A characterization of inscribable 3-poly topes was found by Hodgson, Rivin and Smith using

hyperbolic geometry [HRS92, Riv96].

Six survey and expository papers and 11 research papers explore

hyperbolic geometry, symplectic geometry, geometric topology, and other areas of geometry and topology.

Hyperbolic Geometry appeared in the first half of the 19th century as an attempt to understand Euclid's axiomatic basis of Geometry.

CAIDA researcher Dmitri Krioukov, along with Marian Boguna and Fragkiskos Papadopoulos, have described how they discovered a latent hyperbolic, or negatively curved, space hidden beneath the Internet's topology, leading them to devise a method to create an Internet map using

hyperbolic geometry.

These colloquia held every 3-4 years are dedicated to the contributions of the late mathematicians Lars Ahlfors and Lipman Bers to the fields of geometric function theory,

hyperbolic geometry, partial differential equations, and Teichmuller theory.

Ungar, Analytic

Hyperbolic Geometry and Albert Einstein's Special Theory of Relativity, Hackensack, NJ: World Scientific Publishing Co.

Incidentally, there were some serious attempts at proofs of the parallel postulate, but they all turned out to depend on hidden assumptions that were themselves equivalent to the parallel postulate (as is obvious if one bears

hyperbolic geometry in mind).

11 (A Form of Euclid's Axiom) is replaced by the Axiom of Hyperbolic Parallels, then this system will give rise to

hyperbolic geometry.

In areas where the surface has floppy,

hyperbolic geometry, the algorithm will identify many mesh points; where the surface has more tightly curved geometry, the algorithm will identify fewer points.

Chapters cover

hyperbolic geometry (illustrated with Escher's models of the hyperbolic plane), complex numbers (so essential in quantum mechanics), Riemann surfaces, quaternions, n-dimensional manifolds, fibre bundles, Fourier analysis, G6del's theorem, Minkowski space, Lagrangians, Hamiltonians, and other terrifying topics.

Chapter 11:

Hyperbolic Geometry has been expanded with additional problems and more in-depth coverage by the authors.

The topics are informal topology, graphs, surfaces, graphs and surfaces, knots and links, the differential geometry of surfaces, Riemann geometries,

hyperbolic geometry, the fundamental group, general topology, and polytopes.