Fractals in the Classroom: Koch, Sierpinski, Mandelbrot

Fractals are a rare topic where a picture comes first and the questions follow by themselves: how long is that edge? How much area does it enclose? Built one level per click, they are also a natural lesson on sequences and limits.

The Koch snowflake: infinite perimeter, finite area

Start with an equilateral triangle. At each step, replace the middle third of every side with two sides of a smaller triangle pointing outwards.

Each step multiplies the number of sides by 4 and divides their length by 3, so the perimeter is multiplied by 4/3. After n steps it is (4/3)ⁿ times the original: it grows without limit.
The area increases by smaller and smaller amounts and converges to 8/5 of the original triangle.

Koch snowflake at depth 0, 2 and 5 — Depths 0, 2 and 5. In class, one click adds one level.

That contrast, an infinitely long boundary around a finite area, is a perfect starter for geometric series.

A dimension between 1 and 2

Zooming in on the Koch curve by a factor 3 gives 4 copies of itself, so its similarity dimension is log 4 / log 3 ≈ 1.26. The Sierpinski triangle (3 copies at half size) has dimension log 3 / log 2 ≈ 1.58. Students can compute both from the pictures.

The Mandelbrot set

For each complex number c, iterate z → z² + c starting from z = 0. If z stays bounded, c is in the set (black); otherwise the colour shows how quickly it escaped. Once |z| exceeds 2 it is certain to escape, which makes the test easy to explain.

Julia sets use the same rule with c fixed and the starting point varying. Animating c makes the Julia set break apart and reassemble, a striking way to connect the two pictures.

Build them in Tracé

Open Plot → Fractal and choose the Koch snowflake, Sierpinski triangle, fractal tree, Mandelbrot or Julia set.
For the first three, press Reveal one level of depth per click.
For Mandelbrot, animate Zoom between clicks; for Julia, animate the real and imaginary parts of c.

Next to them in the same menu: the logistic map, cellular automata such as rule 30 and rule 90 (which draws a Sierpinski triangle by itself), and the sieve of Eratosthenes.