Recently,
JeanneE and I attended a talk on fractals. Evan Maletsky
from the Department of Mathematics and Computer Science
at Montclair State College in New Jersey made the case
that the centuries old mathematical view of the universe
is stuck in Euclidean geometry. Maletsky helped produce
the text, Fractals for the Classroom. His thinking is
that Euclid's world which involves points, lines, line
segments, planes, squares, circles, space, etc., is a
world view which while useful is blind to in-depth
explanations of nature and the universe. While we may
remember our secondary school geometry class, we probably
never heard of or know little about fractals.

*Euclid's Blinders *

In our overzealous attempts at describing and explaining
the world around us, we fall into what I call the
Euclidean methodology of the black and white universe. In
this geometry there is no leeway in seeing a triangle as
anything other than a three-sided polygon that contains
three straight line segments which meet at vertices
forming three angles which add up to one hundred and
eighty degrees. This flat two-dimensional space makes
sense as long as the universe we observe is flat and two
dimensional.

Likewise, our thinking, when following along similar
rigid constructs leaves little room for anything outside
the boundaries and definitions of that space. Our
worldview and the process by which we come to conclusions
is stuck in the limitations of Euclidean thinking.

The ego resides and wishes us to function within the
Euclidean space since its priorities lie there. For
example, like the triangle, the ego's black and white
methodology suggests that when it (the ego) is
threatened, then its response is a defensive posture
invoking attack on the suspected perpetrator of the
threat.

*Euclid and the Ego*

Euclidean geometry postulates: if the angles of a closed
figure add up to be one-hundred-and-eighty degrees, then
we have a triangle. Similarly, the ego's thinking
postulates that if I perceive a threat, then I must
attack in order to protect myself. The ego believes, just
as in Euclid's geometry, that a given scenario has only
one possible response. Let's call it the triangle
response, a knee-jerk reaction to events outside
ourselves. The if-and-then triangle response is orderly.
Predictable and definite yet, it leads to turmoil and
lack of inner peace.

As we further examine Euclidean geometry we discover that
although the angles of a triangle all add up to be
one-hundred- eighty degrees, there are many different
triangles. We discover that there are scalene, acute,
right, equilateral and isosceles triangles. These
triangles have specific characteristics. Yet, we easily
recognize them as triangles. Every one of them fits into
our definition of what a triangle is.

Similarly, as we go through childhood our experience with
life adds to our familiarity with threat and fear. These
shapes are ingrained into our psyche through the
intervention of the ego. Our pain and pleasure defines
our responses to a multitude of scenarios. The dangers
that the ego always sees are the triangles, regardless of
the type and, the response is always the same: attack.
The ego is constantly on the alert for triangles and more
than eager to use the triangle response to call it the
way it sees it. Jealousy, fear, suspicion, hate, envy,
guilt are some examples of the ego's vigilance for the
presence of threat.

*Polygon Mechanics *

Consider the diversity in the black-and-white geometric
possibility. As we add more sides to the polygon, we open
up new opportunity. There are many constructs for
polygons. They can take on an increasing number of sides.
Four sided polygons are called quadrilaterals and include
the square, rectangle, rhomboid (diamond), trapezoid,
parallelogram and irregular quadrilaterals which may be
convex.

In Euclidean geometry complexity increases with
similarity. One figure may fit into more than one
category. We find that while the definition of a
quadrilateral doesn't change, one type of quadrilateral
may also be another. A square for example is a rectangle
but, only special cases of rectangles are squares. A
square is also a rhombus but, not all rhomboids are
squares. A rectangle is a parallelogram but, only special
parallelograms are rectangles. Regardless, each is a
quadrilateral.

*It's Still Euclid *

As our life experience in Euclidean methodology become
more involved, that is, our relationships with people and
things become more and more complex and egos interact,
our ego looks to make sense of it all through the
Euclidean constructs that it learned throughout its
existence. When the ego recognizes a polygon for what it
is, no matter how sophisticated the shape, no matter how
multi-sided the human interactions, when it recognizes
the underlying threat as a pattern of fear, it responds
accordingly. It responds with recoil, defensive posturing
through offensive response and the self-reward for seeing
itself acting correctly.

When we delve into Euclidean geometry, we recognize that
all the polygons, no matter what their shape or number of
sides; no matter how complicated, can be broken down into
triangles. While our life situation may involve a
multitude of interrelationships at home, school, work,
etc., no matter the intricacies of our day-to-day lives,
the ego instantly analyzes the complex situation and
inevitably recognizes that just like the triangle is the
basis of all polygons, fear is the basis of its
existence.

Just as the triangle stands out by definition in the
makeup of a multiplicity of polygons and is always
recognized, the ego always responds with attack at the
slightest perception of fear. As the triangle is the
basis of all polygons, fear is the basis of what the ego
is. That fear is not the essence of what we are.

*A
Closer Look *

Throughout the last few years, Metaphoria has examined
the premise that events outside ourselves are reflections
of our internal state of mind. At one point we defined
spirituality as our moment-to-moment internal state of
affairs. If our reality constructs are strictly along the
lines of the Euclidean methodology of black and white and
the triangle response then we are doomed to robot like
mechanical responses to anything and everything around
us. Our operating parameters are thus predetermined,
fixed and pitiful in the sense that events outside
ourselves determine what we see and feel.

What if however, we choose to make a closer examination
of Euclidean geometry? What if the space upon which the
triangle is drawn is curved? If we draw a large triangle
and place it over a large sphere, then when using a
protractor, measuring and adding the three angles, we
come to the startling conclusion that the sum is greater
than one-hundred-and-eighty-degrees! How can the sum of
the angles in the triangle add up to be greater than
one-eighty?

