Let's solve the following problem:

Most people react with disbelief on hearing that the problem has solutions. The four directions (West, North, East, and South) are successively perpendicular to each other. So how can this be? Here's one solution. (The problem has a whole continuum of solutions so that not much will be lost if I give away one of them.) Consider the North Pole. Going 10 mi South from the Pole brings one on a parallel each point of which is located 10 mi South from the North Pole. Walking straight West one stays on the same parallel and, therefore, at the same distance from the Pole. To get there, just stroll 10 mi North.

Poles require a special consideration but everywhere else the four directions do form a cross with four right angles. Our solution to the problem shows that there is a triangle with two right angles at the base (which is already strange) and a nonzero angle at the top. There is no escaping it: there is a triangle whose angles sum up to more than 180^o. This is not exactly what we are taught in high school. Every one who took a Geometry class knows that three angles of a triangle sum up to 180^o.

The high school geometry is Euclidean. Laid down by Euclid in his Elements at about 300 B.C., it underwent very little change until the middle of the 19th century when it was discovered that other, non-Euclidean geometries, exist. I wonder about the source of the above problem. Was it invented in the last century? Before? After?

Discovery of non-Euclidean geometries had a profound impact on the development of mathematics in the 19th and 20th centuries. For more than two thousand years Elements served as a mathematical bible, the foundation of the axiomatic method and a source of the deductive knowledge. Euclid's postulates, however, have been based on our (or his) intuition of geometric objects. With the discovery of non-Euclidean geometries, the Elements were scrutinized and logical omissions were found. As an upshot, axiomatic method has been divorced from intuition and formalized, which eventually led to the development of Metamathematics and Model Theory and ultimately to Godel's Theorems and Abraham Robinson's Non-Standard Analysis. Einstein's Theory of General Relativity is based on the idea that material bodies distort the space and redefine its geometry.

References:

1. J.N.Cederberg, A Course in Modern Geometries, Springer, 1995, Corrected third printing
2. H.S.M.Coxeter, Introduction to Geometry, John Wiley & Sons, 1961
3. D.M.Davis, The Nature and Power of Mathematics, Princeton University Press, 1993
4. F.J.Davis and R.Hersh, The Mathematical Experience, Houghton Mifflin Co, 1981
5. K.Devlin, MATHEMATICS: The Science of Patterns, Scientific American Library, 1997
6. W.Dunham, Journey through Genius, Penguin Books, 1991
7. H.Eves, Great Moment in Mathematics After 1650, MAA, 1983
8. From Five Fingers to Infinity, F.J.Swetz (ed.), Open Court, 1996, Third printing
9. T.Heath, Euclid's Elements, Volume I, pp 202-220, Dover Publications, NY
10. D.Hilbert, Foundations of Geometry, 10th Edition, Open Court, LaSalle, IL, 1971
11. The History of Mathematics, ed J.Fauvel and J.Gray, The Open University, 1987
12. M.Kac and S.Ulam, Mathematics and Logic, Dover, 1968
13. Studies in the History of Mathematics, E.R.Phillips (ed), MAA, 1987

• Non-Euclidean Geometries, Introduction
• The Fifth Postulate
• The Fifth Postulate is Equivalent to the Pythagorean Theorem
• The Fifth Postulate, Attempts to Prove.
• Similarity and the Parallel Postulate
• Non-Euclidean Geometries, Drama of the Discovery.
• Non-Euclidean Geometries, As Good As Might Be.
• The Many-Faced Geometry
• The Exterior Angle Theorem - an appreciation
• Angles in Triangle Add to 180^o

Copyright © 1996-2008 Alexander Bogomolny

Copyright © 1996-2008 Alexander Bogomolny