by Guy Murchie
How the Golden Section relates to Platonic plankton in the sea, however, is deeper than can be casually understood. Indeed it may seem downright mystical when you discover that three identical Pythagorean rectangles with sides in the ratio i :Φ will exactly fit together at right angles (as shown in the illustration) so their twelve corners become the twelve vertices of a perfect icosahedron with twenty equilateral triangular faces, and that Φ turns up as a vital ingredient in the formulas for several other polyhedral functions, including those for the surface and volume of the pentagonal dodecahedron (this last an abstruse 4/5Φ4 counting the edge length as one).
Among still more affiliations of the Golden ratio, the final one I must mention is its almost uncannily intimate identity with phyllotactic leaf and seed patterns. To understand this relation, you need to study the ratios between alternating pairs of successive numbers in the Fibonacci series. If these values are added on both sides of the progression (as I have done in the illustration), it soon becomes clear that the
Fibonacci series is basically a succession of gnomons whose alternating intergnomon ratios are both (from different directions) approaching the mystic Φ value of 1.61803... and that the Golden Section truly lives in flowers and trees as well as swimming in the sea, climbing high mountains, and that it even sires human designs from the playing card (of China) to the Parthenon (of Greece).
Obviously it is a revelation fit to open the kind of fresh, incisive insight into the hidden heart of our universe that thrilled, and would again thrill, Pythagoras and Plato and Copernicus and Kepler and Newton and Einstein. For it is part of the real music of the spheres, parcel of the inaudible, the irrepressible melody of life that pervades everything while known only to the rare few. Awareness of it can reveal the seams of things and the nodes of being that almost all of us miss in this contingent phase of existence.
OTHER ABSTRACTIONS
Of course there are so many worlds and worlds of worlds and sons and grandsons of worlds, all living, spawning and evolving at the same time, some inside others, that it is practically hopeless to discover them, let alone keep track of them or find out what they're made of. And, to some extent, this goes even for Earth. For, if you feel any doubt about this planet's having otherworldly aspects, you may be interested to know that physicists have already found in meteorites (born in other worlds) evidence of elements 111 and 115, which have not yet been really discovered here or named by man but only deduced and dreamed about. And besides man's cities, which, as I've suggested, have various crystalline grains like the rectangular grid of New York or Chicago, the circles of Paris, Washington and Moscow, there are the latticed patterns of country fields and roads, of herringbone clouds and squall lines, of stellar and planetary orbits, of rainbows and sand dunes and animal territories and all manner of invisible and abstract structures, like business cycles, spheres of political influence, lobes of language, even the games of children which have a life of their own sometimes lasting for centuries on several continents. There are visual patterns like the palm-prints of newborn babies too that correlate with vocal graphs of their first cries, both these having been successfully used to diagnose serious abnormalities, from mongolism to rheumatic fever. The subtle, crystalline derivatives of life we call spider webs serve also as a kind of genetic script, revealing each spider's inheritance in the wheel he spins when young (perhaps the very shape of his spinning genes), then showing his developing mind in changes to a more modern design as he matures, to say nothing of the already-mentioned aberrant forms (page 250) when he has been drugged. Regimentation of individual organisms into armylike ranks creates crystal textures in schools of swimming fish, birds flying in formation, viruses and certain rod-shaped bacteria who (for a still unknown reason) have been seen to array themselves in parallel rows in three dimensions without actually touching one other.
It is well known that small creatures from insects to amphibians can adapt to the loss of some of their wings or legs and still fly or run quite well on what is left. But recent laboratory experiments in physiological rearrangement have gone a step deeper into abstraction by transplanting, for example, the front half of one lizard onto where the head of another had been amputated, thereby assembling a six-legged composite lizard that soon learned to run and coordinate all six of its limbs in the natural gait of an insect. And another experiment reconstituted a complete ameba out of parts from three (the membrane of one, cytoplasm of another, nucleus of a third) integrating them so skillfully that the new synthetic ameba lived and propagated normally, apparently without ever dreaming it was basically just a patchwork of spare parts.
Then, shifting from biology to music, I must say that music is so abstract I constantly have to remind myself it is a wave phenomenon lacking not only tangible but even intangible existence until at least one vibration has sounded. For any briefer sound can hardly have a definite pitch or be a note. And when it comes to fitting notes together into a melody, any piece of music can be taken apart and reassembled, indeed a lot more freely than can an animal. Someone has even invented a synthetic symphony that can be played and integrated entirely by one versatile musician, who performs solo on all the orchestral instruments, one after the other, recording each part of the symphony separately before acoustically blending all the parts into one complete performance that, however beautiful, is no less artificially abstract than the reconstituted ameba.
