The answer to the question
What is Geometry ?
The answer to the question, "What is geometry ?", is not as straightforward as you might think. Originally, geometrin, as the Greeks called it, was "earth measure"; the science of measuring land. The Greeks, however, took a collection of useful results which were known to work and strived to understand why they worked. Wanting to know why was revolutionary and mathematical proof was the hot new idea that was to satisfy their curiosity. From five basic ideas called postulates, each of these in itself obviously true, and then by the use of logical reasoning, the Greeks aspired to understand the world about them.
From their five postulates, which can also be refered to as axioms, the Greeks built up an extensive body of knowledge which was all summed up in a thirteen volume textbook written by Euclid in around 300 BC and titled Elements. Most of the geometric results that are taught and studied in schools are in that text and, in fact, this Euclidean geometry, as it is now called, is what Isaac Newton (1642-1727) used to mathematically picture the physical universe, so effectively, in his great works. This, it seemed at the time, was the realisation of the Greeks' dream. Confirmation of their belief that the answer to the question, "What is Geometry ?", was that Euclidean geometry was the mathematics used by nature throughout the universe.
However, Euclid's mighty work, so often held up as an example of brilliant mathematics, contained flaws in the form of hidden assumptions. Right from the word go, before one even got to his five starting postulates, he assumed, for example, that lines exist. Yet, in the real world, it is not possible to draw a line that has no thickness ! Euclid had introduced a mathematical abstraction and this was then applied to problems in a world in which it could not be. The abstract notion of a breadthless line had turned out to be useful as Isaac Newton showed, but there seemed to be more building blocks at the base of Euclidean geometry than the stated five postulates. If man was going to create strange things called lines, then he should be honest about it by including such statements as "lines exist" in the list of basics and also attach the label, man-made.
Figure 1 : Asking secondary schoolchildren what it means for a rectangle to have an area of 15 cm² will generally elicit good explanations of how 15 squares of side 1cm cover the rectangle and that "area rectangle = length times breadth". It can be fun to ask them, "What about the other side ?" for this paves the way for discussion concerning a cone of surface area 15 cm². If the cone is solid, the 15 cm² would include the area of a circular base. But suppose we have just the shell of a cone. You may like to imagine this being made from, say, three quarters of a circle of card with the straight edges seemlessly joined. Clearly, there is no circular base, but should the 15 cm² now include the inside surface of the cone as well as the outside ? Discussing such ideas and concepts is what makes mathematics interesting.
Blemishes in Euclid's work, such as omitting to assert that lines exist, were sorted out by several mathematicians, most notably by the German, David Hilbert (1862-1943). In the process Euclidean geometry grew beyond what it was felt could be grasped from its most elementary starting postulates by schoolchildren and the study of geometry from basics became a University level topic.
It was from this sorting out process that a truly troublesome aspect of the whole basis of Euclidean geometry became most acute. Euclid's five postulates, you may recall, were each supposed to be obviously true. The first four could be amended and enlarged upon so that they were but not so The Fifth Postulate. Euclid himself had been aware that it was the weakest of his five and it has stubbornly refused all attempts to elevate it to the status of "obviously true" for a period of over two thousand years. Euclidean geometry, although down in status from "made by God" to "made by man", was still thought by many to be made such that it imitated perfectly the geometry of the cosmos.
Figure 2 : In schools we teach Euclidean geometry of the plane and yet our pupils live in a world better modeled by a sphere. On the surface of such a world the sum of the angles in a triangle do not sum to 180 degrees. This can be easily demonstrated on a football by travelling "straight" from point A on the equator to point B, the north pole, then making a right angled turn to journey directly to point C on the equator, then turning right 90 degrees again to proceed along the equitorial shortest path back to point A. The sum of this triangle's three angles is 270 degrees. The lesson is all the more relevant for Einstein's realisation that our space is better described using Riemannian geometry, of which the sphere is a model in Euclidean geometry.
The fifth postulate concerned the notion of parallelism. By way of understanding it let us consider figure 3 which shows two rays perpendicular to segment AB. Euclid's fifth postulate stated, in effect, that the perpendicular distance between the rays remained equal to the distance from A to B as one moved to the right. You may feel that this is, indeed, "obviously true" but how could you be sure ? The possibility that such "parallel" rays could diverge leads to a very different non-Euclidean geometry, the first published exploration of which was by a Russian called Nikolai Ivanovich Lobachevsky (1792-1856). The weird new world thus created was every bit as consistent as the more established world of Euclid and, mathematically speaking, an equally valid one.
The upset caused was enormous. Which geometry a true description of the real universe ? Was it Euclidean geometry, or this new fangled Hyperbolic geometry of Lobachevsky, or one of the many other emerging non-Euclidean geometries ?
