Fractals Teach Us That We Matter

Oak MJ DSC03499There is another reason why fractals matter. Fractals teach us that we matter. By becoming aware of the fractal patterns throughout the natural world, we can see that all things are connected. The circulatory system of the human body branches out like the limbs on a tree. The patterns of waves on the shoreline are similar to the patterns of radio waves beaming through space. Even though we are infinitely small in comparison with the rest of the universe, what happens on a small scale reflects what is happening on a larger scale.

Some of these patterns may seem to be unchanging and eternal, but there is also unpredictability in the system. Scientists use the word chaos to describe this unpredictable behavior. Without chaos, there could be no creativity, because creativity means the emergence of something new and unpredicted.

Perhaps you may have heard of the “butterfly effect.” This phrase was used by Edward Lorenz to describe the impossibility of predicting the weather, despite creating complex computer models that looked at multiple variables. Lorenz found that a small change in the initial conditions would produce large changes when the patterned cycles repeated many times. It was expressed in metaphor as the butterfly effect: a butterfly flapping its wings in South America can change the weather in Maine.

We have creative power as human beings. That means that what we do within our patterns has an effect on the rest of the fractal network. We are part of an interdependent web of all that exists. If we change a pattern in our lives, it reverberates through the rest of the web; it ripples out like a stone thrown into a pond. We have the power to create more beauty, more love, more truth, and more goodness in the web. We never really know what greater effect we will have on the future of the universe. We cannot control the ripples that flow out. But human beings for centuries have observed that acts of kindness multiply into more kindness in the world.

There is a fable told by the ancient Greek storyteller Aesop:

A sleeping lion was awakened one morning by a mouse running over his face. The lion became so angry that he grabbed the little mouse in his paws and was about to eat him up. The terrified mouse begged for his life. “Please let me go,” he cried. “If you do, one day I will repay you for your kindness.” The idea of such an insignificant little creature ever being able to do anything for him amused the lion so much that he laughed out loud and good humouredly let the mouse go.

Then, one day the lion got caught in a snare set by a hunter and was unable to get himself free. The mouse heard and recognized the lion’s angry roars and ran to the spot where he was. He went right to work gnawing at the ropes with his teeth. Soon the lion was set free. “You laughed at me when I promised to repay you. But now you see that even a little mouse can help a lion.” So remember: no act of kindness, however small, is ever wasted.

What we do matters. No matter how small we are, we are intimately connected to the vast universe. We are part of its beauty and its creativity and its love. The patterns in the stars live in the patterns of our hearts. We do not walk alone. These are the lessons I learn from the beautiful geometry brought into the world by Benoit Mandelbrot. May these fractal mysteries teach us ever to be mindful of our power.

Fractal Spirituality-The Infinite Within Our Souls

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

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.

Emptying Into Wholeness

My journey into emptiness brought me four kinds of emptying: first, to clear away some outside clutter of unfinished projects; second, to let go of the inner residue of unresolved emotions; third, to let go of habits of character, tools of maneuvering that kept me from merely being with myself and others; and fourth, to let go of the roles I occupy.

Perhaps by numbering these steps, I give a wrong impression that the journey was linear or planned out. As I looked back over my journal writings, I found a winding and twisting path, with bits and pieces of each of these interspersed with the others.

Thomas Merton writes in one of his journals:
“In order to arrive at what I cannot understand, I must go by way of that which I cannot understand.”

I started out on the journey with a hunger. I did not plan the four kinds of letting go. Rather, I encountered them in the pathway, as I meandered down the road, led by my hunger to find the heart of my own soul. If you took a journey into your emptiness, it might be completely different. But perhaps, by my sharing, you might recognize a few turns along the path.

The journey into emptiness led me into a sense of wholeness, a sense of being with and in my complete self. A sense of openness and relationship to larger being. I didn’t stay there. I got busy with work again. But I learned that I need that periodic emptying in order to be happy in my life, and to be happy in my work. That emptiness is the source of creativity and insight and serenity. That emptiness is the place from which to experience Mystery.

Open Water Crosby MJ DSC05331

Letting Go of the Roles We Play

The fourth step on my journey into emptiness was to let go of the roles I occupy in other people’s lives. One of my roles is to be a minister. Part of being a minister is to carry the needs of other people about whom I will be for them. One day, I may help someone with a painful situation, and to them, I become the epitome of kindness and wisdom. That very same day, however, I might not have time to answer another person’s call. So they call me again to tell me how upset they are—at that moment, for them, I may feel like the epitome of an uncaring world.

All of these emotions are perfectly legitimate for people to feel. After the first call, it would be easy for me to believe that I am wonderful; after the second call, it would be easy to believe that I am horrible. But if I pay too much attention to these outer perceptions, I lose sight of who I really am, in the haze of trying to be what everyone wants and needs.

