In this session, you’ll learn how to make Euclidean fractal images like these without needing to rely on the Cartesian plane:

To develop the seedtree kernel, I resorted to recursive geometry. It was a more intuitive way to understand living tree patterns with Infinite possibility than the symbolism of program language.

What I needed were finite parent objects that could nest an arbitrary number of child objects within them: these are “functions;” think Turing machines but the number line is forward-back not left-right. Golden Ratio tessellations are an example of this kind of function, but I needed a potentially Infinite plurality of identities in order to write scripts of multiple functions.

Vera’s Metallic Mean Family — developed out of the [n;n,n,n…] index — solved my identities need. But, by the time PBS Infinite Series exposed me to her work 6 years ago, I was already close to discovering my own identities solution — developed out of the Pythagorean Theorem — through observational experimentation, mostly using WolframAlpha for closed-form discovery and Inkscape for precise geometric measurement.

In the first half of the session, we’ll: watch this video of the Metallic Mean Family, watch this and this video of a geometric series interpretation of the phenomenon, make a spiral together using Inkscape, look at this algorithmic application I submitted to ArXiv on February 29, 2020, contemplate whether base-scale could replace coordinate systems like this one and chat about my architectural plans described here.

In the second half of the session, we’ll have a group discussion. I would like to talk about how the base-scale is related to perspective drawing and “tessellated gradient dissent,” but I’m open to taking the conversation in a different direction.

If you’re interested in the mathart side of this, check out the pitch deck for my active https://artizen.fund/index/p/squaring-the-spiral?season=4 campaign.

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During the class, we looked at this and this wikipedia pages and reviewed these images:

We didn’t get to this and this x posts or how this proof shows how the ‘irrational’ energy is unresolved on the longer side or, not commonly shown, on the shorter sides instead:

We also didn’t go over Einstein’’s boyhood proof here.

We did briefly discuss using these patterns for music, but I didn’t mention this sequence as sounding like random notes.