*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**

ÒA science of
geometric and structural patterns generated by a simple mathematical formula
that repeats on many levels.Ó

ÒA 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.

**Ò**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: ÒIn
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.^{[1][2][3][4]}

Fractals exhibit
similar patterns at increasingly small scales,^{[5]} also known as **expanding symmetry** or **unfolding symmetry**; If this replication
is exactly the same at every scale, as in the Menger sponge,^{[6]} 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.^{[1]} This power is called the fractal
dimension of the fractal, and it usually exceeds the
fractal's topological dimension.^{[7]}

As mathematical
equations, fractals are usually nowhere differentiable.^{[1][4][8]} 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.^{[1][7]}

**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
Bolzano, Bernhard
Riemann, and Karl
Weierstrass,^{[9]}

The word *fractal* appeared
in the 20th century with a subsequent burgeoning of interest in fractals and
computer-based modeling in the 20th century.^{[10][11]}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.^{[1][12]}

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."^{[13]}

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."^{[14]}

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."^{[15]}

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."^{[16]}

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.^{[1][2][3]}

Fractals are not
limited to geometric patterns, but can also describe processes in time.^{[6][4][17][18][19][20]}

Fractal patterns
with various degrees of self-similarity have been rendered or studied in
images, structures and sounds^{[21]} and found in nature,^{[22][23][24][25][26]} technology,^{[27][28][29][30]} art,^{[31][32]} architecture^{[33]} and law.^{[34]}

Fractals are of
particular relevance in the field of chaos
theory, since the graphs of most chaotic processes are fractals.^{[35]}

ÒA 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 canÕt 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.

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