# The Swing’s the Thing

Order emerges from chaos with a triangle or pentagon, but not with a square. That is, if you take a triangle or a pentagon, chose a point inside it, then move the point repeatedly halfway towards a vertex chosen at random, a fractal will appear: Sierpiński triangle from point jumping halfway to randomly chosen vertex Sierpiński pentagon from point jumping halfway to randomly chosen vertex

But it doesn’t work with a square. Instead, the interior of the square slowly fills with random points: Square filling with point jumping halfway to randomly chosen vertex

As I showed in Polymorphous Perverticity, you can create fractals from squares and randomly moving points if you ban the point from choosing the same vertex twice in a row, and so on. But there are other ways. You can take the point, move it towards a vertex at random, then swing it around the center of the square through some angle before you mark its position, like this: Point moves at random, then swings by 90° around center Point moves at random, then swings by 180° around center

You can also adjust the distance of the point from the center of the square using a formula like dist = r * rmdist, where dist is the distance, r is the radius of the circle in which the circle is drawn, and rm takes values like 0.1, 0.25, 0.5, 0.75 and so on: Point moves at random, dist = r * 0.05 – dist Point moves at random, dist = r * 0.1 – dist Point moves at random, dist = r * 0.2 – dist

But you can swing the point while applying a vertex-ban, like banning the previously chosen vertex, or the vertex 90° or 180° away. In fact, swinging the points converts one kind of vertex ban into the others. Point moves at random towards vertex not chosen previously Point moves at random, then swings by 45° Point moves at random, then swings by 360° Point moves at random, then swings by 697.5° Point moves at random, then swings by 720° Point moves at random, then swings by 652.5° Animated angle swing

You can also reverse the swing at every second move, swing the point around a vertex instead of the center or around a point on the circle that encloses the square. Here are some of the fractals you get applying these techniques. Point moves at random, then swings alternately by 45°, -45° Point moves at random, then swings alternately by 90°, -90° Point moves at random, then swings alternately by 135°, -135° Point moves at random, then swings alternately by 180°, -180° Point moves at random, then swings alternately by 225°, -225° Point moves at random, then swings alternately by 315°, -315° Point moves at random, then swings alternately by 360°, -360° Animated alternate swing Point moves at random, then swings around point on circle by 45° Point moves at random, then swings around point on circle by 67.5° Point moves at random, then swings around point on circle by 90° Point moves at random, then swings around point on circle by 112.5° Point moves at random, then swings around point on circle by 135° Point moves at random, then swings around point on circle by 180° Animated circle swing

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