Brief attempt about chaos and beauty – Chaos, order and the ratio in between

You are in „Things and Concepts“

Feigenbaum had been studying turbulence in fluids at Los Alamos National Laboratory since 1974. In the summer of 1975, he heard a lecture on quadratic differential equations similar to those Lorenz had used, and on an abstract level, it was clear to him that they described that mysterious boundary between the gentle flow and turbulence of his fluids. So, using the universal tool of his scientific generation, the first programmable calculator, HP-65, he set about calculating the exact values of the period doublings. In the end, it was likely the slowness of his calculator that ignited his inspiration, as he noticed that the numbers converged in a similar manner to a series of boundary posts on a straight road. If you know how big any two of these reflective fellows are, you know the sizes of all the others. Feigenbaum calculated the ratio of convergence to be 4.669, the 1st Feigenbaum constant.

Related to May’s bifurcation diagram, he had thus found the scale on which its self-similarity takes place: If you reduce the diagram 4.669 times, it looks like the next period doubling.

Until the following year, Feigenbaum investigated other simpler and more complicated equations, all describing systems that pass from a state of order to chaos via a series of period doublings. To his great surprise, they all produced the same ratio value of 4.669 between bifurcations. This was nothing less than revolutionary, for it meant that a whole class of mathematical functions behaved in the same predictable way. Flowing waters, swinging pendulums and electronic oscillators, they all follow a universal law of nature on their way from predictability to unpredictability and therefore form shapes that are all alike on a deep level.

„As we move by our studies of natural works and laws alongside their great Creator and Preserver, we behold that part of truth as he beholds it; we recognize the order of nature the relation of cause and effect as he recognizes it, and the purpose of this exercise must be to bring our senses into grateful harmony with his.“
Horace Mann, Thoughts

„Nature runs its course. And that which appears to us to be an exception is the rule.“
Johann Wolfgang von Goethe, Conversations, December 9 1824

„The forms of life shaped by the adversities of the rugged mountain world seem to show the clearly recognizable brushstrokes of the same great artist. I recognized a hidden sameness in the antlers of the ibex, in the twisted grains of the juniper trees, the features of the old men and women of Baltistan, and the vast arches in the path of an active glacier.“
Galen Rowell, In the Throne Room of the Mountain Gods

Next Ancient teachings

Previous Self-similarity

If you found this post useful and want to support the continuation of my writing without intrusive advertising, please consider supporting. Your assistance goes towards helping make the content on this website even better. If you’d like to make a one-time ‘tip’ and buy me a coffee, I have a Ko-Fi page. Your support means a lot. Thank you!