While not known for a finely coiffed hair and flawless visage, Einstein was motivated by the pursuit of scientific beauty. He looked at the universe with awe, always convinced that a great range of physical phenomena could be elegantly explained by a few simple principles. He obsessively contemplated the deep patterns of the universe through creative thought experiments, trying to see through the eyes and feel through the bodies of imaginary observers in fantastical situations. Chasing a beam of light or being in a windowless box accelerating through space were among his creative “thought experiments.” These could have led him to produce science fiction stories, for he was a very good writer, and early drafts of his theories resembled premises like HG Wells’ stories of time travel. His vignettes, like a traveler on a sub-light-speed train passing two bolts of lightning, remind me of brief art videos or an Escher print, challenging us to go beyond usual conceptions. Einstein’s brother and many others noted that Albert had more the temperament of an artist than a scientist. However, for all his creative fantasizing and meditation, punctuated by frequent violin playing breaks, Einstein was after scientific beauty, symmetry, simplicity and deep coherence in the laws of the Universe.

Mathematician Ian Stewart says of Einstein, “He drew radical theories from the simplest of principles and was guided by a sense of elegance rather than a wide knowledge of experimental facts. The important observations, he believed, could always be distilled into a few key principles. The gateway to truth was beauty.” The beauty of which Stewart speaks is a particular kind, not the beauty of hair salons or fashion runways. When scientists and mathematicians like Stewart speaks of Beauty, he means pattern and symmetry.

**Symmetry in Objects**

In mathematics (the science of patterns), symmetry is an immunity to change. Move an object, or do an operation on it, and if the object does not look different, it is a symmetry. Turn an equilateral triangle one third of a full turn (120 degrees), and it looks the same. Turn it two thirds of a turn and, likewise, it still looks the same. The triangle looks the same because it is essentially the same triangle that is was before the rotations. This is rotational symmetry. By contrast, turn a triangle 90 degrees, a right angle, and the triangle tilts to one side, so we say that move is not a symmetry of the triangle.

In our Beautiful Discovery boxes, we explore such symmetries with Escher tessellations, natural wonders like crystals, honeycombs, and origami. The simplest kinds of symmetries are translational symmetries, which we show with grids, tilings, games with tiles, and artful origami of the type that uses repeated patterns across paper. Physical objects such as cardboard pieces on a game board show simple visual and tangible kinds of symmetries that apply to objects.

**Symmetries of rules and laws**

Not only can objects have symmetries, but so can rules, like rules of checkers on a board, or the board games that come with our Beautiful Discovery boxes. The rules are the same anywhere on the board. In the same way, symmetries apply to physical laws, like the laws of motion. Throw a ball and it will behave according to the same laws of motion, regardless of which direction we face. We can rotate any direction before we test the laws of physics with our ball. The laws of physics have rotational symmetry.

Similarly, move 1000 feet to the right and the laws of physics are the same. For example, we can stand on a pitcher’s mound or over there 1000 feet away. It doesn’t matter to physics where we are in space when we throw the ball. The laws hold even when “translated” in space. This is a “translational symmetry” of the laws of physics. Likewise, the laws of physics are the same at any time. We can throw the ball today or next week and the laws of physics governing the ball stay the same. This is another translational symmetry of the laws of physics. It is a translation in time that doesn’t change the laws.

Finding simple symmetries and patterns is essential to learning at the most basic levels. A developing infant is constantly detecting patterns and regularities to predict and cope with the world. For the same reasons, if you suddenly found yourself in another world, you would naturally seek out patterns and similarities to familiar things in your own world. We made pattern seeking the central theme of our Beautiful Discovery boxes because it is a visual and intuitive way to grasp our world and it leads to the most advanced scientific discoveries. While this search for patterns is as fundamental as the first sights, sounds, and textures of newborn babies, searching for beautiful patterns also has been essential at the most advanced frontiers of scientific research.

Searching for symmetries and patterns has been the key to many great discoveries in the sciences. This includes mathematics, which is the science of patterns, and physics, which uses mathematical models to describe the real world. Amazingly, many real-world phenomena are well described by abstract mathematical models of great beauty. In fact, the search for beauty often has been a key ingredient in the discovery of the nature of our world.

