Chaos as Scientific Study
I will focus on three occasions that help us understand the new science of chaos: the non-solution to the three body debate that gave us a first glimpse of weird order behind disorder; the discovery of Sensitive Dependency to Initial Conditions (“the Butterfly Effect”) and “strange attractors”; and the research of paths from order to chaos and the discovery of self-similarity and scale invariance.
We begin with an ideal of all the physical sciences: accurate prediction. In a dynamic system using Newtonian physics, if you knew all the factors involved and could apply those fixed laws that are relevant to the interaction, you could predict future conditions accurately. From this point of view, the enterprise of science aims to curb the unpredictable nature of future time.
The physics of Isaac Newton and the “age of enlightenment” achieved much, but only up to a point, and from the beginning there was a worm in the apple. Although its crowning success was in predicting the location and movements of planets around the Sun, they could not predict location and movement of three planetary bodies whose gravitational forces influenced each other.
In the late 1800’s there was a contest to award the scientist who could come closest to solving the three-body problem. The young French physicist, Jules Henri Poincaré, one of the greatest scientists of his time, decided to tackle this difficulty.
To find a solution, Poincaré devised a concept called “state space” and an instrument later called the “Poincaré map” wherein all the possible states of a system are displayed as a discrete shape. He conjectured that by placing all the information on one graph, patterns previously undetected could be discerned. He was making good progress, but then he discovered he had made a mistake. Redoing his computations, he came up with a startling result: a curved pattern formed by the permutations of three bodies influencing one another. A pair of curves looped back on themselves, crossing to form an infinity of intersections. No two curves could ever cross themselves, but they would cross the web an infinite number of times.
Here we have an example of orderly chaos: finding qualities, characteristics and rules, but not the anticipated coherent pattern. Poincaré determined that the planets’ changing positions and velocities could not be calculated accurately. He found this simple interaction between three planets to have unpredictable characteristics. There was a pattern inherent within, but none that he could understand.
(There is an important difference between the scientific study “chaos” and quantum physics and Heisenberg’s principle of indeterminacy. Here we are working with a fully deterministic system, whereby present conditions give rise to future conditions. Instead, quantum theory uses indices of probability.)
Poincaré's Natal Chart Using Whole Sign Houses
Here is Poincaré’s natal chart. We first notice many planets in the earth element, but with Mercury particularly strong, angular in Aries and oriental. Mercury is also the midpoint of Sun and Neptune. Poincaré was a plodding person who could stick to one thing well, yet he had a wiry and supple mind.
Once again, we find Uranus and Pluto to be prominent. We first notice the Sun applying towards Uranus, but then we see that the Sun is at the Uranus/Pluto midpoint. Poincaré was a pioneering physicist and mathematician who helped develop the new field of topology: the mathematical study of strange forms such as the Mobius strip.
In early 1887, the year he had completed his project and won the scientific prize, transiting Uranus was opposing his natal Mercury. In the best expression of this transit, Poincaré had produced a wonderful feat of intellectual imagination and thereby established the methodology and outcome of scientific chaos.
Yet he and the world of science moved on from his odd “solution” to the three- body problem. It wasn’t until the early 1960’s that the science of chaos picked up again. It resumed through an investigation into a different chaotic system – the weather. Here we meet Ed Lorenz, a mathematician who became a meteorologist and devoted himself to finding a mathematical means to predict weather. Lorenz devised an elaborate set of equations to model the weather, and with a room filled up with the early computers, he got to work. On one famous evening, he tried to short-cut the data entry process for the start-up conditions and entered rounded off values while re-running a particular model. He rounded his numbers to three decimal places whereas the original model had run with numbers to 6 decimal places. When he returned, he discovered that this small difference made for vastly different weather!
Most scientists might have blamed the problem on bad equipment or faulty inputting (both checked out fine), or attributed the result to having had a tough day, but Lorenz investigated further. Later this phenomenon would be called “Sensitive Dependence on Initial Conditions” (SDIC). This shows the effect of our weather system as nonlinear: a small change can give rise to disproportionately large effects. This turns out to be the case with all interactive systems.
