Math, science & awe

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, 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 Tip and Tolerance, 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.