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Description
Some oldschool compass-and-straightedge construction.
1) Draw four points.
2) With a compass, you are allowed to draw a circle with any point as its midpoint, and with any other point on its edge. Using the four points, draw every possible circle.
3) Mark every point where two circles intersect.
4) Erase all the circles.
5) Repeat steps 2-4 with the new set of points.

Congratulations, with just a single repetition you've already generated several million points! This animation shows point-patterns generated in this exact manner; each frame uses a slightly different choice of the first four points. (In particular, two of the points are fixed, and the other two orbit around them as the animation progresses.)

A high-resolution still: pifactorial.deviantart.com/art…

Description
Some oldschool compass-and-straightedge construction.
1) Draw four points.
2) With a compass, you are allowed to draw a circle with any point as its midpoint, and with any other point on its edge. Using the four points, draw every possible circle.
3) Mark every point where two circles intersect.
4) Erase all the circles.
5) Repeat steps 2-4 with the new set of points.

Congratulations, with just a single repetition you've already generated several million points! This animation shows point-patterns generated in this exact manner; each frame uses a slightly different choice of the first four points. (In particular, two of the points are fixed, and the other two orbit around them as the animation progresses.)

A high-resolution still: pifactorial.deviantart.com/art…