The Imperfect Plane

The Euclidean plane is a strange and unnatural beast. Consider its myriad magical properties: it is perfectly flat and perfectly smooth, highly symmetric, stretching uninterrupted to infinity. This is a rather romantic image, like the surface of some incredibly placid sea, reflecting ghostly moon-circles and point-stars, featureless in itself but capable of showing us whatever we imagine. These aspects of undendingness and perfection must appeal powerfully to humans–they show up in all our stories, from unicorns to deities. Why not at the center of our mathematical mythology as well?

But consider, even the most vast and untroubled sea is warped, bent so that we can spot the masts of approaching vessels before the hulls. Ripples, however small, bend the light out of true and lend a wavery quality to our images. The Euclidean plane is a myth, no such animal can be found in our real experience.

This catalog of virtues–flatness, smoothness, infinitude and perfect symmetry–is merely a more natural way of stating the axioms of our traditional geometry, the instinct out of which those axioms arise, in fact. The flatness is guaranteed by the parallel postulate (a curved plane would have non-Euclidean regions–if the axiom is universal, the plane is flat) the ungrainy, uninterrupted evenness by the continuity axioms, the symmetry by the congruence axioms, the infinitude and simplicity by betweeness and continuity. The incidence axioms allow us to decorate our plane with lines and points (if they do not "define" these terms, they certainly constrict the usages so strictly as to make definition redundant. Gottlob Frege to Hilbert: "Axioms are saddled with something that is the function of definitions. It seems to be that this blurs the boundaries... in a serious manner"

[Frege, 7] ).

Mathematicians struggling to learn the mysteries of this supernatural entity have been accused of mysticism, and certainly they do seem to speak of it like a religion. Think of Plato, and his world of forms. Think of Edna St. Vincent Millay writing "Euclid alone has looked on Beauty bare," and G.H. Hardy saying, roughly "Mathematics is beautiful only when it is useless" (Hardy). These abstractions are better than the real world, they seem to say, or even more important. The properties of things in the real world depend on much that is random, on the temperature and time and the phase of the moon. But geometry is self-contained, pure, arising out of this flatness and smoothness with no exceptions to the rules, nothing redundant or uncertain.

It was, of course, in an ironic attempt to purify the discipline that the first of the complicated, less ideal geometries was discovered. The parallel postulate looked, especially in its original form, as though it should be provable from the others. Redundancy, extraneousness, being anathema to the souls of all true mathematicians, they tried for centuries to do so. If they succeeded, they could remove the suspicious postulate from their art. They failed, but ultimately managed to prove to themselves that the parallel postulate doesn't, in fact, follow from the others, which might have been just as reassuring had they but known to begin with. However, in the intervening thousand-odd years (the timelessness of the mathematics is one of its most awesome qualities, along with the fact that it has been built by peoples all over the globe of every creed and color.) Some tried to manage a proof by indirect means–show that it makes no sense to have the other axioms without the parallel postulate, and you've shown that they entail it.

Since they happen not to entail it, then this method led instead to an experiment no one had dared to try before. What happens when you try to do math on a different plane? In this case, without meaning to, the ancient mathematicians were doing geometry on a warped plane. This they interpreted as a catastrophe.

It implied to them not that their tools had a broader reach than they had known, applying just as well to the more complex, organic curvy surfaces surrounding them, but that their vision of perfection, the plane, was flawed. Unbearable!

They had the misfortune of discovering an idea far before its time, and until the context was there for it, it went no further than their disappointment. (This need for context is probably why certain concepts, when their time has come, seem to appear everywhere at once. Hyperbolic geometry was discovered over and over again by people who feared derision or apathy if they pursued it. Once that fear was gone, there was nothing to stop the most recent discovers from doing that pursuit, rather like Columbus taking advantage of the politics of Europe to make sure the New World stayed discovered this time.) By the nineteenth century, though, philosophy had prepared us for the constructed reality and formal systems, and it had dawned on mathematicians that the rules they had been playing by, though chosen because they seemed "obvious", could be safely treated as arbitrary.

Points and lines were redefined and then "undefined" in that first giddy frenzy, as it became clear that the crippled versions of the plane which had horrified earlier geometers, violating their monotheistic faith in that perfect form, were in fact merely new constructs. They have (it can be proved) the same degree of reality or unreality as the Euclidean plane and are as rich in surprises, unexpected connections, in beauty as it is.

This discovery had a liberating effect. If discarding the parallel postulate has such rewarding effects, what kinds of worlds can we build by discarding some others?

Clearly, we do not expect and inconsistency to arise from any of these, because if one did, mathematicians would be able to use it as a reductio ad absurdum proof that said axiom depended upon the others. This in itself would be a startling and worthwhile piece of knowledge. But since the axioms were chosen to be independent, that is unlikely. The more relevant question is, do these negatives take us any place worth going?

