The Sierpinski Gasket
Join the midpoints, remove the middle triangle, repeat.
Join the midpoints
Take an equilateral triangle. Mark the midpoint of each side. Join the three midpoints.
How many identical small triangles do the three lines make?
What this lesson covers
The idea
The Sierpinski Triangle (Gasket) is made by joining the midpoints of an equilateral triangle to break it into 4 identical equilateral triangles, removing the central one, and repeating this on the remaining 3, and so on.
Join the midpoints
Take an equilateral triangle. Mark the midpoint of each side. Join the three midpoints.
How many identical small triangles do the three lines make?
- 3
- 4
- 6
Make the gasket
Tap the triangle to join the midpoints. Remove the middle triangle. Then break each triangle that is left. Use the button when there are too many to tap.
The triangle version of the carpet
The Sierpinski Triangle (Gasket) is made by joining the midpoints of an equilateral triangle to break it into 4 identical equilateral triangles, removing the central one, and repeating this on the remaining 3, and so on.
Sierpinski made this fractal in a way similar to the carpet. Look at any of the 3 big triangles of the last picture: it is a smaller copy of the whole gasket.
Notes
Sierpinski Gasket: join the midpoints of an equilateral triangle to make 4 identical equilateral triangles, remove the central one, and repeat on the remaining 3, and so on.
Check yourself
How is an equilateral triangle broken into 4 identical equilateral triangles?
At step 1 there are 3 triangles. Each of them is broken the same way and its middle triangle is removed. How many triangles remain at step 2?
Answer: 9 triangles
3 triangles × 3 remaining from each = 9 triangles.
This is step 2 of the gasket. How many triangular holes can you count, big and small?
Answer: 4 holes
1 big hole from step 1, and 3 small holes from step 2: 1 + 3 = 4 holes.
After the central triangle is removed, what do we do next?
- By joining each corner to the centre. That gives only 3 triangles, and they are not equilateral.
- By joining the midpoints of its 3 sides — correct. Yes! The 3 lines join the midpoints, and make 4 triangles, all the same size.
- By cutting it in half with one line. One line gives only 2 pieces.
- Put it back and start again. Nothing is put back. The hole stays.
- Do the same to each of the 3 triangles that remain — correct. Yes! Join the midpoints, remove the middle, and so on at smaller and smaller scales.
- Stop: the gasket is finished. The same rule is repeated again and again, so the pattern never stops.