Martin Gardner The New Ambidextrous Universe: Symmetry & Asymmetry from
Mirror Reflections to Superstrings
Dover, 401 pages, $18.95.
What is mathematics about? That is not so easy to explain. With biology, say, we know where we are. Biology is about living things. Linguistics is about language. Sociology is about the social aspects of human life. But it is hard to point to the objects of mathematics. Numbers? What and where are they? Even if we take the heroic (or foolhardy) Platonic option that they are inhabitants of an abstract world beyond space and time, which we access through a mysterious faculty of intuition, we are left with no understanding of what mathematics tells us about the actual world we live in.
So by default mathematics has often been considered as not about anything at all. Instead it is thought of as a toolkit of methods, formulas, and rules, or as a theoretical juice extractor for getting one truth from another, but not itself actually about any aspect of reality. Sadly that view, understandable enough in physicists and engineers, is reinforced by the rule-based style of teaching that fills the traditional school curriculum: Minus times minus equals plus/ The reason for this we need not discuss.
Neither of those views of mathematics is correct. Mathematics is about certain real properties of objects. The words pattern, structure, and complexity are sometimes used to describe what properties mathematics studies, but they have a certain excessive generality and vagueness to them that creates a mental blank. The easiest object of mathematics to appreciate is symmetry. Dinner tables and the human body have symmetry (not always exact), while palindromes have a perfect symmetry. Symmetry is a property at once highly abstractmuch more so than shape and size, for instanceyet at the same time an easily perceived property of real things. If you are considering a career in politics and have a slight asymmetry in your face, either invest in plastic surgery or change your plans, because it is immediately obvious on television and humans are wired to react badly to it.
A book on symmetry is therefore possibly the perfect tool for giving a general audience a real insight into the heart of mathematics. Yet for all its obviousness, it is not an easy topic to make accessible. On the one hand, bilateral symmetrythe simple left-repeats-right symmetry of an isosceles triangle, a palindrome, or the human bodywould seem to be so simple as to exhaust very quickly what could be said about it. On the other hand, the more elaborate symmetries that interest mathematicians, such as the multiple symmetries of a cube or the elaborate repetitions in the closest packing of spheres in space, quickly become too abstruse for a popular audience (though Hermann Weyls Symmetry is, for a reasonably persistent reader, a brilliant introduction to abstract algebra via symmetry).
Still, it is amazing what can be said about very little. The Seinfelds of the intellectual world are surely the two successful popular books on the topic of zero, which go as far as you can with expansiveness of prose compared to exiguousness of topic. Martin Gardners Ambidextrous Universe is almost entirely devoted to bilateral symmetry.
Thanks especially to biology, there is no lack of material. The biological world is extraordinarily inventive with bilateral symmetry. The following examples just scratch the surface of what Gardner has to tell us. Animals are mostly approximately symmetrical between left and rightwhich no doubt saves on genetic information, as well as making it easy to balancebut there are characteristic departures from exact symmetry. In ourselves, the heart is very slightly to the left, while the right kidney is slightly lower than the left to allow space for the liver. Siamese twins, however, are mirror images of each other, so one has the heart to the left and one to the right. Much has been made, perhaps too much, of left brain and right brain, but there is certainly a difference between them as to the normal location of mental functions. Some animals make much more of a feature of asymmetry, such as fiddler crabs. The only bird whose entire beak twists to one side, Gardner tells us, is the wry-billed plover of New Zealand. It uses the beak to turn over stones when looking for food, and since the beak bends to the right, the bird looks for food mostly on the right.
Even more startlingly asymmetrical are the flatfish with both eyes on one side of the head, such as sole (dextral in European waters, sinistral in tropical and semi-tropical waters). Or rather, they are asymmetrical only as adults after swimming around as normal symmetrical fish in early life, after which they repair in adult life to a sedentary existence on the bottom. The bottom eye then moves around to the top. This condition gave Darwin considerable trouble in The Origin of Species, when perhaps his most intelligent opponent, St. George Jackson Mivart, urged that it was impossible for such a thing to evolve through a gradual series of small steps driven by natural selection. What is supposed to happen when the eye has evolved to be half-way round?
There are thousands of other striking instances of animal asymmetry. The akita dog in Japan with a tail that curls one way on males, the other way on females, the tendency of dolphins to swim counterclockwise around tanks, the asymmetric sex organ of the male bedbug, a fungus called laboulbeniales that grows only on the back left leg of a certain beetleand so on.
Plants and organic molecules do not invest in bilateral symmetry the way animals do, but they produce many phenomena of interest because of the possibility of enantiomorphic pairsthree-dimensional shapes like the left and right hand which are mirror images of each other but which cannot be rotated to fit in the same space as each other. Gardner criticizes science-challenged Shakespearean scholars for failing to understand the point of So doth the woodbine and the sweet honeysuckle gently entwist. Honeysuckle always twines in a left-handed helix and woodbine in a right-handed one, so they cannot grow in intertwined helices and instead produce masses of unruly spirals.
