What Is Geometry?Author(s): Martin HansenSource: Mathematics in School, Vol. 22, No. 4 (Sep., 1993), pp. 9-11Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30215022 .

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What is geometry ? by Martin Hansen, The Schools, Shrewsbury

The answer to the question, "What is geometry?", is not as straight forward as you might think. Originally, geome- trein, 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 iself obvi- ously 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 knowl- edge which was all summed up in one thirteen volume text- book written by Euclid in around 300 BC and titled Elements. Most of the geometric results that we teach and study in school 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.

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 pos- tulates by schoolchildren and the study of geometry from basics became a University level topic.

It was from this sorting out process that a truely trouble- some aspect of the whole basis of Euclidean geometry

Mathematics in School, September 1993 9

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B

A

Ic

Fig. 2(a)

Fig. 1 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 turn to journey directly to point C on the equator, turning right again to proceed along the equitorial shortest path back to point A. The sum of this triangles 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.

Fig. 2(b)

Fig. 2 Using Euclidean geometry to describe even a simple object from nature, such as a plant leaf, involves great complexity. To represent the structure of its veins in a computer, for example, might involve typing in the coordinates of all vein intersections. To model the vein structure of even a single tree by such a process would be insane. In a new type of geometry, called fractal geometry, the mathematics "grows" the object in much the same way as nature does. The film strips illustrate an iterative process at work upon a single elementary building block. Even this most simple of iterative examples has resulted in a leaf like form.

became most accute. Euclid's five postulates, you may recall, were each supposed to be obviously true. The first four could be ammended 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 stubornly 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.

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 dis- tance 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 pub- lished exploration of which was by a Russian called Nikolai Ivanovich Lobachevsky (1792-1856). The weird new

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A B

Fig. 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 prefered the mathemat- ics 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.

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 was what mathematicians had for so long thought geometry should be; 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?

The great mathematician Carl Friedrich Gauss (1777-1855) allegedly performed an experiment to try and 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

Einstein realised that his equations would be a lot more meaningful if he allowed light to follow paths which, although are "curved" in Euclid-speak, are still "straight" (geodesics) in the new geometry. The mathematical "play things" 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, truely of the geometry used to describe it. But at least we have a geometry in which "line" has the interpret- ation 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 engin- eering designs was, according to Mandelbrot, wholely inad- equate 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

1650 1700 1750 1800 1850 1900 1950 I I I I

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Fig. 4

angles in a real world triangle added up to 180 degrees, this being an equivalent statement of 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! Even when shifted to astro- nomical triangles formed by stars the experiment gave inconclusive results. 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 Lock 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.

The mystery deepened when Albert Einstein (1879-1955) developed further a non-Euclidean geometry of Georg Riemrnann's (1826-1866) and used it as a basis for his general theory of relativity. Einstein let go of a treasured physisists belief that light travelled in lines that were "straight" in the Euclidean sense of the word.

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 truely is but, bit by bit, as it reveals it's quirky character- istics and many faces to us, we can at least claim to have come to know it a little better.

Acknowledgement 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.

References The following text are excellent starting points for a first study of

detailed geometry. Gray, J. (1989) Ideas of Space (2nd edition), Oxford University Press. Greenberg, M. J. (1980) Euclidean and Non-Euclidean Geometries:

Development and History. W H Freeman and Company.

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