The hyperbolic plane 
is, up to isometry, the complete simply connected Riemannian
manifold of dimension 2 having constant Gaussian
curvature 
.
Although homeomorphic to the Euclidean
plane, it has a different metric. The Euclidean
plane has zero Gaussian curvature (Anderson
1999, Mann 2017).
The parallel postulate distinguishes its geometry from Euclidean geometry. Through a point
not on a given line, the Euclidean
plane has exactly one line that does not intersect
the given line. In the hyperbolic plane there are infinitely
many such lines, with hyperbolic lines
interpreted as complete geodesics. A hyperbolic
triangle has an angle sum less than 
, whereas a Euclidean triangle
has an angle sum of 
(Anderson 1999).
A pair of hyperbolic lines can be parallel (approaching the same ideal point at infinity in one direction and diverging in the other), intersecting, or hyperparallel (diverging from each other in both directions).
The Poincaré hyperbolic disk represents the hyperbolic plane by the open disk 
equipped, for 
, with the hyperbolic metric
 |
Its geodesics are diameters and arcs of circles meeting the bounding unit circle at right angles. The correspondence with the disk is a conformal mapping, so it preserves angles, but Euclidean lengths in the drawing do not give hyperbolic distances. The bounding unit circle consists of ideal points at infinite hyperbolic distance and is not part of the hyperbolic plane (Anderson 1999, Mann 2017). The Poincaré half-plane model and the Klein-Beltrami model are other representations of the same geometry.
The hyperbolic plane played a central role in the development of non-Euclidean geometry. Lobachevsky and Bolyai independently published accounts in 1829 and 1832, respectively, while Gauss had pursued similar ideas privately. Beltrami's models subsequently established the consistency of hyperbolic geometry relative to Euclidean geometry, showing that the parallel postulate cannot be deduced from the remaining axioms (Milnor 1982, pp. 9-11).
Daina Taimina's crocheted surfaces provide physical models of regions of the hyperbolic plane. Taimina first crocheted such a model in June 1997 (Henderson and Taimina 2001). Increasing the number of stitches by a fixed proportion in successive rows produces a flaring surface on which geodesics and hyperbolic triangles can be explored by folding (Wertheim et al. 2004-2005). The photograph above is by Steve Rowell, courtesy of the Institute for Figuring.
