An Intersection of Fractal Dimension and Pythagorean (et al.) Theorem Generalization | Steve Wait – November 23, 2022

The above titled as applicable to self-similar fractals and satisfying real number exponents respectively. This I first noticed in an exploration of rhombic area units and the postulate that for any triangle ABC where c* is the long leg, an exponent ^ℝ exists that will satisfy a ^ℝ + b ^ℝ = c ^ℝ of non-Diophantine concern.

*Alternatively, a ^ℝ + b ^ℝ = c ^ℝ can be re-written b ^ℝ – a ^ℝ = c ^ℝ at the isosceles inversion point (where leg c becomes equal to leg b) to maintain constancy of convention. This is necessitated as ∠C changes between the degenerate triangle limits of zero and π radians, the transition from obtuse scalene to acute scalene. Constrained by this equation, as triangles of a:b < 1 approach isosceles, ℝ becomes extreme. In the case of a:b = 1, the equation is invalid at the equilateral where c fails to be the long leg.

The power required in the Pythagorean analogy when $∠C=\frac{2\pi}{3}$ happens also to be the fractal dimension of the Koch Curve. The analogous Pythagorean generalization of ${a}^{ℝ}+{b}^{ℝ}={c}^{ℝ}$ from:

$∠C=\cos^{-1}{\frac{{-\left(\sqrt[{ℝ}]{{a}^{ℝ}+{b}^{ℝ}}\right)}^2+{a}^2+{b}^2}{2ab}}$

and the fractal dimension from:

${ℝ}=\frac{ln\ N}{ln\ r}$

Since always an isosceles condition or 1:1 ab ratio exists, then from:

$\cos^{-1}{\left(\frac{-\left(2^{\left(\frac{1}{\frac{ln\ N}{ln\ r}}\right)}\right)^2+2}{2}\right)}$

Here, ∠C (or its compliment) is derived and used at each connection or node resulting in a new angular direction for each segment. Segments do not cross, lest there be self-intersection of the resulting curve. The natural logarithms of only positive integers for r and N are used. Aside from its use as named, the scale factor (r) also defines the number of equal length line segments an initiator of length 1 is divided into. The number of segments (N), as its name suggests, determines how many segments of $\frac{1}{r}$ are used to form the connected generator sequence. Subsequent dilation of the generator sequence by the scale factor then replaces the original segments. See Figs. 1, 2, and 3 below.