**Symmetries of Electricity, Magnetism and Light**

A beautiful discovery in the 19th century was that electricity and magnetism were two sides of the same phenomenon. Building on Faraday’s empirical findings, James Clerk Maxwell was able to show in his equations that changes in electric fields affect magnetic fields in a way that is symmetrical to how changes in electric fields affect magnetic fields. Electric and magnetic fields are simply two symmetrical cases of a more fundamental electromagnetic field (E is electric field, B is magnetic field in the image, a lightwave). Moreover, wave-like disturbances in both fields produce light. Magnetism, electricity and light were not separate and unrelated as was previously assumed. This was a great unification. “It was beauty and symmetry that guided Maxwell and his followers–that is, all modern physicists–closer to truth.” (Wilczek, 2015)

Scientists describe Maxwell’s equations as elegant, beautiful, and symmetrical. Yet, it took much thought to understand the symmetry because it was not like the symmetries of the laws of physics known at the time, in which laws of motion treated time and space as separate. Instead, the equations of electromagnetism led to new ways of describing the relationship of space and time. In fact, deeply contemplating these equations had much to do with Einstein’s revolutionary re-conceptualization of time and space. It was the elegant symmetry of Maxwell’s equations that led Einstein to this dramatic discovery. The symmetry suggested to Einstein that the speed of light was fixed, invariant. This invariance of the speed of light posed a puzzle.

**Symmetries of Motion**

Normally if we throw or shoot an object from a moving vehicle, we add up the speeds. So, if we throw a ball from a moving car the ball goes that much faster to the stationary observer at the side of the road. Say, we have the ability to throw a ball 80 miles per hour from a stationary position. We can get a ball to travel faster by throwing while running. If we can run 20 miles per hour then we can get the ball to reach 100 miles per hour. The ball still is moving 80 mph to the runner, but the ball goes 100 mph to a stationary observer. We add up 80 and 20 mph to get the total speed relative to a fixed observer, say a sitting spectator in a stadium. Relative to a non-moving position, running makes the projectile travel faster. Based on this principle, a javelin thrower can get the javelin to go faster by running while throwing.

If it’s hard to imagine someone able to throw with competency while running, then think of a passenger shooting a bullet straight ahead from a moving car. A bullet goes 1,700 mph relative to the gun itself. If the gun is held by a passenger going 100 mph in a car, the bullet goes 1,800 mph relative to a stationary observer. Similarly, if you throw a ball at 20 mph inside a train traveling at 50 mph, your ball goes 70 mph to a stationary observer at the train station.

**Bizarre Consequences of Light Symmetry**

So, do the same laws apply to the speed of the beams from headlights of a moving car? To a stationary observer standing at the end of the road and facing the headlights. The speed of light is known to be 670 million miles per hour, so that’s the speed of light emanating from the headlights. Since the car is going 100 miles per hour, do we add 100 to 670 milllion miles per hour to get the speed of the light coming from headlights of a moving car?

This question posed a puzzle when physicists first accepted that the speed of light should always travel at the same speed, 670 million mph. Einstein’s reaction to this puzzle was to accept the absoluteness of the constancy of the speed of light. He realized that the absolute constancy of the speed of light must hold also for different observers, for example, an observer at rest or one or an observer moving at a constant speed relative to the observer at rest. This is astonishing because it means both the driver of a car will see the light moving this speed, and so will a pedestrian standing still down the road watching the car approach. We will always observe light to travel at the same fixed speed.

Imagine you are traveling in a car nearly the speed of light, say 80% the speed of light, and you turn on your headlights. How fast do you, the driver, see the light traveling? How fast does a stationary observer see the light traveling?

The answer is that both observers see the light traveling at the same speed, 670 million mph. And no matter how fast we attempt to chase after a beam of light, even if we ouselves are flying at 500 million mph or only 10 mph, the light will be moving away from us at 670 million mph. In the language of symmetry, the speed of light is invariant for all observers.

The way to make this work, Einstein reasoned, was to revise the very model of time and space. We should think of motion not through just space but rather motion through spacetime.

