Self-similarity: a shape that contains copies of itself

0:00Here is a single frond of a fern. Look at one of its leaves. Not the whole frond - one leaf. Ring it, and blow it up until it fills the frame. What you land on is a frond. Same outline, same arrangement, same everything. Do it again, on one leaf of that one. Same answer. So the frond is not a shape with a decoration repeated over it. It is a shape whose parts are the shape.

0:30That is worth saying carefully, because said loosely it lets almost anything in. The claim is not that a pattern recurs. The claim is this. A part of this shape, magnified, is the whole shape. Part, magnified, whole. Each of those words is doing work. Part, because it has to be a piece of the shape and not something laid on top of it. Magnified, because you are allowed to scale it up as far as you like.

1:00And whole, because what you land on has to be the entire shape - not another part, not the motif on it, the whole thing. A shape that passes that test is called self-similar. Apply it and see what survives. A tree. A trunk carries limbs, a limb carries branches, a branch carries branchlets. Take one limb, magnify it, and you are holding a tree. It passes. A coastline. Zoom into a bay and you find smaller bays inside it, with no branching anywhere. It passes too, and it is the more surprising one.

1:41Clouds, mountains, lightning. All of them. One honesty, though. None of these goes on forever. A fern stops after a few levels, a tree after a handful, a coastline at about the size of a grain of sand. They are examples of the idea, not instances of the mathematics, and the difference between a model and the thing modelled is worth keeping hold of. Now what fails, because that is where a definition earns its keep.

2:14A border of identical diamonds, repeated along a strip. Magnify one diamond and you get a diamond. A diamond is not the strip. It fails. Wallpaper is the same story. Its parts are motifs, and wallpaper's parts are not wallpaper. But a blanket whose diamonds contain smaller diamonds, arranged the same way, does pass - and that is the honest contrast. It also has to be every part. A square decorated over half of itself has one part that is the whole and one part that is not, and it fails, which is why glancing at a single part is no test at all.

3:01Of eleven shapes put through this, six pass and five fail. And the weaker thing people usually mean - that the parts are all alike - agrees with the real test on eight of the eleven and disagrees on three. Close enough to fool you. Not the same test. So build one on purpose. There are two ingredients and only two: a shape to start from, and one rule. Start with a solid square.

3:31Cut it into nine equal squares, three across and three down. Each of those has a side one third of the original, and a third squared is a ninth, so each piece is a ninth of the area. Now delete the middle one. One gone, eight left, and eight ninths of the area still there. That is the entire rule. Cut into nine, throw away the middle. And now the move that turns a rule into a fractal, which is the one people get wrong.

4:05You do not apply the rule to the square again. You apply it to what the rule just produced - to each of the eight squares that survived. Watch what the other reading costs you. Fed the original square each time, the rule finds the same square, cuts it the same way and removes the same middle piece. Run it three times and you still have one hole. Fed its own output, three runs give one hole, then nine, then seventy-three.

4:36Same rule, same three runs. One hole against seventy-three. The difference is not in the rule. It is in what you handed it. It also has to reach every surviving piece, not just one of them. Draw a square, then a smaller square inside it, then a smaller one inside that, forever. That shrinks without stopping, and it is not self-similar - because a part of it is a square, and a square is not a nest of squares.

5:08So a shrinking sequence is not automatically a fractal. What the carpet does that the nest does not is apply the rule to all eight survivors, then to all sixty-four, then to all five hundred and twelve. Every piece, every time, or the structure only ever appears in one corner. What you can actually draw, then, is a sequence. Step zero is the solid square. Step one has a single hole. Step two has nine.

5:42Squares surviving: one, eight, sixty-four, five hundred and twelve. Area left: all of it, then eight ninths, then sixty-four eighty-firsts, then five hundred and twelve seven hundred and twenty-ninths. Holes: none, then one, then nine, then seventy-three. And the holes shrink as they multiply. A third of the side, then a ninth, then a twenty-seventh, then an eighty-first. After step two come three dots, and those dots are a promise.

6:16They promise that this never stops. And that is exactly the thing no drawing can show you. Every drawing has a smallest piece. At step four the smallest hole is one eighty-first of the side. Draw step ten and there is still a smallest one. The shape being described has no smallest piece. That is the whole of its defining property. So every picture you will ever see is a step, and the fractal itself is what the steps are heading towards and never arrive at.

6:51A picture that looks finished is not the finished thing. It is a later step. Which means you can date any of these pictures by looking at it. Count the sizes of hole. Not the holes - the sizes. Step one has one size. Step two has nine holes but only two sizes: one large and eight small. Step three has seventy-three holes and three sizes. The number of distinct sizes is the step number, every time.

7:25And it comes out the same way for the other constructions, so it is a property of the machine and not of this one shape. Count the sizes, get a finite answer, and you are looking at a step. One machine, then, and the famous shapes are settings on it. The carpet keeps eight copies at one third the size. Start from a triangle instead, cut into four and keep three at one half, and you get the gasket: one triangle, then three, then nine, then twenty-seven, covering all the area, then three quarters, then nine sixteenths, then twenty-seven sixty-fourths.

8:07Start from a line, and replace every segment by four segments of a third the length, and you get the snowflake: three sides, then twelve, then forty-eight, then a hundred and ninety-two. And its length grows while the carpet's area shrinks. Three, four, sixteen thirds, sixty-four ninths. Same machine. Different settings. The same questions each time: what do you start with, into how many pieces, and how many survive. One last thing, because the definition on its own is not quite the whole story.

8:46Take a plain filled square. Cut it in four and magnify one quarter, and you get a filled square. It passes. A part of it, magnified, is the whole of it. So self-similarity by itself admits a shape with nothing whatever going on inside it. What the carpet has that the square has not is detail that never runs out - a new size of hole at every scale, without end.

9:14Both ingredients, then: parts that are the whole, and structure at every magnification. And the structure is there at every scale for exactly one reason. The rule was fed its own output.