How Can a Mystery as Large as the Universe Connect to a Being Like Me?

Hubble Image of Spiral Galaxy

Hubble Image of Spiral Galaxy

Another reason that fractals matter, and why I want to explore them, has to do with a very old spiritual quandary. Human beings have long imagined the possibility of an infinite being, a divine being, who is creator and sustainer of the universe, commonly known in our culture as God. Not all human beings resonate with this idea, and the details vary as to what God might be like, but most peoples have some sort of divine being or beings as a part of the stories and values of their culture. I have spoken of God as the larger Mystery of which we are a part.

Many human beings have also imagined that they can have a personal relationship with this divine Mystery. Most cultures have forms of prayer to entreat help from God, and forms of prayer to thank God for help given. Many people also directly experience the presence of the divine in their hearts, the intimate presence of the Mystery.

I know that I have had moments in my life when I felt held in the arms of divine love, that I felt cared for by a Mystery greater than myself. Those feelings are so tangible, that they help me get through my most difficult days. When I feel afraid, I can trust that all will be well, because of that tangible presence of love. When I feel overwhelmed, I can keep on walking forward, held in the memory of that love. But if God is infinite, or if the Mystery is all that is, how can that be? How can a Mystery as large as the universe connect to a being like me, small as a speck of dust?

Fractals have given me a new way to think about this spiritual dilemma. A fractal is a pattern that repeats itself, from an infinitely large scale to an infinitely small scale. What if God is a fractal? What if God is a pattern that repeats itself from the infinitely large to the infinitely small.

Here is how I imagine it. The divine pattern is a pattern of life and love and creativity—it expresses itself in the creative unfolding of the universe. It repeats in the attractions of planets and stars, and in the evolution of life itself. Because fractals continue to repeat in self-similar ways at all scales of size, the same divine pattern emerges at the size of our own human consciousness. We can find that pattern in our hearts, the expression of life and love and creativity. Thus we can find the Mystery in our hearts, as well as in the larger whole.

In this way, fractals offer a solution to the old quandary of an infinite God relating to a tiny human being. By understanding fractals, my intellect can make sense of what my heart experiences of the Mystery. It helps me to make sense of the tender feelings I feel, and to welcome their help for the troubles that life brings. I feel less lonely, when I feel connected to the divine love. It becomes possible to believe that I matter, that I am not just a speck of dust in a vast uncaring universe. I have within me the fractal beauty of the infinite MysteryFerns More DSC03600

Fractals In the Forest

A fractal is a pattern that repeats itself, from an infinitely small scale to an infinitely large scale. We see in the patterns and shapes of nature that there is self-similarity at all levels.

This has both practical and mystical applications. I learned about one practical application from a documentary called Hunting the Hidden Dimension. A group of scientists concerned about global warming was trying to determine how much carbon dioxide was absorbed by trees in the rain forest. They could measure the carbon capture of a single leaf, but how could they count the number of leaves in the forest?

They had an idea. They started by measuring the circumference of all of the branches on a single tree. Because of the fractal nature of the tree, the branches form a regular pattern, dividing at certain intervals into smaller and smaller branches. By measuring every branch, they could determine the ratio between branch sizes. Then they took it one step further. They measured the trunks of all of the trees within a given area.

Imagine it with me if you will. If we walk through a forest we see trees of all sizes—small saplings, huge old giants—there is an endless variety of sizes all around us, seemingly in a random pattern.

Trees MJ DSC03686But it turns out it is not so random. The ratio of tree sizes in an area of forest is approximately the same as the ratio of branch sizes on a single tree. There is a pattern to it. And by learning the patterns, the scientists could compute how many leaves were in the forest, and how much carbon dioxide they would absorb.

Now when I walk through the forest near my home, I remember this experiment, and look with wonder at the trees around me. What seemed chaotic and random before, is now bursting with new meaning, full of patterns that start to reveal themselves to me, as I gaze with deeper insight. My experience of the trees’ beauty expands, and I feel a growing sense of awe.

