Search results
Results From The WOW.Com Content Network
Hyperbolic space, developed independently by Nikolai Lobachevsky, János Bolyai and Carl Friedrich Gauss, is a geometric space analogous to Euclidean space, but such that Euclid's parallel postulate is no longer assumed to hold. Instead, the parallel postulate is replaced by the following alternative (in two dimensions):
Despite this difficulty, K. Böröczky gives a universal upper bound for the density of sphere packings of hyperbolic n-space where n ≥ 2. [29] In three dimensions the Böröczky bound is approximately 85.327613%, and is realized by the horosphere packing of the order-6 tetrahedral honeycomb with Schläfli symbol {3,3,6}. [ 30 ]
In geometry, the hyperboloid model, also known as the Minkowski model after Hermann Minkowski, is a model of n-dimensional hyperbolic geometry in which points are represented by points on the forward sheet S + of a two-sheeted hyperboloid in (n+1)-dimensional Minkowski space or by the displacement vectors from the origin to those points, and m ...
A horosphere within the Poincaré disk model tangent to the edges of a hexagonal tiling cell of a hexagonal tiling honeycomb Apollonian sphere packing can be seen as showing horospheres that are tangent to an outer sphere of a Poincaré disk model. In hyperbolic geometry, a horosphere (or parasphere) is a specific hypersurface in hyperbolic n ...
This allows one to study isometries of hyperbolic 3-space by considering spectral properties of representative complex matrices. For example, parabolic transformations are conjugate to rigid translations in the upper half-space model, and they are exactly those transformations that can be represented by unipotent upper triangular matrices.
The model for hyperbolic geometry was answered by Eugenio Beltrami, in 1868, who first showed that a surface called the pseudosphere has the appropriate curvature to model a portion of hyperbolic space and in a second paper in the same year, defined the Klein model, which models the entirety of hyperbolic space, and used this to show that ...
A definition of a -hyperbolic space is then a geodesic metric space all of whose geodesic triangles are -slim. This definition is generally credited to Eliyahu Rips . Another definition can be given using the notion of a C {\displaystyle C} -approximate center of a geodesic triangle: this is a point which is at distance at most C {\displaystyle ...
The metric of the model on the half-plane, { , >}, is: = + ()where s measures the length along a (possibly curved) line. The straight lines in the hyperbolic plane (geodesics for this metric tensor, i.e., curves which minimize the distance) are represented in this model by circular arcs perpendicular to the x-axis (half-circles whose centers are on the x-axis) and straight vertical rays ...