It is easy to notice that we are always moving through time. Just look at a clock or listen to a heartbeat. But would you believe that we also are constantly moving through spacetime? Even if we are standing still on the ground, we still move in spacetime, just with all our motion directed through time. So, we can’t stand still in spacetime. It’s like being a flying drone that can’t stop moving. It can’t even change its speed of 80 miles per hour. The drone can fly straight forward, north, parallel to the ground at 80 mph so long as it doesn’t veer upwards or downwards. It would register as 80 mph to someone watching the drone from the ground, like the police do when searching for drivers’ speed violations on the highway. If the drone veers upwards a bit, then it is moving north at less than 80 mph, despite that its total speed registers as 80 mph on its speedometer. If the drone goes straight upwards at 80 mph, then it makes no progress north, because all its constant motion is diverted upwards.

Likewise, our motion through spacetime is all diverted into time, when we stop moving in space. If we had the amazing ability to divert most of all our motion into space, then we move less through time relative to a stationary observer. If we move through space near the speed of light, then time for us is significantly slower than time is for a person who is just standing on the ground. The faster we move through space, the slower we move in time (a trade-off that is only noticeable for an object moving at extremely high speeds). The wonderful conclusion is that the person moving near the speed of light ages more slowly compared to those who keep relatively still.

**Chasing Moonbeam to Stay Young**

An Einstein chasing a moonbeam is an Einstein who stays relatively young. Is that Einstein’s beauty secret? The beauty Einstein revealed is a deep symmetry in the laws of the Universe. The person standing on the ground is directing all their motion through the time dimension of spacetime. But both persons are moving through spacetime at the same speed, which is the speed of light. This is the absoluteness of the speed of light. For all observers, despite different motion relative to each other, the speed of light is the same, a great symmetry.

The speed of light is absolute, invariant, symmetrical, for all observers, even those moving relative to each other. Einstein added this symmetry to the list of other symmetries in the laws of physics. This was a great triumph for physics and for symmetry.

Many scientists describe Einstein’s special relativity, and Einstein’s reasoning about this topic, in terms of elegant symmetry: “Einstein was led to special relativity by thinking in terms of the symmetry principles that defined light, as it was understood in the late nineteenth century. In fact, this is, in a sense, the greater import of Einstein’s vision. It was Einstein who radically changed the way people thought about nature, moving away from the mechanical viewpoint of the nineteenth century toward the elegant contemplation of the underlying symmetry principles of the laws of physics in the twentieth century.” (Lederman and Hill, 2011)

The symmetry of light is a counter-intuitive concept at first. Most people never learn much about it, and may seem there is little need to, although it is a great joy to think about such symmetries. There are much simpler wonders of patterns and symmetries in our world that also provide ways to understand math, science, art and nature itself, as we explore in our Beautiful Discovery boxes. Even where nature seems very diverse, we often find underlying unities, repeating patterns. What seems to be a dizzying diversity in our world often has underlying symmetry.

As Brian Greene said, “Much in the same manner that they affect art and music, such symmetries are deeply satisfying; they highlight an order and a coherence in the workings of nature. The elegance of rich, complex, and diverse phenomena emerging from a simple set of universal laws is at least part of what physicists mean when they invoke the term beautiful.” (Greene 2003).

**More reading**

Brian Greene. The Elegant Universe. 2003.

Brian Greene. The Fabric of the Cosmos: Space, Time, and the Texture of Reality. Alfred A. Knopf. 2004.

Ian Stewart. Why Beauty is Truth: a History of Symmetry. Basic Books. 2007.

Frank Wilczek. A Beautiful Question: Finding Nature’s Deep Design. 2016. Penguin Books

Leon M. Lederman, Christopher T. Hill. Symmetry and the Beautiful Universe. 2011

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]]>Can you tell what this pattern is?

Does it look like something humans designed, like the inside of a building? A sculpture? Or is it something created by nature?

Here’s a clue: It took a couple of seconds to create it. That time doesn’t include taking the picture (a few seconds) or the time to get the material (easy to find or make). But then to go from a smooth shape to this pattern, the process was almost instantaneous.

