Kepler and Geometry

It’s no secret that I’m a fan of Johannes Kepler, the German mathematician. My love for Kepler goes back to my undergraduate astronomy classes. Here is Exhibit A: the passage from my first astrophysics textbook, by Carroll and Ostlie, which caught my fantasy:

Unlike Tycho, Kepler was a heliocentrist, and it was his desire to find a geometrical model of the universe that would be consistent with the best observations then available, namely Tycho’s. After Tycho’s death, Kepler inherited the mass of observations accumulated over the years and began a painstaking analysis of the data. His initial, almost mystic, idea was that the universe is arranged with five perfect solids, nested to support the six known naked-eye planets (including Earth) on crystalline spheres, with the entire system centered on the Sun. (pp. 22-23)

Kepler Mysterium Cosmographicum Solids
Kepler’s own rendering of his Platonic solid construction of the Solar System, from his book Mysterium Cosmographicum.

I imagined what it would have been like to live back during Kepler’s time (c. 1600 AD), when Kepler was first developing the physical concepts that governed the motions of the cosmos. How fantastic it must have been, to be free to invent such miraculous, invisible, geometric structures that inhabit and guide the universe! It reminded me of some ideas I’d run into in a few science fiction books – the message locked deep in the decimal calculation of π in Carl Sagan’s Cosmos or the Prime Monitor machines at the center of each galaxy, left by God when the universe was created, in Gentry Lee & Arthur C. Clarke’s Rama Revealed. Kepler’s was a time when scientists could believe that they were discovering the scaffolding used by the Great Architect. I secretly hoped that one day we could return to that kind of thinking, because that certainly wasn’t how scientists thought today.

Or maybe I was wrong. Let’s say there is a Creator, and that Creator is the most perfect architect and geometer. He is such a good architect, and has such command of physical geometry, that when he removes his scaffolding, the universe immediately runs perfectly according to all physical principles. The job of science is to discover those physical principles. It shouldn’t matter whether a scientist believes in the existence of the Creator for that scientist to discover the physical principles used by the Creator to make the universe. But, it does matter whether the scientist is a believer for that scientist to take Kepler’s constructions seriously. Note that Carroll and Ostlie labeled Kepler’s Platonic solid construction as “almost mystic.” If you read what Kepler wrote, he certainly thought that his geometric (and musical!) reconstruction of what he believed was God’s creation was necessary to lead him to those physical laws of celestial motion that have now stood the test of over 400 years. Without his “almost mystic” geometry, he would not have been led to those physical laws. But, is it necessary, today, to believe in a geometric Creator in order to discover real physical laws?

Fast forward almost a decade and a half for me. I spent a good chunk of that time with the LaRouche gang, studying Kepler. My initial fascination with the LaRouche movement was that they considered the geometry of Kepler not as an “almost mystic” approach, but as a valid way to make scientific discoveries. How could I not get taken by that! I found that there were other characters in history that also seemed to take a similar “almost mystic” geometric approach. These included Kepler’s followers Leibniz, Gauss, and Riemann, as well as both the famous Russian biogeochemist Vladimir Vernadsky and the famous Jewish physicist Albert Einstein. Among these scientists, only Leibniz and Einstein were explicit, publicly, that they believed they were hunting for the mind of the Creator. The others may have also thought so (Riemann gets pretty spiritual in his personal notes), but never in public. Regardless, each of these scientists leaned on a clearly geometric, as opposed to probabilistic or “spooky action at a distance,” foundation in their work to understand the world. I’m no longer with the LaRouche gang, but am very happy I had the time to study the original works of these scientists under the assumption that they were Keplerian.

I’ll conclude this blog post with a specific example, since I plan to refer to this post in the future. Vernadsky was a geochemist who died old just before the end of the Second World War. He directed his laboratory to catalog the geochemistry of naturally occurring materials, specifically those materials with biological origin. He noted that, without exception, there were clear chemical distinctions between biogenic matter and abiogenic matter. For example, matter generated by biological processes typically contained elements of different atomic weight than matter generated by abiotic processes – inorganic carbon has a different weight than carbon left by plants. He showed that each type of organism actually had a chemical and isotopic fingerprint.

