I’ve added a new feature to my Mandelbrot renderer.

Quick refresher: points are in the Mandelbrot set if, no matter how many times you apply a simple transform, they never escape to infinity.

The new feature shows the path taken while iterating a specific point.

Robot lollipop

This screenshot shows an ever-decreasing spiral. The start of the spiral is pretty close to the centre of the image. That’s point (0, 0) in the plane. The path jumps from there to the actual point being rendered, which I’m guessing is about (0.5, 0.5). From there, as we keep iterating, the point describes a shrinking spiral, never escaping to infinity.

Spirograph

The path taken for this point (a bit up and left of the previous one) is a bit more interesting; it’s still describing a shrinking spiral, but the rate of shrink is a lot lower, and the angle is a lot tighter.

Corporate logo #3414

And this path’s more interesting again – it’s noticeably asymmetrical.

Qix 2010

This is the path taken by a point in the left-hand circle – these tend to be a lot less beautiful looking as screenshots, but seen live, the trajectories (which shift and reconfigure as you drag the mouse around) sweep between different “stable”, “aligned” configurations in a really interesting way. I need to hook up some video capture again.

Here’s an example of a stable configuration that cycles around four nodes (even a long way outside the set proper) without ever actually escaping. Note that this isn’t a precise cycle – each iteration seems to be slightly different to the last.

Escape

None of the five points so far have escaped to infinity, so their corresponding pixel is rendered white, and the path in the above shots is shown in grey. This screenshot shows what happens when a point does escape. The path looks a lot more unordered (although there is some structure), and it ultimately disappears off the top of the screen (well North of (0, 2)).