Feed the machine two completely different pictures and run both. After one pass, the two prints can only differ inside copies half the size of the sheet. After two passes, inside copies a quarter the size. The difference between any two seeds is squeezed by the same factor on every pass — the machine exponentially forgets its input. Twenty passes in, what you fed it is physically too small to see.
So what survives? Only a picture the machine cannot change: a shape A that equals the union of its own shrunken copies, A = f₁(A) ∪ … ∪ fN(A). Mathematicians call it the attractor of the rule — every seed is pulled toward it (the underlying engine is Banach's fixed-point theorem; John Hutchinson worked out the picture version in 1981). Read that equation again: it says the shape is literally made of smaller copies of itself. Self-similarity isn't a coincidence you observe in these images — it's their defining equation.
Counting those copies measures dimension. Double a line segment and it contains 2 copies of itself; double a square, 4; double a cube, 8 — doubling a D-dimensional shape always yields 2D copies. Double the Sierpiński triangle and you get just 3, so 2D = 3 and D = log 3 / log 2 ≈ 1.585. Nothing forces that exponent to be a whole number — and for shapes too rough to be built from equal copies, like real coastlines, you count boxes at ever-finer ruler sizes instead and read the dimension off the slope. Same idea, measured empirically.
And this little machine opens every other door in fractal-land. Run it one point at a time — jump toward a randomly chosen copy, over and over — and the attractor assembles itself out of pure noise: that's the chaos game, and it's exactly how the ∞ stop on the slider is computed. Replace-and-repeat on strings instead of pictures grows L-system plants. And if you feed a number back in instead of a picture — z → z² + c, over and over — the question "does iteration stay small or escape?" paints the Mandelbrot set. Iteration is the engine of all of it.
Key terms
- attractor
- The one picture the copier can't change — where every seed ends up. A = f₁(A) ∪ … ∪ fN(A).
- contraction
- A map that always shrinks distances. Iterate one and exactly one thing holds still — here, a whole picture.
- self-similar
- Made of shrunken copies of itself. For copier shapes this is true by construction, not by luck.
- fractal dimension
- The exponent D in copies = (magnification)D. Whole for lines and squares; ≈ 1.585 for the triangle.