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The Greek geometer Euclid studied the geometry of the plane, and stated 5 axioms that he took as assumptions about the plane (for example, all right angle are equal). These were supposed to the "obvious", but he was unsatisfied with one. This axiom became known as the "parallel" postulate because it states that given a line and a point not on that line, there is exactly one line through the point parallel to the given line. This axiom was more complicated than the others, and mathematicians tried for centuries to prove if from the other four. In the nineteenth century, however, two new geometries were discovered that satisfy the first four axioms but not the fifth. In one of these, there are no parallel lines (this is the geometry of the sphere) and it is called elliptical geometry. In the other, there are many parallel lines through the given point, and this is called hyperbolic geometry. It has several representations within the unit circle, or in the upper halfplane of 2dimensional space. There are also higherdimensional analogs, just as there are higherdimensional Euclidean spaces.
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 237 Tiling of the Hyperbolic Plane
 4 Dodecahedra in Hyperbolic Space
 Borromean Ring Complement Manifold 1
 Borromean Ring Complement Manifold 2
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 Borromean Ring Complement Manifold 3
 Escher Fish
 Hyperbolic Dirichlet Domains
 Hyperbolic Dodecahedron
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 Hyperbolic Kaleidoscope: 235
 Hyperbolic Kaleidoscope: 344
 Hyperbolic Kaleidoscope: S2
 Hyperbolic Space Tiled with Dodecahedra, 1
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 Hyperbolic Space Tiled with Dodecahedra, 2
 The Borromean Ring Complement Manifold
 The Order7 Borromean Ring Orbifold
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Created: Fri Jun 26 14:36:38 CDT 1998

Last modified: Fri Jun 26 14:36:38 CDT 1998
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