ESCHER AND NON-EUCLIDEAN GEOMETRY --------------------------------- At least since Renaissance painters developed systematic perspective, artists have considered mathematics a tool, a means to accomplish effects both strikingly naturalist and unusual. M.C.Escher, as well as a few others, have gone further by representing in their work objects and ideas that come specifically from mathematics, thereby embracing mathematics as an integral part of the world about them, as an important component of their cultural heritage. In this respect, M.C.Escher made no works more striking that the otherwise unimposing woodcuts entitled Circle Limit I,II,III and IV. Looking at the circle limit woodcuts, one may admire them for the precision of their technique, especially fastidious and delicate treatment of detail near the boundary circle. Escher uses merely intricate detail to evoke manifestly infinite detail, details beyond the capabilities of human hands, beyond the power of human reason. One admires precision, but also wit, for example the joke of forming devils out of the negative spaces left by a host of angels. Even more than for precision and wit, these works seem miraculous for their structure, their myriad but gently distorted repetitions which project a mysteriously harmonious web of endlessly interconnected circular arcs. Without much trouble we can imagine Escher's precision as the product of highly honed skill and obsessive patience; his wit as the expression of subversive imagination and long experimentation, but the structures? One doesn't just happen upon such structures accidently, if one didn't have these structures right in front of one's eye, one could easily believe that structures like these couldn't exist. Escher did not invent these structures out of whole cloth. These woodcuts elaborate on illustrations that one can find in various monographs about one of the most important scientific revolutions of the 19th century, the invention (or discovery, if you prefer) of non-Euclidean geometry. THE LANGUAGE AND THE OBJECTS OF PLANE GEOMETRY ---------------------------------------------- The subject of geometry begins with two notions familiar from daily experience, the point and the line. While one may call these familiar as notions, one must also note that we have no actual points or lines about us, for points and lines exist only as idealizations. A dot of ink that may represent a point actually consists of many trillions of molecules and actually encompasses an infinitude of points. Similarly the straightness of a ruler's edge break down under a microscope of sufficiently powerful magnification. Still, we see approximate points and lines wherever we turn and we rarely stop to reflect on imprecision involved. Configurations of points and lines turn out to have some remarkable properties. Though one may choose from many thousands of striking theorem, we give just four examples: i) Morley's Theorem: The trisectors of the 3 angles of any triangle ABC meet in pairs to form the vertices of an equilateral triangle. ii) The adjacent bisectors of a quadrilateral meet in pairs to form four points lying on a common circle. iii) Miquel's Theorem: Pick points x,y,z at random on the 3 sides of any triangle ABC, with x on AB , y on BC and z on CA). Form the 3 circles, one passing through A,x and z, one through B,x and y, and one through C,y,z. These 3 circles always share a point. iv) Suppose the vertices of triangle ABC lie on a circle which also contains the point P. The points P_1, P_2 and P_3 obtained by reflecting P through the sides of ABC lie on a common line. How does one come to believe such facts with complete certainty? Certainly not by mere experimentation, for two reasons: first, no finite number of experiments ever rules out the possibility of an exception eventually arising and second, no single experiment has the precision to verify that, say, three points lie exactly on a common line, or that 4 points lie exactly on a common circle, etc. So not by experimentation, but rather we believe such results on the basis of rigorous logical argument. Now logical argument deduces new facts from known premises. So where does one start? More than two millenia ago, Euclid collected certain statements (called axioms or postulates) about points and lines which he considered self-evident. Nowadays we don't look on axioms as self-evident statements, but rather as statement by which we specify what we mean by, in this instance, the words ``point'' and ``line.'' Thus we hope that our geometry speaks about points and lines in the ordinary sense of the words, but we remain open to the possibility that these words might have other interpretations that fit our axioms equally well. Indeed each new axiom narrows the range of interpretation of these words, a mathematician would say constrains the class of models. Since Euclid, many people have formulated systems of axioms for plane geometry, all more or less logically equivalent in the sense that the axioms of one occur as consequences of the axioms of another. We give one example here, following a recent textbook, Geometry - A Metric Approach Via Models, which emphasizes how each new axiom narrows one's notion of what constitutes a plane geometry. This setup merits careful study. THE RISE OF NON-EUCLIDEAN GEOMETRY ---------------------------------- Now ever since Euclid, mathematicians have wondered whether Euclid's Parallel Postulate (EPP) follows as a consequence of the other axioms. Over the years, hundreds of mathematicians attempted proofs of EPP, some seemingly more convincing that other, but all flawed in the end. Not until the 19th century did three mathematicians Gauss, Lobachevski and Bolyai, independently prove EPP unprovable. Now, to actually prove EPP from the other axioms, if one could, that would constitute a theorem in geometry. But to prove that EPP has no proof, that counts as theorem of logic, the mathematics of proofs themselves. How can one prove a statement unprovable? We sketch the brilliant idea behind the approach in a slightly more modern form, due the Henri Poincare', a French mathematician who worked around the turn of the century. Poincare' would have us reinterpret the words point and line. I'll say ``point'' and ``line'' (with quotes) when I mean these words in Poincare''s strange sense, and I'll leave off the quotes when I want to use these works in their ordinary sense. Poincare' fixes an ordinary circle C within an