I lifted the book (we used a large art history book) over a paper cup, and my three year old daughter then said “One, two, three, go!” I dropped the book over the paper cup. Boom!

The smooth cone-shaped paper cup turned into what you see here.

She was as pleased as I was with the result. “Patterns! You love patterns. Looks like a flower” she said.

The cone was flattened but you can easily pull it back into a cone, except that instead of a smooth cone we have this folded pattern.

Fascinated with this pattern, and the functionality of it as a folded cone, I did some research and found that a woman named Biruta Kresling had discovered something very similar. She made her own cones with origami paper, then quickly crushed it. More interesting to me, I found a scientist, Taketoshi Nojima, had figured out a formal way, an exact pattern, to make origami cones that fold flat when pressed from above.

Now it so happens, that making an origami cone that folds flat is a great mathematical feat. Some scientists had pointed out that many origami folds could not fold flat, some folds lock up, conflicting with other folds, leaving the origami a 3D structure. One has to mathematically orchestrate the folds so that they will work together without conflict to flatten out when pressed. There seems a global coordination that allows it to fold flat, while the individual folds look like little diamonds.

I tried to fold these origami cones, with the happy result below (orange, green, one in white). The origami takes a while to fold, but once finished, it forms a cone that, when pressed, flattens out perfectly. The crushed cones we made (instantly) have a very similar pattern. Likewise, they all can fold completely flat. They all also form spirals, around the cone, that are similar (from the top or bottom view) to what we see in flowers (see our page on golden angle spirals). You may notice that the spiraling shapes look much like the pineapples and pine cones we examine on the spirals page and in our pine cone art projects, book on golden angle spirals, and spiral coloring book.

We consider these just as much beautiful art as beautiful science.

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]]>Slime

Gooey Slime seems the opposite of pattern. Yet there is an underlying pattern called a polymer. Moreover, the polymers are crosslinked, forming yet a larger pattern.

Slime is in the ocean, in our bodies, in our yards. Our bodies make some kinds. Kids make a similar kind out of glue. Slime seems the opposite of pattern, but up close there is a fascinating pattern that makes slime what it is: slimy. Let’s examine the kind of slime kids make out of glue, such as Elmer’s glue.

The glue contains molecules from which we can make polymers. A polymer, in turn, is a pattern made up of repeating molecules, or “monomers” that are connected by relatively strong covalent chemical bonds.

In slime, the repeating unit is poly vinyl alcohol (PVOH). Modeling this with Lego makes features of this PVOH more tangible.

The red Lego blocks show Oxygen atoms, the brown blocks show the Carbon atoms. The small white block shows the Hydrogen atoms.

Two repeating units of PVOH (separated to show how to build with Lego)

Two connected units of PVOH

Slime is made of lots of these polymers, but with with an additional feature, called crosslinking, to give it the sticky, slimy qualities. The PVOH molecules in slime are weakly crosslinked by another chemical (discussed below). PVOH can be crosslinked because this polymer has regions of positive and negative charge. That is, PVOH molecules have “polarity.” In our Lego model, the little white rectangle (Hydrogen atom) has a net positive charge and the red rectangle underneath (Oxygen atom) has a net negative charge. The reason the Oxygen has a net negative charge is because the Oxygen pulls on the electrons (negatively charged) it shares with Hydrogen.

You could say the oxygen “hogs” the electrons. So each repeating unit in polymer has a negatively charged region and a positively charged region. Water (H20) itself is like this, with the oxygen hogging the electrons from the hydrogen.

In the case of water and PVOH, the Hydrogen atom has a net positive charge because its electrons are hogged by Oxygen.

Because the PVOH molecules are polar, as is water, it dissolves in water.

The crosslinking makes a pattern of patterns. The weak crosslinking can break and reform over time, patterns of change, when we move it slowly (move it quickly and it hardens up).

We can show the weak crosslinking using magnets. In a very general and rough way the crosslinking attraction *between* PVOH molecules is weak and breakable, like the attraction between magnets, and is due to opposites attracting, plus and minus charges. This weak crosslinking is what makes slime pliable, sticky, and, well, slimy.

It’s easy to fit magnets in large Lego, or Mega Bloks.