ABSTRACTION IN REVIEW
We will now wind up this chapter on abstraction by trying to enumerate the main evidences of it in a dozen categories:
1. Matter, although it is commonly regarded as concrete because (at any given instant) it is composed of a particular system of atoms, inevitably becomes abstract with the passage of time because metabolism, erosion and other forces, second by second, year by year, millennium by millennium, replace its atoms and molecules with other atoms and molecules, leaving only an abstract pattern to persist indefinitely. In a similar way the spokes of a wheel start to "dematerialize" when the wheel begins to turn, fading farther and farther out of sight as its rotation increases in speed. Billiard balls appear to touch when they bounce off each other but modern molecular science gives us irrefutable evidence that they really interact more like comets rebounding from the solar system with lots of "emptiness" all around and in between. And a hologram image, which can be electronically turned over and looked at from all sides, is obviously an "object" in the abstract, yet in essence it may be no more abstract than a chair or table made of atoms whose real substance is still completely unknown.
2.Subatomic "particles" are being discovered by the hundreds but the most tangible thing they offer physicists is a mysterious new world of ultrahigh energy full of unseen, unproved entities named quarks, antiquarks, unitons, etc. These "things" somehow resonate like musical notes, behave like waves, obey strange laws of symmetry and can be visualized only with the help of such analogies as the "quantum string," whose single dimension gives it zero mass while, under 13 tons of tension, its ends (tipped with quarks) move perpetually at the speed of light. As far as can be determined, neither space nor time reaches much below the magnitude of the atom where a pervasive uncertainty prevails (governed by Heisenberg's Uncertainty Principle) - so there may be no distance anywhere as short as a quadrillionth of an inch, nor any instant as brief as an octillionth of a second.
3.The universe, as far as we can tell, is just as abstract as the atom or its nucleus. Because remote galaxies and supergalaxies of stars are calculated to be receding at speeds proportional to their distances from us, those moving away at the speed of light and at a range of twelve billion lightyears are practically undetectable because their radiation (including light) is carried off as fast as it is generated. And remoter ones, if they exist, would presumably be receding faster than light and therefore not only unknowable but in effect nonexistent as matter in our finite world. Something similar applies to the phenomenon of the black hole, which is so massive its gravitation implodes all its
matter plus all its radiation (including light), making it invisible and almost unknowable. Some cosmologists think the entire universe is a kind of super black hole and that matter draining out of it must therefore appear in a "white hole" elsewhere, perhaps in an anti-universe made of anti-matter where time flows in reverse, these two universes forming what can be represented as mirror-twin cones joined into an hourglass-shaped figure. And at least one of them (page 401) even postulates a third universe shaped like a funnel surrounding the hourglass and inhabited by tachyons, intangible (almost unimaginable) particles that move faster than light - than which I don't think it's possible to be more abstract.
4.Light is the visible octave of a vast abstract spectrum of radiation that includes everything from gamma rays to radio waves and much more, the most important fact about it being that its speed (186,282 miles a second) is the absolute maximum for the propagation of material influence anywhere. It is interesting to note that light is material in the sense that it responds to gravity, for a beam of light will fall like a stone (16 feet the first second, 48 feet the next second, etc.), although the fall is not likely to be noticed since 16 feet is so negligible compared with the 186,282 miles it may go in some other direction. Light can also be twisted like a cable or pumped (through valves) like a gas, and it has been calculated (by computers) that if one could drive along a road lined with telephone poles at speeds approaching that of light, the poles would appear to lean toward you, their tops curling closer and closer as your speed increased, until they seemed to touch you at 186,282 mps. Also that, on a space voyage at comparable speeds, the stars ahead would not only crowd toward your line of sight but also turn bluer and change in brightness (depending on their color), and the stars behind would get redder and most of them disappear - all this because of the Doppler effect of squeezing the approaching light waves ahead and spreading the departing ones behind. The velocity of light or radiation, to posit a generalization, may be a kind of seam in the universe where space meets time, where body meets mind or even, in a sense, where finitude transcends into Infinitude.
5.Mathematics is probably the most obvious abstraction of all, comprising everything from the Pythagorean theorem to Einstein's E=mc2, Bode's Law of planetary distances, Balmer's Ladder in the spectrum of hydrogen, the seven octaves of the Periodic Table of the Elements, the seven shells of the atom, the Fibonacci series, the Golden Section Φ, the rings of planets like Saturn and their harmonic relation, the Platonic solids, V -1 and a lot more (some of it discussed in my Music of the Spheres).
6.Music is mathematical in structure of course and brings abstraction to the earth as a gong, to the atom as a harp, to life as a melody, while the Pythagorean inspiration that celestial bodies have musical relations is virtually a key to the universe.
7.The gene is also abstract as a pattern, independent of the atoms that implement it in any given moment, a meeting point between matter and energy, a message that moves in a wave of meaning through life, an acorn harboring an oak, an egg containing feathers, menu, songs and a map of stars.
8.The interrelation between all creatures and things is abstract too, something we will go deeply into next chapter. And it includes the ecologies of everything from rivers, glaciers, islands and fires to those of wars and worlds.
9.There is also a pervasive polarity between such opposites as good and evil, mass and energy, male and female, predator and prey, cause and effect, body and mind, yin and yang, concrete and abstract - relationships that themselves are abstract, which we will take up in Chapter 18.
10.The concept of mind (Part Two) is likewise abstract, elusive, unlimited by either space or time. For where do dreams reside? Does consciousness begin in the atom? Does it merge or divide in procreation? Does memory depend on a brain? Can a tree yearn? Can the impossible ever become possible?