Because geometry (and all of mathematics) is a formal system, nothing more than a set of rules, it is to be expected that abandoning some rules leaves you with, well, less. Neutral geometry does not have as many theorems as either Euclidean or hyperbolic. This is the price of generality, and the most universal of all geometries are the "incidence geometries", relying on incidence axioms alone. Models for this include everything from stunted three-point systems to Euclidean geometry itself. On the whole, the stunted models must be much more common than the interesting ones, but buried in the infinite permutations are some marvelous discoveries.

Topology itself is built on the incidence, continuity, and betweeness axioms, and by discarding the concept of congruence allows us to say much that is very broad, and certainly not uninteresting. Principle among its manifold virtues is that it can be applied to everything from our old-fashioned plane to a surface with a four-dimensional twist. It predicts the way lines intersect and shapes interact without prejudice, and while it says much less than Euclidean geometry, what it says can be applied to much more. The surface of my desk may approach the perfection of the Euclidean plane, but topology will tell me about the rest of the form, how the wires weave among the crossbeams.

In addition, some of these gem-geometries are built on not the mere casting off of Euclidean axioms, but an actual negation of one or more. The most obvious, almost as old as hyperbolic geometry, is elliptic. This involves the negation of the parallel postulate as well, but this time we take our poor plane, stretch and bend it until the edges touch, form it into a closed surface. And because the lines close in on themselves, the "betweeness axioms" which state that, of any three collinear points, only one is between the other two and that points can always be found "outside" a given set and that a line divides a plane, must be negated in order to describe this new surface. (Greenberg 120). All collinear points are between all others, and lines are finite. This is the geometry of the surface we live upon, where ships appear mast-first over the horizon. With a little stretching of our ideas of "points" and "lines" we can develop something which rather resembles our familiar longitude and latitude. A not-so-exotic geometry born of the negation of multiple axioms.

A newer and more obscure branch of study effectively negates the "continuity axioms", although this is not exactly how it was developed. Discrete geometry fragments the plane. Pictures made of pixels, "points" of measurable size which do not overlap or flow into one another, are one more modern application of the mathematics of finite and separate sets: "The twentieth century has witnessed the creation of many addition subjects in the discrete direction. This development was spurred by the rapidly increasing use of mathematics in the social, behavior, decisional, and system sciences, where the objects under investigation are typically finite in number and not readily approximated by some continuous idealization" (Feder, 10). Graph those equations on your digital computer with it's punctured plane display, and you'll see the results of negating another aspect of the perfect plane.

On a harmonic note, "fractured" planes give rise to some rather shocking shapes–"fractals" grew up alongside discrete geometry by technological coincidence, the computer power needed to produce them happens to display in pixels. But there are no cuts in fractal geometry; zoom in on those pixels and more points appear between. Indeed, fractal geometry depends heavily on the idea of the infinite point density of the plane. They are perfectly consistent with the stated axioms of Euclidean geometry, and yet they surprise us, jar our intuition badly. This is because they violate an aspect of that plane-image which is not explicitly stated in Hilbert's axioms (although it is in a sense assumed in his definition of "inside", which in turn provides part of the "definition" of a line). The thing they violate is evenness, an even density everywhere. The brokenness the name implies exists in the way points are connected. They connect complexly, lines branch infinitely. A fractal shape cannot be properly closed, does not have any reliable measure of "inside" versus "outside", because the borders are always fuzzy to a degree. Bubbles of continuous smoothness are bound by smaller bubbles, which are in turn defined by smaller bubbles yet. This is the "density" variation the branching implies, the placid surface of our plane churned to a froth.

It is illuminating to examine this particular case because it shows us to what an extent our pretty formal system is born of qualitative considerations. Fractal geometry is a revolution not because it violates any of our particular rules, but because it violates the intuition from which those rules are born. It has no place in that spreading silvery Euclidean sweep, and thus it takes us by surprise. Fractal geometry is the trees on the bank.

Elliptic geometry is the curve of the horizon. Discrete math is the pebbles on the beach and the cracks in the mud. Topology, if we may carry the metaphor to a more whimsical extreme, is the inner tube we toss into this majestic scene to make it human, we less solemn worshipers than our ancestors. All of this beauty is here for us to play with, to build castles from or dive into until we are above our heads and splashing. Irregularities we see not as defilements, but as new places to explore.

Tricot says that the mathematical curve is nothing more than the intersection of all the real curves which have been drawn with stubby pencils on rough paper (Tricot 10), a delightful reversal of the Platonic picture. The real world, with its unpredictability an diversity, with all its broken symmetries, usually has all of the properties of those alternative geometries at once. It is grainy and lumpy and warped and punctured and non uniform, and it is stunningly beautiful anyway.

 

Greenberg, Marvin Jay. Euclidean and Non-Euclidean Geometry. W.H. Freeman and Company. New York, 1993.

Frege, Gottlob. On the Foundations of Geometry and Formal Theories of Arithmetic. Translated and with an introd. by Eike-Henner W. Kluge. New Haven : Yale University Press, 1971.

Feder, Jens. Fractals. Plenum Press, New York, 1988.

Hardy, G.H. A Mathematician's Apology. (Essay)

Tricot, Claude. Curves and the Fractal Dimension. Springer-Verlag, New York 1995.