Fractals in art: temple, textile and print built by repeating the whole at a smaller scale

0:00People were building these long before anyone had a word for them. Perhaps the oldest ones still standing are temple towers, and the word perhaps is doing real work in that sentence. Nobody has checked every building tradition there has ever been. What makes a tower one of these is not the carving on it. It is the shape of the building itself. The tower is made of smaller towers with the same profile, and each of those carries smaller ones again.

0:32A wedding blanket, woven a continent away, does the same thing with diamonds. Diamonds holding diamonds holding diamonds. And a printmaker, much later, made a print in which the same interlocking figures come back smaller and smaller towards the middle. Three materials, three traditions, one idea, and this is about what that idea actually says. Start with the word that causes all the trouble. Repeat. There are two completely different things it can mean, and telling them apart is the whole of this.

1:12You can repeat something beside itself. A row of identical diamonds along a border, each one full size, none of them holding anything. Or you can repeat something inside itself. One diamond, holding smaller diamonds, each of those holding smaller ones again. Both of those are repetition. Only the second one is what we are after. Laid beside, or set inside. Everything from here follows from that. So here is a test you can actually run, instead of an impression you either have or do not.

1:52Ring one part of the thing. Blow that part up until it is the size of the whole. Now look at it, and ask what you are looking at. If it is the whole thing again, it passes. If it is something else, it fails. Ring, magnify, compare. That is the entire test. Run it on the border of diamonds. Ring one diamond, blow it up, and you are looking at a diamond.

2:21But the border was a row, and a diamond is not a row. It fails. Now run it on the nested diamond. Ring one of the small ones, blow it up, and you have a diamond holding smaller diamonds. That is the whole thing again. It passes. The same shape on the page, and opposite answers. Of eight structures put through this test, three of them pass. The point of the other five is that every one of them is something people happily call a fractal, and none of them is.

2:57Take the tower first. Ring one of the smaller spires standing against its side. Blow it up, and its outline is the outline of the whole tower. The same silhouette, the same stepping. So it passes, and what passes is the massing. The shape of the stone itself. That matters, because it rules out the reading almost everyone reaches for first. A plain box with a beautiful pattern carved on it is not this.

3:28There, the pattern is on the surface. Here, the building is the pattern. The blanket is the same test run on thread. Ring one diamond in the woven field. Blow it up, and it turns out not to be a plain diamond at all. It is a diamond holding smaller diamonds. Ring one of those, and you get the same answer again. Which is why the interesting thing about that blanket is not that it has diamonds on it.

4:00It is that the diamonds are made of diamonds. Now the trap, because there is an obvious wrong test and it is the one people reach for. The wrong test is: does it have smaller copies inside it? Take a plain box with small flowers carved into its face. Smaller shapes, set inside a bigger one. It passes that test without any trouble. But ring one flower, blow it up, and you are looking at a flower. Not a box.

4:33Of the eight, five have smaller things inside them, and only three are self-similar. The two tests agree on six of the eight and part company on two, and both of those are carved boxes. Smaller things inside is what decoration looks like. It is not what this is. There is a second wrong test, and it is even easier to reach for. Does it repeat? Six of the eight repeat something. Three are self-similar.

5:05They part company on three of them: the border, a tiling, and the carved box again. A tiling covers a surface with one shape over and over, no gaps and no overlaps, every tile the same size. Magnify a tile and you have a tile. The printmaker's shrinking figures are a different thing entirely, and the difference is that one word. The figures come back smaller, inside. Not beside. Underneath all three is a single instruction, and it is worth saying slowly.

5:42Pick a shape. Pick where the copies go, and how much smaller they are. Then apply that same instruction to every copy you just made. That last clause is the whole thing. Not: put some big copies down, and then some small ones. But: whatever you did to the whole, do to each copy, and then to each copy of each copy. Which is how one instruction produces a shape that nobody drew.

6:11The maker chooses a rule. The drawing is what falls out of it. And here is why that is not word-play. Take a triangle. On the middle of each side put a copy at half size. Then do it again to every copy. After two rounds you have one, plus three, plus nine. Thirteen pieces, in three different sizes. Now imagine somebody else made exactly that picture a different way.

6:41They placed three half-size copies, then nine quarter-size ones, and stopped, because that was all they were ever going to do. Piece for piece and size for size, the two pictures are identical. But run each of them one more round. One has forty pieces in four sizes. The other still has thirteen in three. The difference is twenty seven pieces, all of them at one eighth, and there is nothing in the two-level picture that says whether they are coming.

7:15A drawing of two levels does not settle the third. The rule does. Which raises the obvious question. Why does the real thing always stop? Because every material has a smallest mark it can make. A chisel has a smallest cut, a loom a smallest thread, a press a smallest dot. Say the smallest mark is a tenth of the whole. Copies at half size give you three levels. Copies at a third give you two.

7:48Make the material a hundred times finer and you do not get anything like a hundred times the levels. A mark a hundredth of the whole gives six and four. A thousandth gives nine and six. Levels come slowly, however good the material gets. The mathematics has no smallest mark at all, and that single difference is the whole gap between the model and the thing. So build one. Pick a shape, a triangle. Pick a rule: three copies at half size, one on the middle of each side.

8:25Round one gives three. Round two gives nine. One and three and nine is thirteen. Now change one number. Four copies at a third instead, and the counts run one, five, twenty one, eighty five. Same idea. Different object. And that is the thing worth taking away. What the maker chose was never the drawing. Run this rule ten levels and it has placed over eighty eight thousand pieces, quite happily, long after any chisel would have given up.

9:00A mason, a weaver and a printmaker each ran the same instruction as far as their material allowed, and then stopped. Not because the idea ran out. Because the stone did.