It is known that every knot is precisely one of the following: hyperbolic, torus knot, satellite knot. As a consequence, hyperbolic knots can be considered plentiful. A similar heuristic applies to hyperbolic links.
By performing Dehn surgery on a hyperbolic link, one can obtain many more hyperbolic 3-manifolds.
The discovery of hyperbolic space in the 1820s and 1830s by the Hungarian mathematician Janos Bolyai and the Russian mathematician Nicholay Lobatchevsky marked a turning point in mathematics and initiated the formal field of non-Euclidean geometry.
A hyperbolic plane is a surface in which the space curves away from itself at every point.
DH: The discovery of the hyperbolic plane came from the attempt to prove Euclid's fifth postulate, which is also known as the parallel postulate.
We discuss and illustrate tilings of the hyperbolic plane here, but the concepts apply to the sphere and euclidean plane with almost no adjustment, and the underlying theory generalises to higher-dimensional spaces.
The dual of our example hyperbolic tiling by hexagons and squares is therefore a tiling by squares which have two distinct vertices, one of degree 6, and the other of degree 4.
Hyperbolic geometry and the Poincaré disc model from The Institute for Figuring.
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