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May 1995, Volume 2 Nr 9, Issue 21


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.


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.


Self-similarity is a defining criterion of fractals. Without self-similarity the essence or repetitive basic structure of a fractal would be missing. The Sierpinski triangle in three-dimensions would not exist as the tetrahedron would not repeat into new tetrahedra both contained within and without. Such is the nature of the love that is us and the I that is we. If love is the essence of what we are, imagine the good in the world when we realize that I is the essence of what we are? Similarly, we are the essence of what I am. The self-similarity of the larger community that is the world is the same love that we as individuals are.

Richard Moss MD, in The I That is We writes, "By its very nature the dynamics of group energy held in a unified focus at the highest possible quality of consciousness creates a force field that transcends the issues of individual human concern. Areas of the human psyche inaccessible to us through ordinary introspection modes (i.e., by ourselves or in a one-to-one dynamic) can be reached into much more rapidly and safely in the group dynamic."

Dr. Moss is referring to those moments when the love that is I becomes the we that is love. Witness the wonderful response and help offered by the group fractalian dynamic during the Oklahoma City bombing tragedy. In times of great need the large we seems to always be present. It has to be. And, it is made of many I's. What I am suggesting is that we lose track of this dynamic though it actually works most of the time.

"Chaos-induced complexity is also partly responsible for our aesthetic responses. Chaotic feedback makes, for example, the amplified guitar playing of Eric Clapton a more exciting, complex sound. And the spontaneous complexity generated in self-organizing systems makes a tree more beautiful than a telephone pole." Chaos produces both beauty and love.


"In our description of nature the purpose is not to disclose the real essence of the phenomena but only to track down, so far as it is possible, relations between the manifold aspects of our experience." Experience is relationship. Relationship is experience. Our relationship to each other, to money, work, school, church, community... everything determines our experience.

Euclidean methodology prompts us to control the other in our relationships by always being on guard, attacking at the slightest provocation in order to gain advantage or security.

Fractalian methodology brings us freedom through the knowledge that people, events and things are outside the scope of our control. Surprise. We can however, control by choice our outlook by changing our view of what a relationship is. Niels Bohr might easily have referred to love when he searched for the relations between the manifold aspects of our experience. "Look to the essence of a thing, whether it be a point of doctrine, of practice, or of interpretation."

"Before heaven and earth had taken form all was vague and amorphous. Therefore it was called the Great Beginning. The Great Beginning produced emptiness and emptiness produced the universe. The combined essences of heaven and earth became the yin and yang, the concentrated essences of the yin and yang became the four seasons, and the scattered essences of the four seasons became the myriad creatures of the world." Yin and yang is fractalian chaos and order of being.


My old thesaurus equates altruism as selfless, uncalculating, ungrudging, self-forgetful and unselfish. Altruism is the giving of oneself for the betterment or well-being of another. Altruism is both selfless and a fractalian activity. Offering one's services does set in motion a feedback loop where the help given comes back as a good feeling. In turn, this good-feeling fosters the next instance of altruistic activity both in the original giver and the original receiver. The essence of this altruism is love, a self-perpetuating activity which is for giving away. Thus a web of goodness grows in all directions and multiplies. It grows the way a spider's web grows, with self-similarity and feedback in the structure.

Gerald G. Jampolsky and Diane V. Cirincione in Change Your Mind Change Your Life write, "When we remember to nourish the soil with our forgiveness, we experience our gardens as beautiful, lovely, and majestic, with no weeds at all." They are talking about cultivating the gardens of peace and love with the repetitive hoeing of good works..

A Child's Fractal

Recently, our family had a difficult morning. We all occasionally have a hectic start to our day. This was one of those days. Tempers were elevated. When it was all over and the players began to calm down, our son, Dylan said, "You can't control me and I can't control you. But, we can only control ourselves." At that moment we could say that this young man switched from a Euclidean to fractalian point of view.

I explained to Dylan what I was writing in this issue of Metaphoria. I explained to him what a fractal was. Using the computer and a children's artist program, he came up with a  fractal based upon the word love. While simple in nature, it does illustrate what I have been trying to say. Maybe, we should listen to children more often. They seem to be good indicators when we forget who we are and when we slip into the black-and-white Euclidean outlook on life.


We finally come to the title of this issue. Fractuality is a combination of fractals and spirituality. Spirituality has different definitions for different people. I've often thought of spirituality in terms such as: "I am in good spirits" or "I am in poor spirits." When we compare the two methodologies, we see how our spirits and that of our immediate community and the world lift with incorporating fractalian concepts..

It has been difficult writing mathematically on a spiritual topic. Or, am I writing spiritually on a mathematics topic? Regardless, the topic of fractuality has taken me into unique new territory. Metaphoria as the name implies is about metaphor. I would like to hear your responses to this issue.


How do we create peace and happiness?
With weapons?
Of course not.
With money?
In some cases, but not all.
But in love, in sharing other people's suffering, yes.
Good motivation is a sound basis for peace.

The Dalai Lama

1995 Jozef Hand-Boniakowski

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