FRACTALS

by Dr. Lawrence Wilson

© January 2019, LD Wilson Consultants, Inc.

All information in this article is solely the opinion of the author and for educational purposes only.  It is not for the diagnosis, treatment, prescription or cure of any disease or health condition.

Fractals are confusing and the concept is deep and mathematical.  However, you can view fractals easily in nature, such as the shapes of vegetables, fruits, trees and, in fact, every living thing.  Our bodies, and those of all other creatures and plants, are of a fractal design.

It is quite impossible to understand hair analysis interpretation without knowing about fractal design.

Below are some definitions of fractals from various sources on the internet.   Near the bottom of the page is also a link to a video tutorial:

DEFINITIONS OF FRACTALS

“A science of geometric and structural patterns generated by a simple mathematical formula that repeats on many levels.”

“A curve or geometric figure, each part of which has the same statistical character as the whole. Fractals are useful in modeling structures (such as eroded coastlines or snowflakes) in which similar patterns recur at progressively smaller scales, and in describing partly random or chaotic phenomena such as crystal growth, fluid turbulence, and galaxy formation.

Fractals are infinitely complex patterns that are self-similar across different scales. ... Driven by recursion, fractals are images of dynamic systems – the pictures of Chaos. Geometrically, they exist in between our familiar dimensions. Fractal patterns are extremely familiar, since nature is full of fractals.

Wikipedia: “In mathematics, a fractal is a subset of a Euclidean space for which the Hausdorff dimension strictly exceeds the topological dimension

Fractals are encountered everywhere in nature, due to their tendency to appear nearly the same at different levels, as is illustrated here in the successively small magnifications of the Mandelbrot set.

Fractals exhibit similar patterns at increasingly small scales, also known as expanding symmetry or unfolding symmetry; If this replication is exactly the same at every scale, as in the Menger sponge, it is called affine self-similar.

One way that fractals are different from finite geometric figures is the way in which they scale. Doubling the edge lengths of a polygon multiplies its area by four, which is two (the ratio of the new to the old side length) raised to the power of two (the dimension of the space the polygon resides in). Likewise, if the radius of a sphere is doubled, its volume scales by eight, which is two (the ratio of the new to the old radius) to the power of three (the dimension that the sphere resides in).

However, if a fractal's one-dimensional lengths are all doubled, the spatial content of the fractal scales by a power that is not necessarily an integer. This power is called the fractal dimension of the fractal, and it usually exceeds the fractal's topological dimension.

As mathematical equations, fractals are usually nowhere differentiable. An infinite fractal curve can be conceived of as winding through space differently from an ordinary line - although it is still 1-dimensional its fractal dimension indicates that it also resembles a surface.

HISTORY

The mathematical roots of fractals have been traced throughout the years as a formal path of published works, starting in the 17th century with notions of recursion

Then the research moves through increasingly rigorous mathematical treatment of the concept to the study of continuous but not differentiable functions in the 19th century by the seminal work of Bernard BolzanoBernhard Riemann, and Karl Weierstrass,

The word fractal appeared in the 20th century with a subsequent burgeoning of interest in fractals and computer-based modeling in the 20th century.The term "fractal" was first used by mathematician Benoit Mandelbrot in 1975. Mandelbrot based it on the Latin frāctus meaning "broken" or "fractured", and used it to extend the concept of theoretical fractional dimensions to geometric patterns in nature.

There is some disagreement amongst mathematicians about how the concept of a fractal should be formally defined. Mandelbrot himself summarized it as "beautiful, damn hard, increasingly useful. That's fractals."

More formally, in 1982 Mandelbrot stated that "A fractal is by definition a set for which the Hausdorff-Besicovitch dimension strictly exceeds the topological dimension."

Later, seeing this as too restrictive, he simplified and expanded the definition to: "A fractal is a shape made of parts similar to the whole in some way."

Still later, Mandelbrot settled on this use of the language: "...to use fractal without a pedantic definition, to use fractal dimension as a generic term applicable to all the variants."

The consensus is that theoretical fractals are infinitely self-similar, iterated, and detailed mathematical constructs having fractal dimensions, of which many examples have been formulated and studied in great depth.

Fractals are not limited to geometric patterns, but can also describe processes in time.

Fractal patterns with various degrees of self-similarity have been rendered or studied in images, structures and sounds and found in nature, technology, art, architecture and law.

Fractals are of particular relevance in the field of chaos theory, since the graphs of most chaotic processes are fractals.

“A fractal is a non-regular geometric shape that has the same degree of non-regularity on all scales. Fractals can be thought of as never-ending patterns.

Just as a stone at the base of a foothill can resemble in miniature the mountain from which it originally tumbled down, so are fractals self-similar whether you view them from close up or very far away.

The term "fractal" was coined by Benoit Mandelbrot in 1975. It comes from the Latin fractus, meaning an irregular surface like that of a broken stone.

Fractals are the kind of shapes we see in nature.  Science continues to discover an amazingly consistent order behind the universe's most seemingly chaotic phenomena.

Mathematicians have attempted to describe fractal shapes for over one hundred years, but with the processing power and imaging abilities of modern computers, fractals have enjoyed a new popularity because they can be digitally rendered and explored in all of their fascinating beauty.

Fractals are being used in schools as a visual aid to teaching math, and also in our popular culture as computer-generated surfaces for landscapes and planetary surfaces in the movie industry.

The use of algorithms to generate fractals can produce complex visual patterns for computer generated imagery (CGI) applications.

See some fractal image examples.

AN INTRODUCTORY VIDEO

Watch an introductory video tutorial on fractals called Fun With Fractals at: https://whatis.techtarget.com/definition/fractal

AN EXAMPLE FROM HAIR MINERAL ANALYSIS

If one connects the tops of the bar graphs of the first four minerals on an ARL hair analysis chart, at times it forms the shape of a bowl.  We call this the bowl pattern.

What is unusual is the feeling or emotion associated with this pattern is that one is stuck in a bowl, like a spider caught in the sink who can’t climb out.

This is an example of a fractal because it is a biochemical pattern – the mineral levels – that repeats at an emotional level.  This type of repetition or recursion, as the professors of mathematics call it, is common on the hair analysis chart.