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 to maintain constancy of convention). The power required in the Pythagorean analogy when

happens also to be the fractal dimension of the Koch Curve. The analogous Pythagorean generalization of

from:

and the fractal dimension from:

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

∠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 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.

Examples of some geometric fractal patterns are shown below.