This is not only a hazard for ministers: all of us have roles we play for other people, expectations placed on us as workers, as leaders and followers, as parents and children, as lovers and spouses. We wouldn’t be ourselves without our roles, but there is a self beneath the roles.

One day, while writing in my journal, I imagined that I slipped out of my roles, as if I were air or water being sucked right out of a suit of armor. I felt astonished that I could so simply leave behind judgment or praise, and find myself like one hovering behind all the roles I usually occupied. It felt peaceful in that space. I was a spirit that was free to stretch and breathe. In that airy space, I caught a glimpse of my deeper self: I felt myself a spark of light, a caller of spirit, beloved of the larger Mystery. I wanted to be silent, to breathe, to give thanks.

Later, when I was walking on the beach, I saw an empty crab shell. I realized that crabs teach us a wonderful lesson about letting go of our roles: they withdraw from their shells at regular intervals—whenever they have grown too tight—to start fresh and soft again. The roles, like the shells, are good things. But in the journey of the soul, we need to slip out of them periodically, or we will be unable to grow.

Air Balloon DSC04664

Clearing Out the Clutter in Our Character

Fence DSC03309As I continued on my Journey into Emptiness, I discovered a third kind of clutter, which might be identified as the clutter in our character. This is made up of all the tools and maneuverings that we use in our relationships with others.

Does every vacation—for those of us who are partnered—have to include at least one big argument? Mine seem to, and they unfailing arise out of habits that might have some usefulness in completing work projects, but not in being with the one I love.  My colleague, the Rev. Barbara Merritt, drew my attention to the advice of modern theologian, Richard Rohr, who claims there are three things we need to let go of here: 1) being in control, 2) being effective, and 3) being right. Then she told a story that I could have written myself. She said:

Having just read that list, I was a passenger in the car as my husband drove us to the movies Thursday night. I couldn’t help but notice that he’d taken a very slow and, I believed, inferior route. I shared this observation with him. He reminded me that I was not driving (number 1, I was not in control.) I replied that he was not being as effective as was possible (I wanted maximum efficiency) and finally I comforted myself with a smug sense of being right. In less than a minute, I had violated all three precepts of ‘letting go.’ I had struck out.

One day Margy and I were driving to the movies, and I managed to strike out in just exactly the same way. From my point of view, we were rushing to get to the movie, and she was foolishly taking a longer route. There was nothing intrinsically unpleasant about the journey, yet I was worried about wasting time on the way to the movie. I was attached to control, effectiveness, and a sense of myself as being right.

Antoine de Ste. Exupery, in The Little Prince, tells us that it is the time we waste on our friends that makes them so important to us. Of course, there is a paradox here. Being with friends is not really a waste of time. But we cannot enter that time with productivity or effectiveness in mind. Wasting time is the essence of a playful spirit. We are not trying to gain anything, grow anything, accomplish anything. We move from doing some thing, to being some one.

On the journey into emptiness, the third step was to let go of these compulsions of my character, and to waste time. Perhaps because I am partly introverted and partly extroverted, I needed to waste time both alone and with others. I needed unstructured solitude, and I needed unstructured time with friends.

How often do we just take time for being with friends? After one such dinner conversation, I wrote in my journal: I feel saner, restored to myself. This is what I am looking for, finally, this sense of being restored to myself. In order to be in the present moment with others, I needed to stop doing and doing. I needed to let go of control, to let go of being effective, to let go of being right.


The quote from the Rev. Dr. Barbara Merritt was from a sermon “The Spiritual Practice of Letting Go” preached February 25, 2001, at First Unitarian Church in Worcester, MA.

Letting Go of Old Feelings

When we encounter old unfinished feelings within our hearts, we may be tempted to try to sweep them up and move them along, to push them out of the way. But unfortunately, this doesn’t really work. I have found two methods that do help me to let go of old feelings.

Creek DSC01248The first is a devotional practice. I turn over my troubles to the larger Mystery of which I am a part. Sometimes I imagine it with the image of a river—I place my feelings in the flowing river and let them float away. By turning them over, I am relinquishing my control over them. I place them in the hands of something larger, in whom I feel trust.

A second method that works for me is related to a form of meditation. I cultivate an awareness of what is going on within my mind and body, including whatever may be troubling me. I let myself breathe with it, and experience it in a state called mindfulness. I give it my attention, without attempting to change it in anyway. I just keep breathing into the feelings, as they are. I let go of my attachment to pushing the feelings away. Eventually, feelings will change and move of their own accord, if we give them this mindful attention.

There is no way to finish with emotions, just as there is no way to finish our house projects once and for all. As long as we are alive, new emotions keep happening. In the journey into emptiness, I am trying to move from doing something about emotions, to doing nothing, to merely being with the emotions. By the practice of being with my inner clutter, rather than doing something about it, I am getting closer to emptiness.