Tape magnets inside a block, or let them hold themselves together by magnetic attraction.

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]]>Kathy has such a great way of explaining abstraction in art. Watch her YouTube video. Abstraction is a key concept in our modeling, games and art, so will be an ongoing topic for us.

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]]>Working in the 1940s, Turing didn’t have the great computers you and I have (To crack the Enigma code he created a huge computer), but he laid out the basic model that you can now run over the web (you can access this toy model on this page). One reason this model generating spots and stripes was so revolutionary was that it showed how the patterns need not be genetically determined, like a blueprint. Rather the patterns emerge dynamically from positive and negative feedback. Only a few chemical processes of feedback are needed, not an elaborate genetic coding of entire patterns. This is an elegant and general model that, as scientists discovered after Turing’s death, can explain a wide range of patterning, even sand dune stripes. Surface complexity emerges from simple rules of interaction. Turing wasn’t able to take the model further due to his untimely death. He was arrested for having a relationship with a man, and forced to either go to prison or take feminizing hormones. He soon committed suicide.

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]]>Our new board game has the title “Tip A Barely Balanced Ecosystem Game” to suggest that ecological balancing is precarious, but the term balanced can be misleading if it implies that the ecology can resist man’s interventions. We also provide a version of Tip that has simpler rules. The board pieces are smaller so more fit on the board. We call this simpler game, Little Tip. Both board games are models of ecosystem dynamics, which makes them great learning tools as well as fun to play.

We also provide computer simulations of various features of ecosystems. We call these toy models because you can easily manipulate these computer models to see how changing one feature of the model affects the behavior of the system over time.

Our toy models show how periods of balance may last a long time, but are nonetheless punctuated by tipping, much as in earth’s ecological history. Toy models of such ecological dynamics can be great fun, and fascinating, engages mathematical thinking and scientific thinking, and is an adventure into the wonder of microbial, oceanic and terrestrial diversity.

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]]>We also provide a computer model of the Tolerance board game. Our inspiration for the game was Schelling’s model of segregation (1971), so we provide a very brief example of that in this video and discuss this at some length in our rulebook. We also look at hypothetical applications of the model, such as to segregation by gender in classroom seating in this longer video.

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]]>

Below is the original image from the coloring book:

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]]>In a recent post, I discussed how looking at forests and fractals evokes a sense of awe. The research studies found that natural fractal structures, like trees, and man-made fractals, like certain art, stimulates awe, altruism. It might be that looking at these structures that repeat patterns across very large and small scales evokes a sense of infinite. The abundance of the infinite might even support altruism, since giving any amount of the infinite still leaves one with the infinite. When I see the same rich patterns in larger and larger (or smaller and smaller) scales, I feel a sense of infinity. I have the same feeling when I participate in Thich Naht Hanh’s meditations on a tangerine. When we eat a tangerine, Thich explains, we can be aware of the sun, soil and people that nourished the tangerine. When we peel the tangerine, we can find a seeming infinity of sensations, smell and sight, if we peel mindfully. Eating the tangerine, slowly, we can find an infinity of sensations in our mouth as well as tastes. In these practices, I feel awareness of the very large and the very small, and feel a wonderful sense of abundance. It struck me that those enjoying coloring mandalas might feel something like this.

Mindfulness meditation gives me this rewarding feeling of abundance in the present. While, there is a huge amount of scientific literature on the positive effects of mindfulness practices (Farb, Anderson and Segal 2012), for me the reward is a fuller experience of life in the present moment. Similarly, I think that the coloring is its own reward. Nonetheless, I think it is helpful and interesting to know about the research on how coloring brings other rewards. Several studies randomly assigned people to various coloring activities and found that coloring resulted in significant mood improvement (Eaton and Tieber 2017, Babouchkina and Robbins 2015, Carsley, Heath and Fajnerova 2015, Van der Vennet and Serice 2012, Curry and Kasser 2005). Interestingly, these studies involved coloring in circles or mandalas. One study reports that the circle shape is important. People randomly assigned to color circle mandalas showed significantly greater mood improvement compared to those assigned to color within square shapes (Babouchkina and Robbins 2015). But after reading this I wondered if there are other aspects of the mandalas that might have an effect similar to fractals that research demonstrated to have such a positive effect on mood (as we reviewed in our post on math, pattern and awe).