I find myself looking for fractal patterns everywhere. This new understanding has changed the way I see the world. And it is not only visual. I can feel the patterns in bark with my fingertips, and I start to listen for patterns in the sounds I hear as well. Next time you look at a spider’s web, or gaze into the clouds in the sky, watch for the fractal patterns.

Our ability to measure fractal patterns in the natural world has also given us the ability to create digital worlds that remind us of our own. Fractal formulas are used to generate computer graphics that look realistically like mountain ranges, and rivers, and forests, and clouds. That wasn’t possible just a few decades ago.

Fractals have been used to design antennas in greatly reduced sizes, which enabled the creation of the next generation of cell phones and other electronic communicators. Fractal geometry is enlarging our ability to create new devices that work better, because they follow patterns that resonate with the natural patterns around us.

Why Fractals Matter-Reading the Book of the Universe

Why should fractal geometry matter to those of us who are not mathematicians? First of all, fractals give human beings a new way to look at the universe. When we can describe something, we can see it better than if we cannot describe it. Because we are better able to see the natural world, fractals enable us to have a deeper relationship to the natural world.

It reminds me of learning to read a book. In order to read, we need to understand the patterns of squiggly lines that form the letters of the alphabet. And then we need to understand how those squiggly lines are combined in multiple ways to form words, and then sentences, and so on. A person who cannot read may look at a book, and it might seem beautiful, or there might be pictures in it to be curious about, but that person cannot understand what it means. When we learn to read the patterns of squiggly lines, the book becomes a doorway into a whole story, and suddenly we have access to a wealth of ideas and thoughts and understandings.

The natural world is like a sacred book; it is the place where we search for truth and beauty and goodness. We might say that the universe itself is our bible. We don’t have to understand the world to appreciate its beauty. Even a baby can laugh with delight at the bright colors of flowers, or try to catch a butterfly. But the more we understand the natural world, the deeper can be our appreciation, and the more its mystery opens up to us. Fractals help us to read the book of the universe.

Fractals give us a way to measure and describe the complex patterns in the natural world. Fractal geometry, in fact, reveals to us the inherent patterning that permeates the universe. A fractal is a pattern that repeats itself, from an infinitely small scale to an infinitely large scale. Complex entities are created from simple designs extended out to many dimensions. We see in the patterns and shapes of nature that there is self-similarity at all levels. Ferns DSC05288

Fractal Spirituality-The Infinite Within Our Souls

Every
Child
Has known God,
the God who knows only four words.
And keeps repeating them, saying:
Come Dance with Me.”
Come
Dance.
                                                     Hafiz

How can a Mystery as large as the Universe find expression within the smallness of our souls? How can we tiny beings experience the Infinite? I found a new way to think about this question when I learned about fractal geometry. Fractals are never ending patterns, with self-similarity at all sizes.

Benoit Mandelbrot was the mathematician who first coined the word fractal, and brought to our attention the possibility of exploring the geometry of the natural world. Fractal comes from the word for broken, and Mandelbrot wanted to explore the rough shapes of nature. Traditional Euclidean geometry could not describe these shapes. Mandelbrot wrote: “Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in straight lines.” Fractal geometry enables scientists to describe the world through complex mathematical formulas.

Mandelbrot Set by Wolfgang Beyer, Wikimedia Commons

Mandelbrot Set by Wolfgang Beyer, Wikimedia Commons

I am not a mathematician, but I was curious to see if I could make sense of the math. Perhaps you have heard of the most famous image associated with fractal geometry, what is called the Mandelbrot Set. It has a dark area that looks a bit like the shape of a bug, with a large round spot, and a small attached round spot. But the edge is what makes it fascinating. It is filled with beautiful complex curlicues that continue to be complex curlicues no matter how much the set is magnified. In fact, it continues through infinite magnification.  (For more images of magnification, see here.)

Mandelbrot Set Magnification by Wolfgang Beyer, Wikimedia Commons

Mandelbrot Set Magnification by Wolfgang Beyer, Wikimedia Commons

But “What is it?” I wondered.

If you have math anxieties, I promise you, I am only going to give a simple explanation with ten sentences. You are also welcome to skip the next paragraph.

