The Sierpinski Carpet

The idea

The Sierpinski Carpet is made by breaking a square into 9 smaller squares and removing the central one, then repeating this on each of the remaining 8 squares, and so on.

Break a square into nine

Take a square. Break it into 9 equal smaller squares, 3 rows of 3. Then remove the middle one.

Now the same thing is done again to every square that is left.

How many squares will be broken in the next round?

• 4
• 8
• 9

Make the carpet

Tap the big square to break it. Then break each square that is left. Use the button when there are too many to tap.

The same pattern, smaller and smaller

The Sierpinski Carpet is made by breaking a square into 9 smaller squares and removing the central one, then repeating this on each of the remaining 8 squares, and so on.

The Polish mathematician Sierpinski found it. At every step you see squares of one size that remain, and square holes. The squares get smaller and smaller, so we see the same pattern at smaller and smaller scales.

Notes

Sierpinski Carpet: break a square into 9 smaller squares, remove the central one, then repeat on each of the remaining 8 squares, and so on.

Check yourself

A carpet starts from a square of side 27 cm. At step 1 each remaining square has what side, in cm?

Answer: 9 cm

27 cm ÷ 3 = 9 cm.

For the same carpet, what is the side of each remaining square at step 2, in cm?

Answer: 3 cm

9 cm ÷ 3 = 3 cm. Step by step the side shrinks: 27, 9, 3, 1, …

This is step 1 of the carpet. To make step 2, which squares are broken?

Why do we see the same pattern again and again, smaller each time?

• Only the hole in the middle. The hole is empty. There is nothing there to break.
• Each of the 8 blue squares — correct. Yes! The same thing is done to every remaining square: break into 9, remove the middle.
• Only the 4 corner squares. Every remaining square is broken, not just the corners. The 4 side squares too.
• The squares keep getting bigger. They get smaller each step, because every square is broken into 9 smaller ones.
• The same rule is repeated on every square that is left — correct. Yes! Break into 9 and remove the middle, again and again.
• The middle square is removed only once. The middle is removed from every remaining square at every step.