More importantly for the discussion here, Vernadsky recognized a few distinctions that were of clear geometric character. Matter of biological origin tends to be left handed. Handedness, or chirality, is a complicated concept, but permeates living matter. Pasteur first found this to be true. In his experiments on wine, Pasteur showed that a solution of tartaric acid produced by living organisms would rotate the plane of polarized light. A solution of the same chemical, produced without the action of organisms, however, would leave the polarized light unrotated. Pasteur concluded that the abiotic solution contained two forms of the same chemical – a left and a right – which were mirror images of each other. The action of the organisms selected out one of the two forms, leaving a chemically equal, but optically distinct, solution behind.

Left and Right handed crystals
Sketches of left and right handed versions of a crystal.

Vernadsky claimed that the ubiquity of this “left handedness” of living matter was one hint that living matter was geometrically distinct from abiotic matter. He imagined a universe where two Einsteinian spacetimes cohabited: one that governed abiotic matter, and one that governed biotic matter. Vernadsky located several other distinctions which he thought helped characterize this biological spacetime. For instance, abiotic minerals are always composed of infinitely extendible crystal lattices. Organisms, on the other hand, are never geometrically extendible to infinity but are always bounded. Vernadsky believed that, in this way, abiotic spacetime may be fundamentally Euclidean, while biological spacetime may be fundamentally Riemannian. He employed the help of several Russian mathematicians to help fully characterize such a Riemannian spacetime that had a preferred orientation built in, but died before such a task was completed to his liking. It may be that such a geometry already existed, as hinted at by the Geometric Algebra of David Hestenes.

The idea that life has a different geometry is not new to Vernadsky. Even the Pythagoreans recognized that life exhibits five-fold symmetries, while nonlife rarely if ever exhibits such symmetries. But, Vernadsky was one of the few moderns that tried to scientifically extend that idea. Unfortunately, to be so explicit today is considered by some scientists to be a dangerous thing, which is not helped by the fact that many truly mystical cults today also adopt “sacred geometry” beliefs. It implies that life is a different type of substance than nonlife, and could lead some to try to posit the existence of a Creator. I guess the fear is that, once you bring life into the cosmos as a non-dead principle, you will then urge a relaunching of the Roman Catholic crusades against the infidels, starting with that now infamous atheist, easy target Stephen Hawking.

I don’t plan to run a crusade against fellow scientists, but I am a believer in a cosmos constructed according to unseen geometric principles, principles that are knowable through rigorous investigation. I also believe I have found a few modern examples of scientists who are carrying on this tradition, though perhaps not consciously. This is a subject that I shall explore in future posts.

2 thoughts on “Kepler and Geometry”

  1. Thanks Pete, this is is a pretty tantalizing post! I think it might be overstating it a bit to say that Vernadsky found “without exception” that biogenic matter was different from abiogenic. From what I understand he cited examples where ratios of naturally occurring isotopes can differ due to living processes. Definitely looking forward to the follow up!

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    1. True, I can’t quote him on that. But, he certainly had overwhelming evidence of a big, old, fundamental difference between living matter and inorganic matter.

      On the isotopic differences: There’s a speech he gave in 1926, and reissued in 1940, specifically on isotopes and living matter. There, he elaborates what he mentioned in he 1938 paper “Problems in Biogeochemistry, II” – that the “well known” change of an element’s atomic weight once it enters the biosphere is a characteristic of living matter. He says that, since it must require an expenditure of energy to change the weight, it must be difficult for matter to exit the biosphere once it enters. Therefore, what enters the biosphere stays in the biosphere for a long, long time. This may be related to his belief (which I think he dropped later on) that the amount of matter in the biosphere was a constant. In any case, by 1940, Vernadsky cited experiments that demonstrated the biogenic weight change in calcium, carbon, and nitrogen.

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