Most mandalas I have seen are symmetrical, but they are not fractals. Fractals are symmetrical (or at least similar) across scales, large and small. But some spirals are fractals and also have an overall round shape like a mandala. So I wondered about spirals of the type seen throughout nature. These spirals are fractals because if we look at smaller and smaller scales, we see the same spiraling patterns. I find the natural spirals found in a sunflower particularly fascinating, because we see many overlapping clockwise and counterclockwise spirals. It is quite hypnotic, and reveals a fascinating underlying dynamic. So we created a coloring book with these images.

Also, the hypnotic arrangement of seeds on a sunflower reveals an elegant underlying process. A sunflower often has 21 spirals clockwise and 34 spirals counterclockwise, or 55 clockwise and 34 counterclockwise. Larger ones can have 89 and 55. Smaller ones might have 21 spirals clockwise and 13 counterclockwise (as do some other flowers and plants). All these are “Fibonacci sequence” numbers. The sequence includes: 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, and so on. The number of spirals is usually a number from this sequence. The reasons for this are so fascinating, I wrote it all up in a few pages that I added to my coloring book, Patterning Spirals. You can see these Fibonacci numbers in the spirals in the coloring book.

To make this coloring book, I created images with a computer mode of floral spiraling.The model shows how various spiral patterns emerge over time, and how very small changes in the turn angle can drastically change the final patterns. Here’s a long movie of the process. Here’s a very brief version of the movie. I found the spiraling animation so beautiful that I wrote a waltz to go with it.

The coloring book is for all ages, and the mindfulness exercises in it also work for all ages. If you are a parent or teacher, you will appreciate that this coloring book combines science, technology, engineering, art and math (STEAM). Likewise we can increase are scientific and aesthetic awareness of the generation of spots and stripes n nature. The material in this coloring book thus addresses STEM and STEAM educational objectives, while capturing the elegance of a natural process, and raising both mindful and scientific awareness about the natural process generating spirals.

Understanding the math and science of the spirals is one way to appreciate the beauty of the whole structure that emerges. An alternative way is to focus on the separate parts, the diamond shaped seeds. One could meditate upon each seed, one at a time, like one would do with prayer beads or a rosary, focusing on one bead at a time. If you turn inwards to contemplate the coloring activity itself, you can cultivate mindfulness. A mindful coloring session is a focused attention on the sensations of the act of coloring in the present moment, without judgment of the final outcome. So I included in the coloring book two meditations that I adapted from mindfulness meditations by Thich Nhat Hanh, a Buddhist monk since he was sixteen. He has been teaching many forms of mindful living, mindful breathing, mindful walking, mindful eating, even mindful peacemaking, for many decades. Bringing gentle mindfulness to his life of action in a world full of conflict, Thich Nhat Hahn has inspired countless numbers, and inspired Martin Luther King to nominate him for a Nobel Peace Prize. I have practiced his meditations for many years and they have brought much joy to my life. So, I naturally tend to bring his approach to this coloring book.

I think many people might find mindfulness of coloring spirals to be very different from the awareness of the science of spirals. Letting all concepts slip away and turning attention inwards is at the heart of mindfulness meditation, typically. Nonetheless, I think mindfulness and appreciation of the math and science of our world ultimately can support one another. Thich Nhat Hanh brings into his tangerine meditation the nourishing of the sun and soil, and ecological interdependence shows up in many of his guided meditations. The Dalai Lama has an insatiable appetite for science, particularly brain science. Now I see a recent research piece, published in the respected Journal of Developmental Psychology (Schonert-Reichl et al), found kids put into a mindfulness meditation course ended up with higher math scores. The study assigned children to either four months of a mindfulness program, or four months of the standard “social responsibility” program used in (Canadian) public schools. The meditation included mindful eating and social perspective taking as well as mindful breathing. The kids with the mindfulness intervention had 15% higher math scores, demonstrated 24% more appropriate social behaviors and 24% less aggressive behaviors. We may do mindfulness for the joy it, yet reap other benefits too.

