context

Taking inspiration from copyright information theory, math must be fixated in electromagnetism.

The following article attempts to explain how geometric rotation relates to the Pythagorean Theorem and the Golden Ratio:

In brief, this "Base Scale" scale-symmetry is a vanishing point alternative to Euclid's 5th postulate that utilizes the logarithmic bases between 0 and 1 as relative origin points:

Important to computer science, the Base Scale is gradient descent ‘energy’ converted into “tessellated gradient descent” energy:

Please visit Nohalt.art/forum to find more articles about this mathematical observation.

the “Sacred Geometry” question

Ramona shared this book with me during Infinita.city Forever:

I noticed this pattern in the book that is close to, but critically different from, the Base Scale:

This Drawing 6.3 pattern is also "Squaring the Spiral" with a vanishing point decay rate, but it does not use the Base Scale.

I presumed that this Drawing 6.3 pattern was real, before I confirmed with the algebra shared below, as there is no algebra to accompany it and it is not quite robust enough to be considered a ‘proof without words.’

“Sacred Geometry” solution

Surprisingly, the solution does not rely on the Pythagorean Theorem. It does rely on 4 variables a, b, g, h, rather than the Base Scale’s a, b, c 3 variables, however.

The first step is to recognize the double-spiral is more fundamental, algebraically, than the single spiral:

Instead of Base Scale's (c - b)/a = x scale-symmetry formula, this scale-symmetry formula is (a - 2h)/a = y.

a/(2b + g) = h/b and (a - 2h)/a = y are the common algebras in this expansion of the π/2 Base Scale. The other definitions depend on what h; g we want to find.

For instance, in the Diagram 6.3 example above, we want to find when h = g. To translate that into decay rate scaling, the ratio is h/g = 1 .

As such, the scale-symmetry answer to the Diagram 6.3 example is here a = 100; b = 50; a/(2b + g) = h/b; h/g = 1; g; h; (a - 2h)/a:

more examples

In this example here, a = 100; b = 50; a/(2b + g) = h/b; h = a/3; g; h; (a - 2h)/a, h = a/3:

In the Base Scale special case, the spiral fits on the unit circle here a = 100; b = 50; |(a^2 + b^2)^(1/2)| = c; a/(2b + g) = h/b; b + g = c; g; h; (a - 2h)/a.

In this example here, a = 100; b = 50; |(a^2 + b^2)^(1/2)| = c; a/(2b + g) = h/b; g = 2c; g; h; (a - 2h)/a:

Base Scale collapse

These observations show that “tessellated gradient descent” does not need to rely on the Pythagorean Theorem.

However, when the decay rate does harmonize with the longer and shorter-sides relationship of Thales’s right triangle theorem, then the scale-symmetry is a π/2 rotation that comports with electromagnetic spin. I have shown this scale-symmetry as x and the non-rotating scale-symmetry that I discovered in what you have read here now as y = x^2 for the special case where b + g = c.

more derivations

Beyond π/2, there are ‘flower of life’ derivations (for instance, this article’s banner image, and the image directly above, include the 4^(1/2) - 3^(1/2) flower derivation) and π/n dimensional derivations such as:

dedications

MIDABI for coining "Base Scale."

Ece for coining "Squaring the Spiral."

Einstein for creating polycentric origin point gedankenexperiment.

King William III & Queen Mary II for showing us the middle-way of patriarch & matriarch co-parenting so that we can upward double-spiral into the narrowness of harmony as a ‘nuclear multifamily’ species -- a.k.a. polycentric atomic sovereignty -- germinating out into the stars.

Próspera, the Honduran Free City, for hosting me while I witnessed this observation of mathematical law.

Edit: Fair Market Value statement by ChatGPT for this article:

PDF backups:

ipfs://bafybeidbdur4tflman2kesbh7q4e44qaitt5ac6plsmrlz3cynuwuk2v7e

ipfs://bafybeidyd5twiltrph32ehtfyuyter45xfkbug2gifw3wj5pfokn6atmui

ipfs://bafybeicvxlcqo66qftxayfym2nq7jfgefmxg3tdjdk5t6oyezy3z3265my