The flaw in our thinking is that the universe is not so
simple. It is more along the lines of infinite shades of
gray. In Faust, the poet Goethe (1749-1832) writes:

*Gray, dear friend, is every theory.*

When we choose to accept that Euclidean methodology,
though useful, has limited us in our ability to deal with
the universe, we open up the incredible resources and
possibilities of multidimensional mathematics and fractal
geometry. We open up the possibility that the ego's
geometric methodology of right and wrong is just wrong!

*Fractal Possibility *

Theoni Pappas in The Joy of Mathematics, writes,
"The discovery of non-Euclidean geometries has
introduced new objects that depict the phenomena of the
universe. Fractals are such objects." It is my
intent in this issue of Metaphoria to suggest that while
fractals may be a mathematical concept, they are also, by
the inherent nature of their existence, definition and
description an important psycho-spiritual metaphor of
what we are and what we might become.

Theoni Pappas writes,

"It had been felt that the orderly
shapes of Euclidean geometry were the only ones
applicable to science, but with these new forms nature
can be viewed from a different perspective. Fractals form
a new field of mathematics - sometimes referred to as the
geometry of nature because these strange and chaotic
shapes describe natural phenomena such as earthquakes,
trees, bark, ginger root, coastlines, and have
applications in astronomy, economics, meteorology and
cinematography."

I wish to add that fractals describe who
we are. It is my contention that fractals have vast
psycho-spiritual implications. They offer a rich insight
into the essence of our being. Let us begin with a brief
examination of the fractal view of the universe and
transition into the fractal methodology of inner peace.

*Mandelbrot *

Benoit Mandelbrot first described a fractal as an object
whose detail cannot be diminished nor lost as we magnify
it. Neither can the essence of the large entire construct
be lost as we examine the smaller and smaller detail of
the original.

The larger detail of a fractal's entirety is duplicated
and contained within the smaller and the smallest subset
of the original going down toward infinity and the
infinitely small.

A classic example of a fractal is the Sierpinski triangle
which is generated by taking an equilateral triangle in
three dimensions, shrinking it, flipping it, rotating it
and placing it inside the original and then repeating the
process forever. The figure below shows the Sierpinski
triangle. If we magnify the figure, we come up with a
larger triangle that contains all the detail of the
original with the smallest elements previously
indistinguishable now clearly containing the original.
Enlarged to the point where we can recognize the smallest
shape within, we see yet another copy of the original.
And on to infinity.

Let us imagine we are like the basic three-dimensional
equilateral triangle (called a tetrahedron). Only one
face of this tetrahedron is fear, the remaining sides and
the depth are love, the essence of what we are. Wherever
fear is perceived let us respond with love as the
fractalian essence of our being which is in endless
supply. Love is the answer. Rather than recognize fear,
we supplant it with love and its infinite iterations
which include: kindness, friendliness, helping,
forgiveness, forgetting, letting go, guiltessness,
self-esteem, etc.

*The Coastline*

Consider for a moment, the coastline of an island nation,
say Madagascar. The Euclidean view of the coastline's
length would be akin to simply measuring off the distance
around the island. The fractal view suggests however,
that the coastline is far from straight. The fact that a
coastline is usually rugged with twists and turns both
small and large leads us to conclude that the actual
length of the coastline is much larger than the
straight-line distance. The more we examine the twists
and turns, the more we realize that they are smaller and
smaller copies of the larger original, making their way
deeper and deeper into the coastline itself. Depending
upon how small a yardstick we choose to measure the
coastline, the numeric outcome becomes larger. The final
length of our measured coastline becomes bigger and
bigger as the essence of what we measure becomes smaller
and smaller.

Benoit Mandelbrot describing such a coastline in
Fractals: Form, Chance and Dimension writes, "We
will see that...the final estimated length is not only
extremely large but in fact so large that it is best
considered infinite."

It is best then, when using mathematics as a model, to
perceive the universe as fractalian instead of Euclidean.
People are not simple, easily measured entities. Neither
is what we see. Our experience of external events
heretofore easily interpreted or explained through the
framework of our black and white reactions, i.e. the
triangle response, is highly irregular and mostly
incorrect.

We are capable, infinitely capable, of taking control of
what we see and how we respond. Just like the actual
twists and turns of the complicated fractalian coastline
of Madagascar, our being is an apparent series of random
events initiated by the basic sameness in design that
continues indefinitely. The essence of what we see is the
essence of what we are. And, the essence of what we are,
when seen through the fractal lens, is what we see: love.
Only when we slip into the ego's pervasive and
quantitative Euclidean notion of survival do we tend to
straight-line measure the people and events around us.
The outside world has an event and we robotically react.
We perceive a threat, thus we respond with attack.

How magnificent to think that the inherent beauty,
intricacy and infinite length of the fractalian coastline
can be transferred onto the recognition that love is
infinite and it is what we are. There is chaos in the
coastline. Yet, there is beauty and orderliness.

When we respond to fear with the fractalian certainty of
love, we may experience chaos as we are on unfamiliar
territory. The ego tries desperately to seduce us into
our old patterns. The ego does not like taking risks nor
is it comfortable with the ambiguous or chaotic. With
firm aplomb, we soon see the beauty of the infinite
coastline and the chaos takes on a structure that expands
the love in the universe. The love that is us is
self-similar, ever present and self-organizing. We simply
lose sight of it from time-to-time.