The Sierpinski Carpet

What’s going on

Start with a square. Cut it into a 3 by 3 grid of nine equal squares and throw away the middle one. Eight squares are left. Now do the same to each of those eight: cut it into nine, throw away its middle. Then again, and again, forever. What is left is the Sierpinski carpet.

Press Next step above and watch the counters. Each step multiplies the number of squares by 8 and divides their side by 3:

Step	Squares left	Side of each	Area left
1	8	1/3	88.9%
2	64	1/9	79.0%
3	512	1/27	70.2%
6	262,144	1/729	49.3%

The area disappears

Each step keeps 8 of every 9 pieces, so it keeps 8/9 of the area. After n steps the area left is (8/9) × (8/9) × … = (8/9)ⁿ. After 6 steps less than half is left, after 12 less than a quarter. Keep going and the area falls below any number you name. The finished carpet has area zero.

But something is still there

Is the carpet just empty, then? No. The edges of the holes are never removed. A point on the edge of the first hole, such as (1/3, 1/3), sits on the border of a hole, never inside one, at every later step. The four corners of the square survive too, and so do infinitely many points that are on no edge at all, as you will see in the first puzzle.

The holes also add more and more edge. The first hole adds an edge of length 4/3, the next eight holes add 8 × 4/9 = 32/9, and every step adds 8/3 times as much as the step before. The total length of the edges grows without limit: zero area, but infinitely long edges.

Between a line and a surface

A line segment has length but no area. A square has area. The carpet has no area but infinite length, so it sits somewhere in between. Mathematicians can make this exact.

Make a square three times wider and it holds 9 = 3² copies of the original: the 2 says a square is two-dimensional. A line segment three times longer holds 3 = 3¹ copies: one-dimensional. Press Show the 8 copies: the carpet, made three times wider, holds 8 copies of itself. So its dimension d is the number with

3ᵈ = 8, which gives d = log 8 ÷ log 3 ≈ 1.893.

That is more than 1 and less than 2. Shapes with dimensions like this are called fractals.

The endless zoom

Press Endless zoom. The picture zooms in, without end, towards the point (1/4, 1/4). Every 9 times closer, the dashed square grows to fill the view and looks exactly like the whole carpet again. That is what “made of copies of itself” means: there is no level where the holes stop.

Try it yourself

1. A point that is never thrown away

The zoom heads for the point (1/4, 1/4), which is not on the edge of any hole. Why is it never thrown away?

Answer

Look at its x-coordinate, 1/4, inside the first column of the 3 by 3 grid (from 0 to 1/3). Stretch that column three times to fill the whole square: 1/4 becomes 3/4, in the last column. Stretch the last column (from 2/3 to 1) three times: 3/4 becomes 1/4 again. So the point is always in the first or last column, never in the middle one, and the same goes for y. A point is thrown away only when both coordinates are in the middle at the same step.

2. Ninety-nine percent gone

After how many steps is less than 1% of the area left?

Answer

After 40 steps. You need (8/9)ⁿ to drop below 1/100. After 39 steps about 1.01% is left, after 40 about 0.90%. Each step removes only one ninth of what is there, so it takes a while.

3. A different carpet

Cut a square into a 3 by 3 grid, but this time keep only the four corner squares and the centre square, and repeat that forever. What is the dimension of the result?

Answer

Three times wider holds 5 copies, so 3ᵈ = 5 and d = log 5 ÷ log 3 ≈ 1.465. That is smaller than the carpet's 1.893, because it keeps fewer pieces each time. Its area also drops to zero, faster: (5/9)ⁿ.

Go deeper

• The Polish mathematician Wacław Sierpiński described the carpet in 1916.
• Its 3D cousin is the Menger sponge: cut a cube into 27 and remove the centre cube and the six cubes in the middles of the faces, keeping 20. Its dimension is log 20 ÷ log 3 ≈ 2.727.
• In 1967 Benoit Mandelbrot asked “How long is the coast of Britain?” The answer depends on the size of your ruler, because a coastline, like the carpet, has more and more detail the closer you look.