is bigger than one, and that single comparison is the whole reason the perimeter has no ceiling. Multiplying by something bigger than one, over and over, gets you past anything you care to name. You only have to be willing to wait. Name ten. Five steps.

6:21Name a hundred. Thirteen steps. Name a thousand. Twenty one steps. Now start with a triangle whose sides are one centimetre, small enough to draw on a fingernail. Thirty steps in, its boundary is roughly a hundred and seventy metres. Not the shape. The shape is still on the fingernail. Only the edge. Which sounds impossible, so let us pin the shape down. Draw the circle through the three corners of the starting triangle.

6:55Every step fits inside it. Not a bigger circle drawn to be safe. That one. The star's points sit exactly on it. Six of them, and three of those six are the triangle's own corners. And nothing later ever gets out. Each new peak rises above the side it stands on by a fixed fraction of that side, and the sides are shrinking by a third every time. Add up every rise that will ever happen, all of them, and the total is less than half of one starting side.

7:31The boundary gets longer by getting more crinkled, not by spreading out. One more thing the rule has to say, and it is easy to leave out. The triangles stand outward. Build them inward instead and you get a different figure, one that bites into itself rather than growing spikes. And here is the uncomfortable part. That figure has the same number of sides, the same side lengths, and the same perimeter, at every step.

8:04The perimeter cannot tell you which of the two you drew. Only three corners out of twelve are different. Those three are the whole of the difference. So a growing boundary is not, on its own, a description of a shape. So why does the boundary run away when the figure does not? Shrink something by a third. Its lengths shrink by a third. Its areas shrink by a ninth, because area lives in two directions at once.

8:35This rule makes four pieces and shrinks each of them by a third. For length that is four times a third. Four thirds, bigger than one, so it grows. For area the same rule would give four times a ninth. Four ninths, smaller than one. Same four. Same third. Opposite answers. Put six such rules side by side and three of them lengthen the boundary while only one enlarges the region, and the two questions disagree about two of the six.

9:10A boundary with no ceiling, drawn inside a circle you could cover with a thumb. Nothing there is a paradox. They were two different questions, and nobody ever promised they would agree.