Copy!

The machine

The multiple reduction copy machine is a thought experiment from Heinz-Otto Peitgen, Hartmut Jürgens and Dietmar Saupe, who used it in Chaos and Fractals (1992) to explain iterated function systems. Picture a photocopier fitted with several lenses instead of one. Each lens reduces the original and lays it somewhere on the page. Feed the page back in as the next original and let the machine run.

Each lens is an affine contraction: it scales the page by less than 100%, then may rotate, skew, flip and shift it. Here a lens is one frame on the blueprint.

Why the source stops mattering

Every lens shrinks, so any two starting images move closer together with each pass. John Hutchinson proved in 1981 that the process settles on exactly one shape, the attractor, whatever you begin with. A lotus, a letter F and your own photograph all become the same Sierpiński triangle. What survives of the source is its color and texture at the finest scale.

Michael Barnsley's collage theorem runs the idea backwards: cover a shape with shrunken copies of itself, and that blueprint will regrow the shape from anything. The Fern preset is four lenses, one of them squashed almost flat to make the stem.

The Sierpiński triangle

Wacław Sierpiński described it in 1915. On this machine it is three lenses at 50%: one on top, two below. Each pass triples the number of copies and halves their size, so its dimension is the number D for which three half-size copies exactly rebuild the whole.

The dimension readout solves the same equation, Σ rᵢᴰ = 1, for any blueprint you build, where rᵢ is each copy's reduction. When the copies overlap enough that the sum still exceeds 1 at D = 2, it shows 2.00+.

How to play

Touch

The canvas is the instrument. Dragging, pinching, twisting, double-tapping and pressing each change the blueprint in a different way. There are eight gestures to discover; the machine names each one as you find it. Once you find the one that picks a single copy, a row of colored dots appears above the dock so you can choose which copy you steer with one tap.

Play

The round button sets the blueprint drifting so the fractal slowly morphs. Tap it again to hold still.

Source

Choose what gets copied from 91 symbols (spiritual signs, nature, everyday objects, faces and shapes) or upload your own. Images with a transparent background work best.

Blueprint

Arrange the copies precisely, pick presets, undo, and save or load compositions.

Adjust

Change the look (Living Canvas, Optical Lab or Playground), the iterations, the print style and the motion.

Explore

Dive into the attractor, move the view, record a clip, capture an image or share a link.

Calm

After a few seconds the controls fade and leave only the image. Tap once to bring them back. Tap the COPY! title to return to the Sierpiński triangle.

Two ways to print

Copies draws the source image at every leaf, so you can watch it shrink into the pattern. Dust plots the attractor point by point and colors each point by where it falls on the source, like projecting the original image onto the fractal.