Fractals: self-similarity generates a GP

0:00Start with a solid triangle. All three sides the same length, filled in completely. Now find the middle of each side, and join those three midpoints with straight lines. The triangle is cut into four smaller ones, and all four are the same size and shape. Three of them sit in the corners. The fourth sits in the middle, upside down. Throw the middle one away. What is left has a triangular hole in it, and that is the entire instruction.

0:33Nothing about it is subtle. The only interesting question is what happens when you keep doing it. So do it again. Every black triangle that survived is a smaller copy of the one you started with, so the same instruction applies to it. Join its midpoints. Cut out its middle. Do that to all three at once and you get the next picture. Do it to all nine of those and you get the one after.

1:04Each round the holes get smaller and there are more of them, and the black breaks into finer and finer specks. The shape has a name. It is the Sierpinski triangle, after the Polish mathematician who described it. Call each round a stage, and call the untouched triangle stage zero. That last choice matters more than it looks, and we will come back to it. Count the black pieces. Stage zero has one. Stage one has three. Stage two has nine. Stage three has twenty-seven.

1:42Carry on and you get eighty-one, then two hundred and forty-three. Divide any of those by the one before it and you get three. Every single time. So the counts form a geometric progression, and the common ratio is three. That is a fact about a list of numbers, and it would be a thin fact if noticing were all we had done. The better question is whether we could have known it before counting anything at all.

2:13We could have. Look at what one round does to one piece. One black triangle goes in. Three black triangles come out. That is the instruction, and it says nothing about which triangle, or how big it is, or where it sits. So whatever the pieces are, and however many of them there are, every one of them becomes three. Nine pieces become twenty-seven. Twenty-seven become eighty-one, and eighty-one become two hundred and forty-three.

2:46Multiplying by three is not something the counts happen to do. It is the instruction, restated. The ratio comes from the rule. Counting only confirms it. You multiply by three once per stage. So the count at stage n is three multiplied by itself n times, which we write as three to the power n. Check that at both ends. Stage zero has had no rounds at all, and three to the power zero is one, which is exactly the single untouched triangle you started with.

3:25Stage five has had five rounds, and three to the power five is two hundred and forty-three. The exponent is not a number you look up. It is the number of rounds you have carried out. Now stop counting pieces and start measuring paper. Call the starting triangle one unit of area. The cut makes four pieces of equal size, so each one is a quarter of the whole. You throw one of them away.

3:55So a quarter of the area goes, and three quarters stay. Be careful here, because two different quarters are easy to confuse. The piece you removed is a quarter of the area. But its sides are not a quarter as long. They are half as long, because you cut at the midpoints. Halving every length quarters the area. Length and area are not the same measurement. Three quarters stay at every stage, and again the rule does not care which piece it is applied to.

4:29So the black area gets multiplied by three quarters each round, in exactly the way the count gets multiplied by three. Stage one keeps three quarters. Stage two keeps three quarters of that, which is nine sixteenths. Stage three is twenty-seven sixty-fourths, just over two fifths of the paper. Stage four is just under a third, and stage five is under a quarter. Same kind of sequence, same kind of argument. A different ratio, and this one is below one.

5:07One instruction has produced two progressions, and they run in opposite directions. Both ratios come out of the same two numbers. The rule makes three copies, and each copy is half the size. Three copies gives you the count ratio, three. Half the size means each copy carries a quarter of the area, so three of them carry three quarters. There is the area ratio. Three is more than one, so the pieces multiply. Three is less than four, so the paper drains away.

5:42And if you kept all four pieces instead of three, the area would sit at one forever. The draining is the throwing away. It is not the cutting. There is a reason the same argument works at every stage. Take one corner of a late picture and magnify it by two. What you get is not merely similar to an earlier stage. It is that stage, triangle for triangle, exactly.

6:12Blow up a corner of stage four and you are holding stage three. Blow up a corner of stage three and you are holding stage two. That is self-similarity. The part, magnified, is the whole. And it is why one fixed replacement rule can govern every scale at once. The area falls at every stage. So where does it end up? It never reaches zero. Every term is three quarters of a positive number, and three quarters of something positive is still positive.

6:49But it gets as small as you please. It drops below a half at stage three. It drops below one hundredth at stage seventeen. And at stage seventeen the black pieces number one hundred and twenty-nine million, one hundred and forty thousand, one hundred and sixty-three. That many separate specks, holding less than one per cent of the paper between them. More pieces has never meant more paper. Counting and measuring are two different questions about the same picture.

7:25One warning now, and it catches almost everybody. We numbered the stages from zero, so the count at stage n is three to the power n. But when you write a rule as a step instead of a formula, you usually seed it at term one. Seed the first term at one, then multiply by three to get each next term. Now term one is the value at stage zero, and term five is the value at stage four, which is eighty-one, not two hundred and forty-three.

7:59So the step rule's index always runs one ahead of the stage it is reporting. Both descriptions are right about the sequence. They disagree about what term one names. So pick one, say which one you picked, and stay inside it. Finally, the method transfers. The numbers do not. Take a square. Cut every side into three, which slices the square into nine equal small ones, and remove the middle one.

8:31Eight survive, and each is a third as wide as the square it came from. Eight copies gives the count ratio, eight. A third as wide means each carries a ninth of the area, so eight of them carry eight ninths. The counts go one, eight, sixty-four, five hundred and twelve. The areas fall by a factor of eight ninths a stage, which is a far gentler fall than three quarters.

9:00The square is still holding more than half its area at stage five. The triangle was under half by stage three. One rule, applied to every piece at every scale. That is all a fractal is, and everything else follows from it.