Figure 3 : In Euclidean geometry we may begin by defining parallel lines as having the property that the perpendicular distance from A to B does not change as one moves to the right. This is all very well but do light rays, the "straight lines" or rather, geodesics, of our universe, have this property ? Newton assumed they did but Einstein preferred the mathematics that resulted from assuming they did not. It has been speculated that just as a new geometry better enabled man to understand the very big, perhaps some other geometry needs to be found to cope with the subatomic.
The great mathematician Carl Friedrich Gauss (1777-1855) allegedly performed an experiment to try to settle the matter. Using surveying equipment atop three mountain peaks, each of which he considered to be a vertex of a large triangle, he tried to determine if the sum of the angles in a real world (plane) triangle added up to 180 degrees, this being an equivalent statement to the parallel postulate. The results were inconclusive because of experimental error. He could only state that the sum of the angles of a triangle could indeed be 180 degrees, although it could also be a little more or a little less ! Indeed, such an experiment could never prove that Euclidean geometry was the geometry of the real world because of the impossibility of measuring any angle exactly, but it might have proved it false had the sum been well above or below 180 degrees.
The non-Euclidean geometries were becoming the Loch Ness Monsters of mathematics ! Mysterious creatures, rumoured to exist, but no one had actually seen one. Looking to see a non-Euclidean geometry was like looking for Nessy. Never could you prove she did not exist but, just possibly, you might get lucky and actually see her, thereby proving that she did.
Figure 4 : Non-Euclidean geometry became the Loch Ness Monster of Mathematics.
The mystery deepened when Albert Einstein (1879-1955) developed further a non-Euclidean geometry of George Riemann's (1826-1866) and used it as a basis for his general theory of relativity. Einstein let go of a treasured physicists belief that light travelled in lines that were "straight" in the Euclidean sense of the word. Einstein realised that his equations would be a lot more meaningful if he allowed light to follow paths which, although "curved" in Euclid-speak, are still "straight" (geodesics) in the new geometry. The mathematical "playthings" of non-Euclidean geometries suddenly were a vital part of explaining, in an elegant manner, a universe that was proving to be vastly more complicated than the Greeks had ever envisioned.
Figure 5 : A time line to show the approximate dates during which the named mathematicians lived.
This is not to say, however, that space is, even now, truly of the geometry used to describe it. But at least we have a geometry in which "line" has the interpretation we want; that of "light ray". Here, we seem to glimpse a deep truth; the geometry that we have chosen to use is a matter of convenience.
Back to the question; "What is geometry ?". It seems that it is not a something that is connected to the real world. In other words, all of every geometry is abstract. Man has made many geometries. Those that are the most useful are those that seem to have a physical interpretation in the real world but if a geometry is not easy to apply to the objects under study then we are free to discard it and search for another. This, exactly, was Benoit Mandelbrot's opening tactic in his famous book The Fractal Geometry of Nature in which, in the first paragraph, he discards Euclidean geometry with the words that, "Clouds are not spheres, mountains are not cones and lightning does not travel in a straight line". Euclid, still ideal for man's engineering designs was, according to Mandelbrot, wholely inadequate at describing elegantly the complexity of nature's shapes.
Figure 6 : Using Euclidean geometry to describe even a simple object from nature, such as a flower head, would be extraodinarily tedious. One could, for example, type in the coordinates of every branch intersection and every blossom. A more appropriate geometry, fractal geometry, allows mathematicians to more elegantly capture the apparent complexity of the biological world. The key mathematical ideas involved are iteration and recursion.
That we can play this game of "find a geometry to describe what we see in reality" is quite remarkable. That finding a good set of consistent axioms sets us off on long trails that then diverge so little from what is real is even more remarkable. That these trails of logical thought mirror our reality whilst being disconnected from it is most remarkable of all. Is it that the mathematical universe is so vast that we can find all of our own universe within it, or is it that our own universe includes all of the mathematical one such that all mathematical discoveries apply to some aspect of our reality ? We still do not know what geometry truly is but, bit by bit, as it reveals its quirky characteristics and faces to us, we can at least claim to have come to know it a little better.
I would like to thank The Open University for inviting me to a Non-Euclidean Geometry Day in March 1992, which opened my eyes to the exciting worlds of such geometries.
The following texts are excellent starting points for a first study of detailed geometry.
Gray, J (1989) Ideas of Space (2nd Edition), Oxford University Press.
Greenberg, MJ (1980) Euclidean and Non-Euclidean Geometries: Development & History. WH Freeman & Co.
Mathematics In School, September 1993, Volume 22, Number 4.
© The Mathematical Association. ISSN 0305-7259
Figures 2, 3 & 5 redrawn, © Martin Hansen, April 2008
New figures 1, 4, 6 & 7, © Martin Hansen, April 2008
Figure 7 : The author's software being used to produce new graphics for the article.
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