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.

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 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 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.

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.

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.

References :

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.

© The Mathematical Association. ISSN 0305-7259

New figures 1, 4, 6 & 7, © Martin Hansen, April 2008