Autumnal Equiknot Animation (Friday Fractal LXI)

This week’s fractal has been delayed slightly, to coincide with the Autumnal Equinox. It isn’t your usual Friday Fractal, either.

I was fiddling around this week, thinking about ancient symbols which may have represented some sort of dynamic changes. Were our ancestors fascinated by the relentless cycles of nature on which their lives were so dependant? Changing seasons meant changes in food supply to a hunter-gatherer culture, and thus changes in survival strategies. Rhythmic patterns were the way of life. (Is today really any different?) It stands to reason that some ancient artists would have expressed some awe and reverence for repeating cycles, feelings which were then expressed in symbolic form. There are likely many examples, but today, I’d like to consider one: the Celtic knot.

The Celtic knot is one part aesthetic, one part mathematics. It follows simple rules, which can vary from knot to knot. The simple rules typically involve uniformity and repetition; that is, the same thickness of line is usually used in a knot, and the patterns repeat. Celtic knots also tend to be space filling: as they weave in and over themselves, the lines fill all free space within the pattern, lending to the uniform balance of the whole.

Fractals and Celtic knots have quite a bit in common: they are self-similar, repeating, space-filling patterns. Imagine what the ancient artists could have done if they’d had some way to add scaling repetition to their knots. (I wouldn’t be surprised if they’d thought about it.) Luckily, today we have computers which can give knotted designs just such a twist. Let’s take a shape, and plot it along the scaling, logarithmic spiral of a Julia set. I chose a seven-sided shape for this project... it seemed complex enough, but not overpoweringly so. It also seemed the best number to create the spade-like corners found in Celtic knots. When the shapes interact, if they are the same color, they attach to one another. Otherwise, they fall behind or in front of the other. This way, the shapes are sort of woven together:

An Autumnal Equiknot

But the fun has just begun. The beauty of this digital lacework doesn’t become apparent until the knots are "tied" and "untied". To do that, all we need to do is vary the size of the shapes. The only way to display this occurring is with an animation:

Unfortunately, the complexity of this video gives us two choices: the crappy low-resolution version shown above via you-tube, or the high-quality version you can download here. Since it takes a few minutes to download, even at high speeds, and then only plays for 13 seconds, I’d suggest saving it to your computer (right-click on the link, click ’save target as’... you know the drill) and use a program on your system that can play it in a repeating loop. Oh... and have a pleasant first day of Autumn!

Image from the Book of Kells via the D. B. Weldon Library. Fractals created by the author using ChaosPro.