References

Babouchkina A, Robbins SJ. 2015. Reducing negative mood through mandala creation: A randomized controlled trial. Art Therapy: Journal of the American Art Therapy Association, 32: 34–39.

Carsley D, Heath NL, Fajnerova S. 2015. Effectiveness of a classroom mindfulness coloring activity for test anxiety in children. Journal of Applied School Psychology, 31: 239–255.

Curry NA, Kasser T. 2005. Can coloring mandalas reduce anxiety? Art Therapy: Journal of the American Art Therapy Association, 22 (2): 81–85.

Douady S, Couder Y. 1996. Phyllotaxis as a dynamical self organizing process Part I: the spiral modes resulting from time-periodic iterations. Journal of Theoretical Biology, 178: 255–273.

Eaton J, Tieber C. 2017. The Effects of Coloring on Anxiety, Mood, and Perseverance: Journal of the American Art Therapy Association, 34 (1): 42-46.

Farb NA, Anderson AK, Segal ZV. 2012. The mindful brain and emotion regulation in mood disorders. Canadian Journal of Psychiatry 57 (2): 70-7.

Nhat Hanh T. 1991. Peace Is Every Step: The Path of Mindfulness in Everyday Life. New York: Bantam Books.

Schonert-Reichl KA, Oberle E, Lawlor MS, Abbott DT, Thompson K, Oberlander TF, Diamond A. 2015. Enhancing cognitive and social–emotional development through a simple-to-administer mindfulness-based school program for elementary school children: A randomized controlled trial. Developmental Psychology, 51(1): 52-66.

Swinton J, Ochu E. 2016. Novel Fibonacci and non-Fibonacci structure in the sunflower: results of a citizen science experiment. Royal Society Open Science 3: 160091.

Van der Vennet R, Serice S. 2012. Can coloring mandalas reduce anxiety? A replication study. Journal of the American Art Therapy Association, 29 (2): 87–92.

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]]>Many scientists have confessed to experiencing awe when beholding nature’s vast complexity of forms. But only recently have researchers systematically studied causes and consequence of experiencing awe. One cause of awe is vastness of scope, size or complexity, experiments revealed. The same experiments found that one consequence of awe is altruism (Piff et al 2015). Beholding a forest of cypress trees evoked a feeling of awe which, in turn, stimulated altruistic behavior. The researchers (Piff et al 2015) credit the awe to witnessing vastness. They also found that small things that were vast in complexity had the same effect as the cypress forest. A drop of colored dye spreading in milk also triggered the altruism after evoking a feeling of awe. It is a vastness of complexity, not sheer size, since the swirling droplet provoked awe-altruism, and awe-altruism didn’t result when persons looked up at a tall and relatively plain building. Neither can we just credit positive feelings, since witnessing tornadoes evoked fear and, at the same time, had the awe-to-altruism effect.

**Emerging patterns may stimulate awe.**

What do these awe-striking images have in common? The droplet of colored dye splashes outwards in milk, breaking into smaller droplets projecting in all directions, but ultimately merging back into the milk in a new homogeneity. In all these experiences evoking awe-altruism, it seems to me, patterns emerge. In cases such as the cypress forest, the patterns may emerge because our perspective is changing. When I’m in a forest, my eyes travel upwards and back, looking at trees close, then up to far away treetops. On the way up, I see branching out into ever more diversity. Patterns emerge. The tops of the trees, due to their distance from us, converge towards a vanishing point.

Another feature of many of these images is that the patterns repeat in the smaller and smaller parts. For example, we see the same level of branching off, at smaller and smaller levels (or larger and larger) sizes. Many growth processes in nature, like branching out of growing trees or developing lungs, result in patterns repeating across scales, where the shape of the whole is reflected in the smaller and smaller parts. These patterns are called fractals. The spirals of cyclones are fractal, similar across scales, and can aggregate into tornadoes. Very often a simple natural process aggregates into patterns that repeat at different sizes. An irregularity at one size, repeats at larger and larger scales, such that there is a symmetry across scales, as in a snowflake, or in spirals.

