# hyperbolic geometry

## Definitions

• WordNet 3.6
• n hyperbolic geometry (mathematics) a non-Euclidean geometry in which the parallel axiom is replaced by the assumption that through any point in a plane there are two or more lines that do not intersect a given line in the plane "Karl Gauss pioneered hyperbolic geometry"
• ***

## Usage

### In literature:

Riemann discovered elliptic metrical geometry, and Lobatchewsky hyperbolic geometry.
"Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 6" by Various
***

### In news:

A journey to an alternate universe based on hyperbolic geometry can serve as an entrée to eye-popping wallpaper designs.
***

### In science:

Three-dimensional manifolds, Kleinian groups and hyperbolic geometry.
$\ell^2$-homology and planar graphs
The main focus was to explore hyperbolic geometry beyond the class room and to see if we could ﬁnd the smallest possible volume for this class of ob jects.
Volumes in Hyperbolic Space
To have a better and more formal understanding of Hyperbolic Geometry, we were introduced to Curved Spaces, a program which visualizes Hyperbolic Geometry, and Orb, a program which calculates volumes of hyperbolic polyhedra.
Volumes in Hyperbolic Space
In Hyperbolic Geometry, all of Euclid’s postulates hold, except the Parallel Postulate.
Volumes in Hyperbolic Space
The revised 5th postulate for Hyperbolic Geometry goes as follows: “Given any point P in space and a line l1 , there are inﬁnitely many lines through P which are parallel to l1” [ab12].
Volumes in Hyperbolic Space
***