The microscopic world of the organic molecules that form the basis of life is the scene of some of the most fascinating enantiomorphs, called in this context stereoisomers. The complexity of carbon chemistry means that there are many molecules that have a mirror image counterpart that is essentially different, that is, the two cannot be rotated into one another. They are in one sense chemically the same, but living things can tell the difference. The difference in the smell of oranges and lemons is caused by right and left forms of limonene. In many cases, such as with amino acids, only one stereoisomer of the pair is found in living things. The other only occurs when the compounds are made artificially or extracted from meteorites that have brought amino acids from elsewhere in the universe, where they are apparently made by some random process not involving life. That has been taken as evidence that life on earth has only evolved onceor at least that all present life forms are descended from only one primitive form. If life had evolved many times, one would think, there would be both left-leaning and right-leaning micro-organisms, one producing amino acids of one handedness and the other the opposite. The argument is not watertight, since it may be that any life form of the wrong handedness was or is killed off by the environment created by the life forms of the dominant handedness. But the phenomenon is still one of the few windows we have on the scientific mystery of the origin of life.
The subtle same-yet-different aspect of enantiomorphs has been the subject of philosophical argument too. In his 1768 paper, On the first ground of the distinction of regions in space, Kant argued that space was a kind of stuff, substance, or ether (as opposed to being merely a way of describing the relations of distance between the things in it). Take your right hand. Now take it away from where it was and try to fit your left hand in the space that your right hand formerly occupied. It is impossible. But all the relations of distance between parts of your right hand are the same as for your left hand. Therefore, Kant said, there is more to space than just relations of distance. The space occupied by your right hand is a real thing, with the power (so to speak) of excluding your left hand. Gardner is not impressed with this argument and gives some interesting replies to it, but it is arguable that the discoveries of modern physics concerning the curvature of space and its ability to support gravitational fields have at least confirmed Kants conclusion that space is a real thing. Having variable curvature is a property not easily attributed to something that does not really exist.
The book is not as successful on physics as on mathematics. That is not Gardners fault. It is the fault of physics. Physicists keep changing their story about what is happening at the fundamental subatomic levels of reality, so a book with editions over fifty years needs extensive rewriting. With admirable honesty, Gardner recalls in the preface his 1952 article Is nature ambidextrous?: In it I considered the possibility that someday a basic law of nature might prove to be left-right asymmetric [normal laws like Newtons force = mass x acceleration do not distinguish between left and right, in that if a situation is permitted by the law then the mirror-image situation is also permitted by the law]. I ruled this out as almost unthinkable. Five years later, the unthinkable occurred. It was the shock of this discovery, the fall of parity, that prodded me into writing an entire book about mirror-reflection symmetry. He does give an account of that discovery that is just intelligible to the general reader and a readable story of the careful experiments of Madam Chien-Shiung Wu that established the result (after some other researchers had dismissed the experiments as not worth performing since the chance of asymmetry was so low). Although there is no inherent difference between the left and right hand or the south and north pole of the earth, for example (so that we could not communicate to an alien civilization in words which was which), there is a difference between one end of a cobalt 60 nucleus and the other: the south end is the end that is most likely to emit an electron. So, somewhere in obscure parts of the subatomic realm, the universe can tell the difference between left-handed and right-handed. That remarkable result is firmly established, but a mass of later developments in such controversial areas as string theory are less so, and Gardner is hard put to tell a consistent story through three editions. The present edition is a reprint of that of 1990 with only a preface and some references added, so it cannot be taken as a sure guide to the current state of physical theory.
Gardnerborn in 1914 and now with a lifetime of achievement in the popularization of science and the exposure of pseudosciencehas a style of exposition that is a relief after the overwrought efforts typical of contemporary popular science. He feels no need to sell his book by appeal to the mind of god, to concepts that hang between metaphor and paradox like the selfish gene, or to overstated claims about this or that discovery that changed the world. He does include some carefully chosen human interest stories, like Pasteurs well-justified excitement when his experiments on polarized light revealed the phenomenon of stereoisometry in organic molecules. But nearly all the book is an unvarnished account of the scientific and mathematical facts themselvescarefully selected and organized, but told with a confidence that those plain and nourishing ingredients will be palatable without overspicing by hype. It is a voice from another century. So much the better for that century.
- The New Ambidextrous Universe: Symmetry & Asymmetry from Mirror Reflections to Superstrings, by Martin Gardner; Dover, 401 pages, $18.95. Go back to the text.