The gasket and the snowflake

Sierpinski found a second fractal with almost the same rule, but on a triangle.

Take an equilateral triangle and join the midpoints of its three sides. This cuts it into 4 identical equilateral triangles. Remove the middle one. Then do the same to the 3 that are left, and so on. This is the Sierpinski Triangle, also called the Gasket.

Why are the 4 pieces identical equilateral triangles? Look at a corner piece. Two of its sides are each half a side of the big triangle, so it is isosceles. The angle between them is the big triangle's 60°. An isosceles triangle with a 60° angle between its equal sides is equilateral. All three corner pieces are the same, so the middle one is equilateral too.

Counting works as before. Each remaining triangle becomes 3 at the next step, so the count goes 1, 3, 9, 27: that is 3ⁿ.

Now the area. Start with area 1. At every step you keep ¾ of what you had, because 1 of every 4 triangles is removed. So the area left at step n is (¾)ⁿ. It shrinks towards nothing, even though the number of triangles keeps growing.

A third fractal, and this one grows. The Koch Snowflake is named after the Swedish mathematician Von Koch, who first described it in 1904.

Start with an equilateral triangle. On each side: divide it into 3 equal parts, raise an equilateral triangle on the middle part, then remove that middle part. The straight side becomes a 'bump' made of 4 shorter pieces. Repeat on every new side, for ever.

Count the sides: 3, then 12, then 48. Each side becomes 4, so the number of sides is 3 × 4ⁿ.

Each new piece is a third as long, so the perimeter is multiplied by 4 × ⅓ = 4/3 at every step. Starting from a perimeter of 3, it is 3 × (4/3)ⁿ. For example, P₂ = 3 × 16/9 = 16/3. It grows without limit, yet the shape stays small enough to cover with your hand.