Begin by knowing that a fractal is one of the two kinds of geometry present in our world. And yes, no matter what people say, in our world, there are only two major categories of geometry. Fractal. And other than fractal. Which, to simplify our discussion, we'll refer to as fractal (non linear) and classical (linear).

Now before proceeding, let's first define the word, "geometry." What is "geometry?"

Geometry is the measurement of visual shapes and patterns. Spaces, originally. Moreover, you can know this for yourself by dissembling the word. The "metr" part of the word comes from an old english word which meant "limits" or "boundaries." Thus the "metr" part tells us what we do with geometry; we measure the limits of something. As in meters, and metrics, and metronomes.

The "geo" part then comes from a word referring to the space in which we live; Earth. Thus, the "geo" part tells us what we're measuring; we're measuring the visual patterns of the spaces in our world, as in, "geography" (measures of the visual patterns of earth spaces) and "geology" (logical words about the visual patterns of earth spaces.)

Expanding on these ideas a bit, it might be helpful to know that in ancient Greece, measuring the visual patterns of spaces (geometry) was considered sacred knowledge. A science which the ancient Greeks believed allowed them to peer into the very nature of the spiritual world. Or at least, a way in which they could know there was a spiritual world. And in truth, I am saying very similar things here about fractal geometry. Specifically, that it has sacred qualities, one of them being, that it reveals the underlying structure of natural objects. Or the true visual essence of natural relationships, if this is an easier concept for you to grasp.

As for what makes these two geometries different, know I have spent several decades looking for a way to define fractals. To my surprise, I discovered that we can completely define the essence of these two things using only three qualities. Moreover, two of these qualities are the same for both geometries, while the third is what makes them different. Let's start with the two qualities which are the same.

The first quality. Both classical geometry and fractal geometry refer to recognizable shapes. This is the first quality. In order for it to be geometry, we humans must be able to recognize these shapes. At least, in theory. And while it might take a bit of work to learn how to do this, this quality must always be present. No recognizable shape. No geometry.

The next quality? Both classical geometry and fractal geometry must refer to recognizable shapes which always repeat. This is the second quality. In theory, all classically geometric shapes can manifest in an infinite amount of ways. So too naturally occurring fractal shapes, which can also manifest in an indefinite variety of ways.

The thing to know here is that when I say, "always repeat," I am basically referring to that geometric shapes can manifest over and over and over and still not exhaust the possibilities. This potential for repetition is what this second quality refers to. Moreover, it also refers to that in geometry, these repetitions are ongoing. Squares to circles. Waves to water erosion. Cubes to trapezoids. Pendulum swings to mood swings. All these geometric repetitions recreate themselves in endless ways.

Note that it is this second quality which disqualifies as fractal most man made objects, such as couch covers and wall paper. At some point, the detail within these things ends. Whereas, with the nature of something like stock market sales, while humans do initiate these transactions, the resultant cost variations are indeed fractal. Why? Because there is an infinitely detailed nature hidden within these variations, a natural complexity which defies linear analysis. As well as that there are endlessly recurring, recognizable patterns within what appears to be an infinite quantity of detail.

So "always repeat" is the second quality. Then there is the third quality, the one which makes these two geometries differ. What makes these two geometries different?

Classical Geometry refers to "recognizable shapes which always repeat identically," while Fractal Geometry refers to "recognizable shapes which always repeat differently." This is the entirety of geometric definition. Simple, really. However, like E=MC2, this very simplicity is what makes it so hard to grasp.

All this said, what exactly makes these two geometries so important to education? Sacred, even?

Their importance lies in that they are the only two ways in which we can truly learn to know and recognize things. Moreover, in order for us to know the nature of anything in our world, we must be able to recognize the visual patterns of relationships present within these things. No visual recognition. No real understanding.

In essence then, both geometries hold the potential to be our teachers. They are, in fact, the essence of being a teacher, regardless of what subject is being taught.

Why this need to be able to visualize a thing's nature in order to learn what it is? Because all things within our world are made of light. And while much of this light falls outside the range of our natural eye sight, with analog aids (like televisions and radar), we can see all of these visual patterns. Which is why we Emergence Practitioners define "human consciousness" as "the skill of picturing movement."

Here, "picturing movement" is just a way to refer to that we know things only by measuring light. No coincidence, "picturing movement" is yet one more way to refer to what happens in fractal geometry.

Some would now ask, but what if an oak leaf (which is fractal) were perfectly still. Would this mean it was no longer fractal?

My answer? No. The leaf would still be fractal. Why? Because the thing which enables us to see this leaf is that something in this measurement process must be moving; either us, or the leaf, or both.

What I am saying here is, the movement I have been referring to here is "relative" movement. The very word which defined Einstein's genius, in fact. This means, whenever we look at a thing, the quality which enables us to see this thing is "movement."

What I am saying is, either us, or the thing we are looking at, or both, must be moving. No movement. No access to the nature of the thing.

With oak leaves then, if an oak leaf were ever to be perfectly still, and if we could ever keep our eyes perfectly still as well, we would quickly lose our ability to see this leaf. Why? Again, no movement, no vision. No vision, no consciousness. This, in fact, is the nature of the screen of the mind going blank.

In real life, though, everything is always moving in relation to something else, molecules to solar systems. Thus, this stillness never actually occurs. Except in our minds, of course. Moreover, because it does not, it never occurs to us what we would happen if it did occur. Invisibility, to be exact. Again, the blank screen of the mind.

Finally, there is one very important idea you must know in order to access the power of fractal geometry. This idea? That in order to see the truth within things, you must, in some way, be able to picture the visual pattern of movement underlying this thing; peoples' faces to academic grades; personal psychology to astrophysics. No moving picture. No understanding. Period. No exceptions.

So what about logical understandings? Can't we logically understand things? Good question. To see the answer, imagine you are visiting a world famous art museum, and as you stand in front of what should be the Mona Lisa, you see only an empty frame with a descriptive plaque beneath it. Now add to this that you are one of the few civilized human beings whom has never seen a visual representation of the Mona Lisa. And that you have been asked to visually describe the Mona Lisa. How well do you think you would do?

The truth is, even if you were to have learned every logical truth about this painting, and about the painter, and about his times, you still would fail miserably. Why? Because even with the myriad of logically true things you could say, given the rest of eternity, you could not know the visual essence of this painting. Why not? Because human consciousness relies entirely on how our minds fill in that frame. Moreover, in order to know the nature of any naturally occurring thing, we must be able to fill in our minds with recognizable patterns of movement.

What I am saying here is, the most beautiful logic in the world is still mere captions to these pictures. Why? Because the very nature of beauty itself is rooted in recognizable visual movements. Including in the Mona Lisa herself. As well as in the nature of each and every human virtue, including the nature of truth.

Herein lies the power and the beauty of fractals. Amazing really. But then, we live in an amazing world, now don't we?

Warmly,

Steven