The Problems of Nineteenth Century Geometry
Is there one 'geometry' or many? Since Euclid, geometry had meant the geometry of Euclidean space of two dimensions (plane geometry) or of three dimensions (solid geometry). In the first half of the nineteenth century there had been several developments complicating the picture. Mathematical applications required geometry of four or more dimensions; the close scrutiny of the foundations of the traditional Euclidean geometry had revealed the independence of the parallel postulate from the others, and non-Euclidean geometry had been born. Klein proposed an idea that all these new geometries are just special cases of the projective geometry, as already developed by Poncelet, Möbius, Cayley and others. Klein also strongly suggested to mathematical physicists that even a moderate cultivation of the projective purview might bring substantial benefits to them.
With every geometry, Klein associated an underlying group of symmetries. The hierarchy of geometries is thus mathematically represented as a hierarchy of these groups, and hierarchy of their invariants. For example, lengths, angles and areas are preserved with respect to the Euclidean group of symmetries, while only the incidence structure and the cross-ratio are preserved under the most general projective transformations. A concept of parallelism, which is preserved in affine geometry, is not meaningful in projective geometry. Then, by abstracting the underlying groups of symmetries from the geometries, the relationships between them can be re-established at the group level. Since the group of affine geometry is a subgroup of the group of projective geometry, any notion invariant in projective geometry is a priori meaningful in affine geometry; but not the other way round. If you add required symmetries, you have a more powerful theory but fewer concepts and theorems (which will be deeper and more general).
Read more about this topic: Erlangen Program
Famous quotes containing the words problems, nineteenth, century and/or geometry:
“She has problems with separation; he has trouble with unityproblems that make themselves felt in our relationships with our children just as they do in our relations with each other. She pulls for connection; he pushes for separateness. She tends to feel shut out; he tends to feel overwhelmed and intruded upon. Its one of the reasons why she turns so eagerly to childrenespecially when theyre very young.”
—Lillian Breslow Rubin (20th century)
“I delight to come to my bearings,... not to live in this restless, nervous, bustling, trivial Nineteenth Century, but stand or sit thoughtfully while it goes by.”
—Henry David Thoreau (18171862)
“The wound thats made by fire will heal,
But the wound thats made by tongue will never heal.”
—Tiruvalluvar (c. 5th century A.D.)
“I am present at the sowing of the seed of the world. With a geometry of sunbeams, the soul lays the foundations of nature.”
—Ralph Waldo Emerson (18031882)