The Sierpinski carpet and gasket: what repeated removal leaves behind

0:00Two constructions, side by side, and almost everything about them is the same. The carpet starts with a square. Cut it into nine, throw away the middle one, and do the same to each of the eight that survive. The gasket starts with an equilateral triangle. Join the midpoints of its three sides, throw away the middle piece, and do the same to each of the three that survive. Same machine. What changes is the shape you begin with, how many pieces the cut makes, and how many of them you keep.

0:37Nine and eight for one. Four and three for the other. Everything in this video follows from those two pairs of numbers. But the gasket's rule has something hidden in it that the carpet's does not. Cut a square into a three by three grid and you can see the nine pieces are equal. Join three midpoints of a triangle and you have to argue it. The claim is that those three segments cut the triangle into four identical equilateral triangles.

1:12Not four triangles. Four identical ones, and every one of them equilateral. And the piece that takes work is the middle one, because it is the only one that does not inherit a corner from the triangle it came out of. Here is the argument, and the way in is the corners. Take one corner triangle. Two of its sides are half-sides of the parent, so they are equal to each other, and the angle between them is the parent's own sixty degrees.

1:46Two equal sides means the two base angles are equal. They have a hundred and twenty degrees to share between them, so each one is sixty. All three angles sixty. So the corner triangle is equilateral, with side half the parent's. Now the middle one. Every joining segment is a side of some corner triangle, so all three of them are half the parent's side. Three equal sides. The middle triangle is equilateral too, and exactly the same size. Four identical pieces, each a quarter of the area.

2:26And one thing worth noticing: the four-identical-pieces part is true of any triangle at all. It is the equilateral part that needs the parent to be equilateral. Now count. Write R for the number of surviving squares, and R at step n for how many there are after n steps. One square at step zero. Take any single survivor and apply the rule to just that one. It is cut into nine, it loses one, and it leaves eight.

3:01And that happens to every survivor, independently of what any other one is doing. So the count at the next step is eight times the count at this one. That is not a pattern spotted in a table. It is one sentence about one square. From that rule to a formula. R at step zero is one. R at step one is eight times one, which is eight. R at step two is eight times eight. R at step three is eight times that again.

3:36Multiplying by eight exactly n times, starting from one, is eight to the n. So R at step n is eight to the n - and the reason is the repeated multiplication, not the fact that one, eight and sixty-four happen to look like powers of eight. Holes next, and this one needs two observations rather than one. The first: each surviving square gets one new hole punched in it at the next step. One apiece, no more and no fewer.

4:11The second: a hole that is already there stays there. Nothing ever heals. So the holes at the next step are the holes you already had, plus one for every survivor. Not plus one for every hole. The number you add is the survivor count, and confusing those two is the commonest slip in the whole topic. Run it out. No holes at all at step zero. At step one, one hole. At step two, one plus eight, which is nine.

4:44At step three, one plus eight plus sixty-four. Seventy-three. Every term added is a power of eight, and the running totals are one, nine, seventy-three, five hundred and eighty-five. That sum has a formula of its own: eight to the n, less one, over seven. At step three that is five hundred and eleven over seven, which is seventy-three. It agrees. The two wrong readings are worth taking seriously, because both of them can be answered rather than just warned against.

5:21Read it as a new hole inside every hole. You start with no holes, so you add no holes, and the figure never gets a single one at any step. Nothing at all happens. Read it as though old holes healed, and count only what this step made: one, then eight, then sixty-four. That is right at step one and wrong at every step after it. Against six steps it agrees twice and disagrees four times.

5:52The recursion adds survivors. Every other plausible thing it could add gives you a different figure. Now the gasket, and there is almost nothing new to say, which is the point. One triangle becomes three, so the survivor count multiplies by three each step: one, three, nine, twenty-seven. Each survivor punches one hole and no hole heals, so the holes go one, then one plus three, then one plus three plus nine. One, four, thirteen.

6:28And the formula is three to the n, less one, over two. At step three that is twenty-six over two, which is thirteen. The same two sentences did all of that. Eight becomes three, and everything else follows unchanged. Area now. Take the shape you start with as one unit of area. Each step keeps eight of the nine equal pieces of every surviving square, so whatever area is left gets multiplied by eight ninths.

7:01After n steps that is eight ninths to the n. One, then eight ninths, then sixty-four eighty-firsts, then five hundred and twelve seven hundred and twenty-ninths. For the gasket the multiplier is three quarters: one, three quarters, nine sixteenths, twenty-seven sixty-fourths. And this is not a second result. Eight to the n pieces, each one a ninth of a ninth of a ninth, is eight ninths to the n. The area was already inside the count. It was never a separate question.

7:39Which brings the two halves of this together. At step three the carpet has five hundred and twelve surviving squares and seventy-three holes, and it still covers over seventy per cent of where it started. At step ten it has over a billion surviving squares - and it covers just over thirty per cent. The gasket at step ten has fifty-nine thousand and forty-nine triangles, and is down to between five and six per cent.

8:11Eight pieces at a ninth each is less than the one piece they came from. More pieces and less area are not two facts fighting each other. They are the same fact said twice. Although growing and shrinking are not automatically the same thing, and it is worth knowing why they travel together here. A rule that cut into nine and kept all nine would multiply the count every step and never lose a scrap of area.

8:43A rule that kept one piece out of nine would shrink away to nothing without the count ever growing at all. Of five rules put side by side, three grow and three shrink - and they are not the same three. Only these two do both. Which leaves one last question, and it is harder than it looks. The area goes to zero. Does that mean nothing is left? Not obviously. Removing pieces forever is not the same thing as removing everything, and settling which of those has happened is a longer story than this one.