Watch this (courtesy of Harvard Natural Sciences Lecture Demonstrations), but also see that here is a principle that is a very powerful key to aesthetics.
Its a visual form of what we experience usually in sound when two or more notes sound in harmony.
If you were an ant and experienced time slowed down, you might just “see” harmonious tones as patterns like this.

This way of governing motion so that every part is changing at a numerical multiple of the slowest
(which can be 15 as here, or 30,000 points in experiments I have done) was first discovered by the late animation/computer film maker John Whitney (senior). John discovered the basic principle which he called differential dynamics. I developed it way further and call it harmonic math, as it is not just governing motion.

I have applied Harmonic Mathematics to dynamic computer graphic art, to music and to dynamically changing tones and healing timbres.

In the 1980s with the help of friends, in particular Mike Tusch and later Dave Snowdon in the 90s and 00’s, this technique was greatly extended. Not just using simple pendulum motions that you see here, we used waveforms themselves also consisting of harmonics to control the motion of each element.

The video Helix uses 5 or 6 harmonics to govern each point moving up and down vertically. If oyu study it you will see each point is moving to trace a complex waveform – several superimposed waves not unlike a toy boat bobbing on the sea, subject to many motions overlaid.

Computer graphics are not “unplugged” like this video which is getting on for 10 million views (the public responds strongly, but likely does not realise why).
But we can choreograph the individual motions of teeming multitudes of points, and that is an intricacy and exhilaration uniquely obtainable from a computer.

I applied this “harmonic mathematics” (as I call it) not only to visual images, but also to the defining points in the shape of a tone (see UFO, and Sonic Shapes (healing tones developed in the 80s with Isobel McGilvray), especially “Helix” – see the video), and to melodic loops and cascades of canons as such loops went out of phase, also governed by harmonic relationships.
(Number rays 3, Bubbles, Silverstream, Crystal Mists In Silver Air, and a double album Pete Townshend helped me record, “Method Music”).

I have also used HM to produce musical scores. Approximately 1/3 of my 170 or so compositions use HM, including orchestral, string quartet and solo piano. I also used it for the Method Music project, realised with Pete Townshend and Dave Snowdon, which generated 10,500 pieces of music over the net 2007-8, as well as for the double album Method Music.

It is better suited to magical moving graphics than fractals because it deals with change, fractals are inherently static, although zoomable.

HELIX:

SPIRAL DANCE (with the music Number Rays 2)

Harmonic maths in visual and audio, here in an extended one hour sequence, permitting a fuller experience of what the maths can do. Its a challenging journey, so make a special hour for it. The harmonic pendulum video has over 9 million views. That is very primitive in comparison to this, its much shorter, as well as being “unplugged” – after one cycle the pendulums begin to go out of sync. There is great power and beauty in this math.

more videos:

Triangles – another Sonic Shape

Here you hear what you are seeing as a waveform

The original audio-visual of Triangles, created by Lawrence Ball and Mike Tusch in 1984.
This is unique, a digital display of the waveform as you hear it as a low drone, with modulating harmonics, created by the shape you witness as it changes.
The audio and visual were created on Apple II desktop computers, with 64K of memory, and the processors were made to run their legs off.

6502 assembler code programmes were written by Mike Tusch and Lawrence Ball.

The sounds here were later released as “Triangles” in the set of “ShapeTapes” in 1988, then later renamed “Sonic Shapes” in 2011.

These are healing sounds developed in their use by Lawrence and Isobel McGilvray, who herself carried out a research project on their effects, including on the chakras.