11.Life and death are both very abstract besides being aspects of a profound transcendence, as we shall see in Chapter 19. This concept derives from an understanding of life as a river or flame with currents and eddies where organisms evolve into superorganisms, mortality is a phase of immortality and finitude steadily unfolds toward Infinitude.
12.Evolution is another abstraction of life (Chapter 21) involving time, place, tributaries, multiple trunks, branches and the continuous development of matter, mind and spirit, particularly as exemplified in creatures, beings, machines, ideas, virtues, worlds ... Evolution in the cosmos progresses from the circle to the helix in a new dimension, creating genes, muscles, rivers, galaxies - also cycles of fermentation and germination from cells to worlds (Chapter 22) to metaworlds - culminating in the spiral grain of the universe.
13. The divinity aspect of the world may be its most profound abstraction (Chapter 23), especially if, as I surmise, the world turns out to be in essence a Soul School. There is ample evidence, for example, that some unfathomable influence has somehow permitted the atmospheric flow of oxygen to evolve breathing, solar radiation to initiate vision, nerves to induce thought, thought to ignite spirit. And whatever the whole of this mysterious creative force may be, as with other wholes, it shows signs of being greater than the sum of its innumerable parts. Could any creator possibly design and bring into being a world better than Earth for teaching life and spawning spirit? Whatever the answer, there seems little reason to think the universe will soon, if ever, yield any concrete solution to its abstract Mystery.
Chapter 13
Second Mystery: The Interrelatedness of All Creatures
* * *
WHEN YOU LOOK at a planet whole, from outside, for the first time, certain things come clear that you never could be quite sure of before. One is a persistent feeling that all the inhabitants of that world must be related. Surely this is an intuitive inference from the obvious oneness before you, bestowed by the spatial perspective of the astronaut, which is, in the case of Earth, a crucial step in her dawning consciousness of herself.
But there is a profound mystery in Earth's inner relatedness, for relatedness implies common ancestry, just as first cousins share grandparents, and all life here must ultimately have come out of the womb of Earth. Yet if in fact this happened, how did it happen? How could the steaming, volcanic planet, or even her later, salty seas, spawn the vast and vital proliferation we know today?
COUSINHOOD OF MAN
I will begin our exploration of this Second Mystery of Life with the most familiar of species: Man. I know man is something of an upstart on busy Earth, having existed for only the most recent one tenth of one percent of her history. And it is almost certain that he is now entering a critical period which may show him up as a catastrophic bungler. For his position is precarious, he being but a single species, only one of millions of species, most of whom have not only been around a lot longer but are much more solidly established. His total numbers who have lived their lives within the couple of million years of his existence add up to a few dozen billion individuals - which, in planetary perspectiye, isn't much. Specifically the anthropological estimates of those numbers range from about twenty to fifty billion, depending on how far back he can be said to have attained his station of being human, on the duration of his average generation and on how widely and densely he may have populated the world.
Although few of his members seem to realize it, man's relation to himself is fairly easy to measure and is surprisingly close. In fact, no human being (of any race) can be less closely related to any other human than approximately fiftieth cousin, and most of us (no matter what color our neighbors) are a lot closer. Indeed this low magnitude for the lineal compass of mankind is accepted by the leading geneticists I have consulted (from J. B. S. Haldane to Theodosius Dobzhansky to Sir Julian Huxley), and it means simply that the family trees of all of us, of whatever origin or trait, must meet and merge into one genetic tree of all humanity by the time they have spread into our ancestries for about fifty generations. This is not a particularly abstruse fact, for simple arithmetic demonstrates that, if we double the number o
f our ancestors for each generation as we reckon backward (consistently multiplying them by two: 2 parents, 4 grandparents, 8 great-grandparents, 16 gr. gr grandparents, etc.), our personal pedigree would cover mankind before the thirtieth generation. Mathematics is quite explosive in this regard, you see, for the thirtieth power of two (1,073,741,824) turns out to be much larger than was the earth's population thirty generations ago - that is, in the thirteenth century if we assume 25 years to a generation.
But you cannot reasonably go on doubling your ancestors for more than a very few generations into the past because inevitably the same ancestor will appear on both your father's and your mother's sides of your family tree, reducing the total number (since you can't count the same person twice), and this must happen more and more as you go back in time. The basic reason it happens is that spouses are not just spouses but they are also cousins (although the relationship is usually too distant to be noticed), which means they are related to each other not only by marriage but also by "blood" because somewhere in the past they share ancestors. Another way of saying it is that your father's family tree and your mother's inevitably overlap, intertwine and become one tree as their generations branch out backward, a process forced by geometry until eventually all your ancestors are playing double, triple, quadruple and higher-multiple roles as both sides, all quarters, eighths, etc., of your tree finally merge with others into one common whole and the broad tops of the family trees of everyone alive and his family become identical with their ancestral world populations. These populations of course are the fertile portions of past societies and naturally cannot include "old maids," cautious bachelors or anyone who fails to beget at least one continuous line of fecund descendants to disseminate his genes into mankind's future.
