Friday, July 17, 2009

Poincaré's Great Topological Papers

Henri Poincaré had learned of non-Euclidean geometry from Beltrami who first noticed that hyperbolic geometry was a constant curvature geometry in which the curvature is negative. Beltrami pointed out that a surface called the pseudosphere carried such geometry. He also discovered the disk model and realized that Riemann's conception of geometry provided the link that united the two conceptions. Poincaré worked out the details and discovered more models. Moreover, Poincaré did not stop at two dimensions. He extended this work to higher dimensional geometries. Here he is, in his own words, describing his model of three-dimensional hyperbolic space:

"Suppose, for example, a world enclosed in a large sphere and subject to the following laws: The temperature is not uniform; it is greatest at the centre, and gradually decreases as we move towards the circumference of the sphere, where it is absolute zero. The law of this temperature is as follows: If R be the radius of the sphere, and r is the distance of the point considered from the centre, the absolute temperature will be proportional to R2-r2. Further, I shall suppose that in this world all bodies have the same coefficient of dilatation so that the linear dilatation of any body is proportional to its absolute temperature. Finally, I shall suppose that a body transported from one point to another of different temperature is instantaneously in thermal equilibrium with its new environment. There is nothing in these hypotheses either contradictory or unimaginable. A moving object will become smaller and smaller as it approaches the circumference of the sphere. Let us observe, in the first place, that although from the point of view of our ordinary geometry this world is finite, to its inhabitants it will appear infinite. As they approach the surface of the sphere they become colder, and at the same time smaller and smaller. The steps they take are therefore also smaller and smaller, so that they can never reach the boundary of the sphere. If to us geometry is only the study of laws according to which invariable solids move, to these imaginary beings it will be the study of laws of motion deformed by the differences of temperature alluded to..

Let us make another hypothesis: suppose that light passes through media of different refractive indices, such as the index of refraction is in R2-r2. Under these conditions it is clear that the rays of light will no longer be rectilinear, but circular.... If they [the beings in such a world] construct a geometry, it will not be like ours, which is the study of movements of our invariable solids; it will be the study of the changes of position which they will have thus distinguished, and will be 'non-Euclidean displacements', and this will be non-Euclidean geometry. So that beings like ourselves, educated in such a world, will not have the same geometry as ours."
from The Poincaré Conjecture Donal O'Shea

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