Take three dots on a page. Put a fourth dot anywhere. Now roll a die: if it says 1 or 2, walk halfway to the top dot; 3 or 4, halfway to the left one; 5 or 6, halfway to the right one. Mark where you land, roll again and repeat.
It sounds like it should end up as a grey smudge. It doesn't. After a few thousand rolls a sharp, hole-ridden shape appears, the same one a mathematician drew by hand in 1915: the Sierpiński triangle. Pure randomness, and an exact picture.
Look at one step. Whichever corner the die picks, your new point lands exactly halfway between the old point and that corner. Seen from the whole picture, a step is a shrink to half size, pinned at the chosen corner.
So every roll squeezes everything into one of three half-size copies of the triangle, one tucked in each corner. After two rolls you are inside one of nine quarter-size copies, after three rolls one of 27 eighth-size copies, and so on. The middle of each triangle is never part of any copy, so the hole in the middle appears again at every scale.
Where you started stops mattering fast. Each roll halves the distance between you and the pattern, so after ten rolls your starting position is off by about a thousandth of what it was. That is why the first few points are not drawn, and why the die can never fill a hole: it only picks which tiny copy comes next, never where the copies are.
Each time you cut out the middle triangle you keep three quarters of the area. After n cuts, the area left is (3/4)n of the original, which shrinks towards zero. Yet the shape is not nothing: it is a web of infinitely many thin threads that still touches every corner.
Mathematicians give it a dimension of log 3 / log 2, about 1.585. Doubling its size gives you three copies of the original, not two (as for a line) or four (as for a filled shape). It sits between a line and a surface.
The slider changes how far each jump goes. At 0.50 you get the Sierpiński triangle. Push it below 0.50 and each jump is shorter, so the three copies are bigger and overlap, and the holes shrink and fill in. Push it above 0.50 and the copies get smaller and pull apart into separate islands of dust, with gaps you can see between them.
The special value is 0.50, where the three copies exactly touch at their corners. That is why the holes look so crisp there.
The triangle is named for the Polish mathematician Wacław Sierpiński, who described it in 1915. The random "chaos game" way of drawing it was popularised by Michael Barnsley in his 1988 book Fractals Everywhere. It is the simplest case of a bigger idea: a few simple rules, applied again and again, can generate a shape with detail at every scale. The same idea underlies fractal image compression and many models of ferns, coastlines and clouds.
Try dragging one corner far away. The shape bends with it, and every tiny copy bends the same way. The die never knew where the holes were. It only knew the corners.