A Mandelbrot Set is a diagram of a mathematical equation. The equation is: Z = Z2 + C. You insert a number into the equation, and the equation computes it to a new number. Then you start the equation all over again with the new number. Now here’s the interesting part—we don’t care about the answer. We care about how many times you can repeat the equation, with the number you started with. If you can repeat it only a limited amount of times, that number is part of the Mandelbrot set—and it becomes a black dot on your diagram, part of the black spot. If you could repeat it an infinite amount of times, that number is outside the Mandelbrot set. Depending on how quickly it gets to be infinite, it can be given a different color. Only computers can actually do all of these calculations, but they do them very well, and so we can see the images formed by the equation.

Okay, I’m done with the math part now. (I didn’t go into complex numbers or imaginary numbers, so my apologies to anyone who really knows about all of this. But for the rest of us, it is probably more than enough anyway.) The thing is, when Mandelbrot computed his formula, it created a picture filled with beautiful complex curlicues. And no matter how many times you magnify the picture, you will continue to see similar complex curlicues.

More tomorrow…

 

 

For those who would like more detail about the Mandelbrot Set, see the website Introduction to the Mandelbrot Set: A guide for people with little math experience by David Dewey.

A Meditative Journey Into Your Own Emptiness

Walking MD DSC05318

Margy Dowzer Photo

I invite you now to take a small journey into your own emptiness.  I invite you to sit comfortably and notice your breathing.

Imagine that you are walking at a leisurely pace down a path in a meadow.

Ahead of you, you see an old oak tree by the path.  As you slowly come near to it, you find a pile of stones in the path blocking your way.Stones in Path MJ DSC04228_2

I invite you to take a minute to pause and remember the unfinished projects in your life, the clutter you want to clear away. Imagine them like these stones in the pathway.

As you remember each project, imagine that you lift a stone from the road, and place it in a huge pile under the oak tree.

After a few minutes, if you haven’t finished clearing all your projects and stones, I invite you now to move just one more stone under the oak, to represent all the rest.  

Then slip around the outside of the path to continue on. And imagine the wind of your breathing gently blowing over the top of the stone pile.

Now you are walking further on the trail, past the oak tree, through a field. Ahead of you, you notice a majestic willow tree next to a peaceful brook. As you reach the brook, you find a large empty basket floating in the water.

I invite you to take a minute to notice the unresolved emotions you carry in your heart, the old conflicts and entanglements. Imagine these old emotions like objects in a back pack, weighing you down.

As you notice each old emotion, take it out of your back pack, and place it gently in the huge basket in the brook.

After a few minutes, if there are still more unresolved emotions in your heart, I invite you to place one more object in the basket, to represent everything else you cannot resolve at this time.

And now imagine the water of the brook gently taking away the basket in the wind of your breath, until you can no longer see it.

River MJ DSC00578

Continue on the trail, past the willow tree. Ahead of you, you now see a garden of rose bushes, with brilliant red flowers, and thorns. As you come close to the roses, you see that they are in a circle, around a large compost heap.

I invite you to take a minute to notice all the ways that you try to get other people to behave according to your expectations. Notice the ways you try to control, to be effective, or to be right.

Red Rose MJ DSC00893Imagine each of these maneuverings like old rotten fruit in your backpack. As they come to mind, toss the rotten fruit lightly over the roses and thorns, into the compost.

After a few minutes, I invite you to take your backpack itself in your hands, to represent all that is unconsciously driven in your heart, and toss it too over the roses into the compost.

Imagine that the roses are growing taller and more beautiful, and they are swaying slightly in the wind of your breathing.

Then you continue on the trail.

Ahead of you there is an apple tree. Under the apple tree there is a shady spot.

I invite you now to think about all the roles you fill in people’s lives. Imagine each of these roles as a heavy coat you are wearing.

As you think of a role, take off that coat, and place it on the ground beneath the apple tree.

Finally, see yourself in a comfortable summer outfit. You have nothing to carry.Apple DSC01750

You reach up and take a bright ripe red apple from the tree. You take a bite, and it is sweet and crisp. You sit down on the ground cover formed by the coats.

You hear the wind of your breathing blow like music across the grass. It is enough.