Fractals.

Step 1 of 8

Copies of copies

A fractal is a self-similar shape: each small part of it looks like the whole. The Sierpiński pyramid is built in steps: swap a pyramid for four pyramids at half size, sitting in its corners, then do the same to each of them. The Depth is the number of steps. At depth $3$⁠ there are $4^3 = 64$⁠ pyramids.

Step 2 of 8

A growing perimeter

The Koch snowflake starts from an equilateral triangle. At each step every side is split into thirds, and a small triangle is added on the middle third, pointing out. Each side so becomes four sides, each a third as long. The Shape and Depth are fixed now.

Step 3 of 8

Without end

At depth $5$⁠ the steps are already hard to see. Zoom in goes into a corner, and the same picture comes back again and again. Suppose the building goes on for ever.

Step 4 of 8

A broken dimension

Halve a line: $2$⁠ copies. Halve the sides of a square: $4$⁠ copies. Of a cube: $8$⁠. These are $2^1$⁠, $2^2$⁠ and $2^3$⁠, and the exponent is the dimension. In general, if a shape is made of $N$⁠ copies of itself, each $\tfrac{1}{s}$⁠ its size, then $s^D = N$⁠, and the fractal dimension is

$D = \frac{\log N}{\log s}$$D =$$\frac{\log N}{\log s}$⁠

For the Sierpiński pyramid $N = 4$⁠ and $s = 2$⁠, so $D = 2$⁠, though it is a solid in three dimensions.

Step 5 of 8

How many copies

Back to the pyramid, one step only.

Step 6 of 8

How much smaller

Each copy is like the whole, only smaller.

Step 7 of 8

The volume

At each step part of the pyramid is taken out. How much is left?

Step 8 of 8

The snowflake’s area

The snowflake’s perimeter grows without end. What happens to its area?