Mathematics · Simulator 055
Fractals repeat their shape at every scale. Zoom into the Mandelbrot set, morph a Julia set by moving its parameter, grow a Koch snowflake whose perimeter keeps lengthening, and branch a fractal tree. Change the depth and watch the detail emerge.
Mandelbrot set: for each complex number c, iterate z ← z² + c from z = 0. If |z| stays bounded, c is in the set (black). Colours show how quickly points escape past |z| = 2. Julia sets use the same rule, but fix c and vary the starting z; each c gives a different shape. Connected Julia sets correspond to c inside the Mandelbrot set.
Koch snowflake: replace the middle third of every edge with two sides of a triangle. Each step multiplies the perimeter by 4/3, so it grows without limit, while the area converges to 8/5 of the original triangle. Its fractal dimension is log 4 / log 3 ≈ 1.26: more than a line, less than an area.
Fractal tree: each branch splits into two shorter branches at a set angle and length ratio, the same recursion that shapes real trees, lungs and rivers.
Escape-time rendering with a maximum iteration count; deeper zooms need more iterations and eventually run into floating-point limits (about 10¹³×).
A Koch snowflake starts from a triangle with sides of 1 m. How long is its perimeter after 10 steps? Could you paint its inside with a finite amount of paint?
Perimeter = 3 × (4/3)¹⁰ ≈ 53.3 m, and it grows without limit. The area converges to (8/5) × (√3/4) ≈ 0.69 m², so finite paint covers it: an infinitely long boundary around a finite area.