In these patterns that nature produces, the whole has symmetry, an order not seen in isolated parts. The number of scales the pattern repeats can be very large, and that is a source of complexity, while the fact that the patterns are the same are a type of order, a self-similarity across scales. Thus, fractals have a certain level of complexity, while the repeating of the same pattern across scales, the symmetry, makes the complexity more intelligible, pleasing and, as recent research revealed, relaxing (Hägerhäll et al 2015). Many of these natural patterns that stimulate awe and altruism are fractals, associated with a certain degree of complexity. Hägerhäll and colleagues (2015) found that fractals in nature stimulated brainwaves associated with positive affect and relaxation. They believe that our brains and visual perceptual system are hard-wired to understand the level of complexity associated with such fractals. Images with complexity greater than a fractal in nature, such as seen in a forest, overwhelm our minds. They conclude that humans need to see this type of fractal by getting out into nature, spending time in forests where we are surrounded by fractals. Fascinatingly, they found the same fractal patterns in Jackson Pollock paintings. They created their own abstract art that also would match the fractal patterns of forests. This art had the same soothing effect on human nervous systems, their experiments revealed. Now, I’m particularly interested in this topic, partly because I found fractal patterns with this level of complexity in my simulations of human interactions (Keane 2016, Journal of Theoretical Biology). More fundamentally, I feel a great sense of awe, as well as relaxation, meditating on fractals in nature and art. I suspect that patterns emerging at many scales may be the source of awe of the experiences in the experiments (by Piff and colleagues 2015). Placing ourselves in the context of these larger scales of size and complexity may be what evokes the altruism, the selfless giving.

These patterns are also surprising. Patterns emerge in the whole that you would not expect based on the simple rules of interaction of the parts. You can see this in our two board games, Little Tip and Tolerance. Tolerance shows how segregation emerges even when no one wants it. Blue and red crabs start out completely integrated: Blue crabs are next to more red crabs than blue, and red crabs are next to more blue crabs than red. Players take turns moving crabs, just trying to find a few similar neighbors for most of their crabs. There are no extra points for finding mostly same neighbors, since crabs are very tolerant of difference. Yet massive segregation usually results. Thus the board game Tolerance shows a pattern emerging that you would not expect based on the simple game rules (video). We show an alternative game that destroys segregation. In this way the game introduces a practical application of pattern emergence and offers some insight into an important social issue. Our game Tip also shows patterns of segregation emerging, but also shows coexistence often results from gameplay, even when the players are trying to eliminate each other. You try to take over the ecosystem, but coexistence is hard to overcome. Again a surprising pattern emerges. I give a quick glimpse of how this happens in Tip and Tolerance in this video. You can explore this in more depth with our computer models of both games, and many more system models, all of which you can run through a web browser.

One more thing about these emerging patterns that inspires me. Math is about patterns of relationships. Science explains what gives rise to the patterns. With computers, we can create system models that generate this patterning, very often with very simple code, and use the models to solve problems. We do this with spirals, and use the images in our coloring book. The beautiful patterns that emerge inspire art. There’s a movement called the art of emergence, art in the age of emergence, or generative art. Studying the emergence of these patterns, as we do in Little Tip and Toleance, can engage Science, Technology, Engineering, Math (STEM) and Art (STEAM).

More to read:

Bies AJ, Blanc-Goldhammer DR, Boydston CR, Taylor RP, Sereno ME. 2016. Aesthetic Responses to Exact Fractals Driven by Physical Complexity. Front. Hum. Neurosci.;10:210. doi.org/10.3389/fnhum.2016.00210

Hagerhall CM, Laike T, Küller M, Marcheschi E, Boydston C, Taylor RP. 2015.Human physiological benefits of viewing nature: EEG responses to exact and statistical fractal patterns. Nonlinear Dynamics Psychol Life Sci. 19(1):1-12.

Keane C. 2016a. Chaos in Collective Health: Fractal Dynamics of Social Learning. Journal of Theoretical Biology, 409: 47-59.

Piff PK, Dietze P, Feinberg M, Stancato DM, Keltner D. 2015. Awe, the Small Self, and Prosocial Behavior. Journal of Personality and Social Psychology 108(6): 883-899.

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