Marden, Albert School of Mathematics, University of Minnesota, Minneapolis, Minnesota.
Last reviewed:August 2020
- How we can think about geometry
- The familiar 2D and 3D geometries
- What, exactly, is a geometry?
- What one can do with hyperbolic geometry
- The universe
- Properties of hyperbolic geometry
- Comparison with Euclidean geometry
- Discrete models of the hyperbolic plane
- Hyperbolic manifolds
- Hyperbolic space
- Chimney example
- Hyperbolic isometries
- Hyperbolization theorem
- Statement of the theorem
- Some common examples of hyperbolic manifolds
- The 3-sphere S3
- Hyperbolic knots
- Related Primary Literature
- Additional Reading
A geometry that obeys a consistent set of axioms differing from Euclid's in implying that the angle sum of a triangle is less than 180°. In the early nineteenth century, K. F. Gauss proved the existence of such a geometry. He questioned which geometry—the new geometry, which is now called hyperbolic geometry, or Euclid's—was the “real geometry of the natural world.” Gauss made numerous measurements before confirming that the “real” geometry was Euclidean. Instead of listing axioms, we will approach hyperbolic geometry by introducing a model of hyperbolic space with its distance formula, lines and planes, and distance-preserving motions. See also: Euclidean geometry; Non-Euclidean geometry
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