# Advanced: Harmonic Evolutionary Standing Waves

Inspiration strikes often in mysterious ways. While encoding the evolute algorithm, a sneaking suspicion had grown in my consciousness, that the evolute wanted to reveal something hidden… something deeper… a fractal or harmonic nature, which required an added dimension of representation. The evolute needed a dimension of “flow”, so we added the functionality of spheres moving along the evolute path called “trons”. Immediately patterns emerged with almost every configuration I threw at the newly baked algorithms.

A more in-depth study of one evolute configuration was required. In this case, a 24 loop evolute was used as the base evolute pattern, to explore the symmetries created with different numbers of “trons” sent down the evolute curve. * The hypothesis being that dynamic symmetries may exist which have a corresponding relationship of “turns to trons”*… Turns being the number of loops in the evolute path, and Trons being the number of spheres sent down the path.

Practically speaking, it was an inquiry into the symmetry and asymmetries of toroidal dynamics, albeit from an artistic and deeply visionary perspective. Yet, I felt there would be a host of meaningful corollaries… a spectrum as diverse as toroidal coil pulse frequencies to orbital spin patterns… Could it serve as a toroidal pulse dynamics simulator? Could it predict harmonic frequencies of a specific toroidal geometry? Or help visualize toroidal orbital spin dynamics?

The approach was simple… I started with one sphere, and iterated through more than 100, taking screen grabs wherever something interesting appeared. Some principles could be anticipated; yet others were altogether surprising.

For example, notice how the symmetry changes dramatically, when a slight change is made to the number of “trons”. When animated about the evolute path, the dynamic symmetry is further pronounced. Incrementing higher and higher revealed what I can only describe as ** harmonic evolutionary standing waves**. For me this was

*freaky cool and totally unexpected*. 😉

This extraordinary observation of dynamic symmetry shift can be demonstrated by starting with something expected.. in this case of 24 loops, 24 trons would create an expected single circle of symmetry orbiting about the radial axis, or horizontal plane…. But when you add one more, for a total of 25 spheres.. it morphs into a circle orbiting AROUND the core. Amazing! Higher numbers seem to increase the density effect, and other patterns emerge…

24 turns with 48 trons = 2 radial orbits.

24 turns with 49 trons = a phi-spiral like single evolute!

24 turns with 50 trons = 2 circles orbiting the core.

Click through the entire collection, and you will get a feeling for the simulation potential of the Cosmic Geometry Toolkit’s evolutes and trons.