ordinary plane. The for Poincare', ``point'' means _point lying within circle C_ and ``line'' means _circular arc within C crossing C at right angles_. Now, of course, you could choose to mean by ``point'' and ``line'' anything you damn pleased, but Poincare''s reinterpretation has some very special features. Namely, if you go back to the axiom for a NEUTRAL geometry and put quotes around all the occurences of the words line and point, you'll obtain only true statements about ``points'' and ``lines''. How makes us sure of the truth of these statements? By replacing occurence of ``point'' by the phrase _point within C_ and of ``line'' by _circular arc within C crossing C at right angles_, we'll get a perfectly ordinary sequence of statements in plane geometry. Turns out that one can prove each of these statements using the EUCLIDEAN GEOMETRY axioms, INCLUDING EPP itself. (Verifying this claim involves lots and lots of hard work, so it took ingenious insight to see in advance that something like this could work out!!) In contrast, putting quotes around point and line in EPP leads to a manifestly false statement. Indeed, given a circular arc A in C crossing C at right angles and a point p in C not on the arc A, we can always find infinitely many circular arcs through p in C crossing C at right angles and missing A. Now we can explain what makes EPP unprovable. Suppose someone came to you with a neutral geometry proof of EPP. Their proof would speak of points and lines. You could add quotes whereever they mentioned point or line. This wouldn't alter the logic at all, since all the axioms about points and lines apply equally well to ``points'' and ``lines'' (by all the hard work that we've skipped). So one could add the quotes to the conclusion too, to EPP. But that leads to a false statement. So we have a proof of a false statement. That can't happen, of course, so we must reject the idea that anyone could come up with a neutral geometry proof of EPP. NON-EUCLIDEAN GEOMETRY AND ESCHER --------------------------------- A casual inspection of Escher's Circle Limit woodcuts suggests a relationship to plane symmetric patterns. Like the latter, a finite motif undergoes an infinite repetition which displays at least combinatorial regularity. Nevertheless, one finds few reflections, glide reflections and gyrations, at least in the Euclidean sense. It turns out that non-Euclidean geometry requires us to measure distance in a different way. One may compare Poincare''s model, the hyperbolic plane, to a map of the earth. In a typical Mercator projection one finds small distances stretched profoundly near the poles (indeed such maps must omit the poles lest they stretch infinitely up and down). In Poincare''s model one has just the opposite: large hyperbolic distances appear compressed as one travels toward the boundary. The exact formula for computing hyperbolic distance lies beyond the scope of this lecture, but an exotic notion of distance requires change in our notions of translation, reflection and gyration, since these woulds should specify various types of (hyperbolic) distance preserving motions. Describing the distance preserving motions actually turns out easier than describing the distance, and we may eventually return to this theme when and if we take up the complex numbers. The richness of Poincare''s geometry, the hyperbolic plane, arguably exceeds Euclidean geometry itself. The theory of symmetric patterns gives one indication of this. While symmetric patterns on the Euclidean plane all possess one of 17 symmetry times, symmetric patterns on the hyperbolic plane show inexhaustable variety. Escher's Circle Limit woodcuts show just 4 examples. One may extract templates from these patterns and fold them into polyhedra, just as we did with plane patterns. Computing V-E+F gives negative values in all four cases (try it!). [I will put these patterns on-line soon.] A difficult theorem says that any polyhedron with V-E+F negative occurs as the polyhedron of a symmetric hyperbolic pattern. For all we know, our physical space manifests a hyperbolic geometry on a large scale, so we can't simply prefer Euclid's geometry. Even if ordinary space turns out basically Euclidean, the mathematics of non-Euclidean geometry show up in fundamental, if indirect ways in theoretical physics, and so bears on our understand of the real world. The richness of hyperbolic geometry also makes it a tool for other parts of mathematics, and in Andrew Wiles' recent and spectular proof of Fermat's Last Theorm n n n (the unsolvability of a + b = c for integers n>3 and integers a,b,c >0). BEYOND GEOMETRY --------------- With EPP, do we have all the axioms for geometry that we need? It turns out that these axioms prove exactly all the true statements about the Euclidean plane. While there do exist exotic interpretation of ``point'' and ``line'' that satisfying *all* the axioms of Euclidean geometry (not just the axioms of neutral geometry), they also satisfy all the theorems, so anything you can say about these models you can also say about the ordinary Euclidean plane and vice-versa. The set of counting numbers constitute another structure nearly as old as mathematics itself. At first sight they seem simpler, more primitive than the Euclidean plane, but it turns out that we have no systems of axioms about the counting numbers from which we can prove exactly all the true statement about them. Nor will we ever have such axioms, as Kurt Godel showed in the 1930's. One can say that a complete understanding of even the basic properties of the counting numbers lies beyond human reason, even in principle. Mathematicians have extended the method of giving familiar words exotic interpretations in order to discover the limits of proof which first arose with non-Euclidean geometry to many domains. In particular, the last 30 years has seen a detail investigation of so called non-standard arithmetics, systems of numbers which mimic all the obvious properties of ordinary counting numbers, but differ in significant ways that allow us to show the unprovability of certain true statements from various conventional (but inadequate) collections of axioms. Philosophers who seek to understand the nature of knowledge and truth examine and reexamine Godel's work very closely because the notion of counting number seems so fundamental to our experience, perhaps even independent of our experience, and yet already presents such fundamental difficulties.