Lorenz’s discoveries and his first publications occurred before the famous Uranus-Pluto conjunction of the mid-60s, yet ten years afterwards SDIC became known as “the Butterfly Effect” to the scientific community and in the popular imagination.
Does a natal astrology chart also exhibit Sensitive Dependence of Initial Conditions? Not in the same way as Lorenz’s “Butterfly Effect.” Changing an Ascendant from 29° Scorpio to 00° Sagittarius would have helped Friedrich Nietzsche be a more socially and emotionally secure person, but this would have been the case from birth. Strictly speaking, the “Butterfly Effect” displays increasing divergences over time. It is similar to the 1998 movie “Sliding Doors” where a woman missing the subway or not would have consequences that increasingly diverged throughout the movie.
Lorenz moved from complex weather equations to simpler ones using convection: the patterns of water rising and falling from a heated surface below and a cooler surface above. This is also illustrated with moving water wheels. As the wheel fills with water, the wheel will stay still, or oscillation will occur periodically or haphazardly, depending on the amount of tamping of the wheel’s motion. (Do a search on YouTube for “Lorenz Water Wheel” for a video demonstration.) Our interest here is in the haphazard back-and-forth movements of the water wheel.
Lorenz found that the seemingly random movements of the wheel did settle into a form, but what a weird form it was! It was three-dimensional, non-returning and a-periodic. This became known as the “Lorenz Attractor”, one of many such shapes that were later called strange attractors. (His definitely looks like a butterfly’s wings.) A strange attractor is different from a point attractor or basin, where things converge at one point, or a periodic attractor that resolves into a regular oscillation. Because the system is a- periodic and the foci are irregular, this attractor certainly qualifies as “strange”.
All three kinds of attractors are found in astrology. The stars, planets and angles move around a fixed point that is the Earth in a geocentric chart, or the Sun in a heliocentric chart. Additionally, the planets are periodically “attracted” to its conjunctions with the Sun and other planets. This is because the positions of the angles and planets are deterministic and also linear.
According to the art of astrology, the always-changing and never-repeating configurations of the skies give rise to the patterned but unpredictable circumstances in our lives. These circumstances may have their own repeating situations, coincidences, and omen-like occurrences with hidden meaning. Astrology gives a conceptual framework for this to occur. The applications of astrology help us understand these occasions – they provide context. Although the configurations of the sky are linear, we use them as strange attractors in their ability to signify people and events on Earth.
Let’s take a brief look at Lorenz’s natal chart. Unfortunately, we only have a date of birth. Although it appears that Lorenz has a close Moon-Pluto conjunction, this is not trustworthy because of the unknown birth time.
Natal Chart for Edward Lorenz using Whole Sign Houses (Note: Birth time is unknown)
Lorenz’s Sun in Gemini sextile to Neptune qualifies him as a bright and unusually imaginative person. As with Poincaré, Uranus figures prominently: it is in close applying square to both Mercury and Jupiter in Taurus. We see the doggedness of the sign Taurus, and the brilliance and eccentricity associated with Uranus. During the time of his research into the chaotic nature of weather prediction, unsurprisingly, transiting Uranus was in opposition to his natal Mercury; his initial article was published in 1963 when transiting Uranus was in opposition to his natal Sun. Lorenz’s trailblazing work was practically unnoticed when it was first published in the 1960’s. Today he is considered a pioneer of the study of chaos.
Our next development takes us into systems and patterns that reveal themselves on the way into chaos. The foremost name associated with this development is Robert May, a theoretical biologist working in the mid-1970’s.
At that time May was working with the mathematics of insect population growth. This is also an example of a nonlinear system of interacting components: a growing population will compete for limited resources, variable predator populations, etc. As a result, population will decrease and increase again, and so on. According to the mathematical model used, if the growth rate is too small the species dies out; if the growth rate is a little higher, the population resolves into a continuous oscillation; larger rates of growth, however, result in population swings that are irregular and seemingly random.
The diagram below shows the results of increases of population growth from what’s called a “logistic map.” It gives a different version of the “state space” discussed previously. The vertical “x” stands for the population; the “r” horizontal axis is the rate of growth. The map begins simply and then things start to get weird.
The single line to the left (rate of growth (r) =2.4 to about 3.0) is an example of a point attractor – there is a steady population growth. We first see bifurcations into two paths at r= 3.0 to about 3.45. Here, the population is oscillating between a high and low population level. This would be a periodic attractor if they lead to a regular oscillation but they do not. Instead they lead to another bifurcation into four values (two different highs and two different lows) and then another one into eight and this can continue. Then we reach the “accumulation point”, an amount a shade less than r=3.6, where things seem to go out of control.
However, when we look to the right of the accumulation point, we see a darkened area but one not without some form. If we could look closer at the white areas, we would find further bifurcations that look quite like the visible ones. This is not the mess that it appears to be; instead, it obeys principles of self-similarity – we get the same general shape of horseshoe, though they are not quite the same. They are not quantitative identities but quantitative similarities.
There is another strange factor to the shape: its rate of decrease of the horseshoe shapes that are the bifurcations. A mathematician named Mitchell Feigenbaum discovered that the rate of decrease horizontally, and the difference in values horizontally between “period doublings” was 4.6692016091. These same numbers would show up in other chaotic systems, from iterating a function on a hand calculator to chemical and physical systems of values or qualities that interact with each other. The scientific study of chaos leads not to an affirmation of randomness, but rather to the discovery of a kind of order that is very hard for us to comprehend. Can anybody find a rationale for Feigenbaum’s number?
In our lives, do we encounter bifurcations and period doublings? When do we not encounter them? Any moment in which we are not acting out of habit we meet bifurcations. Indeed, when we examine our responses to specific situations in our lives, we always encounter a moment to go one way or the other. Tomorrow’s route is based on the one we take today, and so bifurcations and period doublings lead to unpredictable but deterministic outcomes. Perhaps this is what free will is.
As astrologers working with clients, we can spot times of new possibilities for the client, but not know what direction he or she will choose. From knowing astrological symbolism and having some understanding of the circumstances and issues our client is facing, we can make some good conjectures, but the outcomes are unpredictable. The client may be influenced by our advice, but the actions always belong to him or her.
In astrology do we encounter self-similarity? We certainly encounter self- similarity in time. Consider the range of sensitive points in a natal chart: every day the Ascendant and Midheaven transit the same sensitive points; each month the Moon and each year the Sun will transit these points; Saturn will take almost thirty years. This is also the case with directions, many ancient chronocrator systems profections of 30° a year, and the movement of the age point according to the Huber method of progression.
Moving on to the next development in Chaos Theory, we will now look at fractals. The person most associated with this is Benoit Mandelbrot, who coined the term from the Latin word for “irregular”. It turns out that fractals are far more prevalent in nature than are the standard geometrical shapes. Their presence ranges from the twisting shapes of many jagged shorelines, to the intricate surfaces of our lungs and brains, to the distribution to leaves on a tree, to the contours of geological formations. They also are found in various art forms. Fractals obey principles of self-similarity and scale invariance – the same shapes appear regardless of larger or smaller scales.
Certain mathematical equations that use “complex” numbers (merging real and imaginary numbers) can give rise to stunning forms. The best known is the “Mandelbrot set” that contains forms that continue infinitely. I trust that interested readers will pursue the story further on their own. An animation of the Mandelbrot set is a great place to begin and is easily found on the internet.
To summarize, the study of chaos is the study of systems of apparent chaos. “Chaotic” phenomena may be non-periodic and unpredictable, but they are not random. Instead, they show evidence of peculiar kinds of order. There are many other systems that also appear random that are not yet understood by the science of chaos. For example, we cannot quite determine highly complex interactive systems like the general economy, although we can perceive patterns of self-similarity and scale invariance in the stock market. Social trends among highly interactive beings like ourselves are at the high end of seeming randomness.
As a logos, astrology is not about chaos but order. Certainly the sky and earth we look at as astrologers have an order to it. Otherwise we couldn’t predict planets’ positions in the sky and would have nothing to base our predictions upon. This is the case whether we use earth maps or star maps, geocentric or heliocentric positions, or a Hellenistic or Vedic chart. Yet its relationship with chaos and complexity gives astrology its relevance and importance in peoples’ lives. People seek the advice of astrologers (among others), not because their lives proceed randomly, but because there is unknown order behind these seemingly random happenings of life.