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An interesting example is the modular group = (): it acts on the tree given by the 1-skeleton of the associated tessellation of the hyperbolic plane and it has a finite index free subgroup (on two generators) of index 6 (for example the set of matrices in which reduce to the identity modulo 2 is such a group).
Hyperboloid_Print.ogv (Ogg multiplexed audio/video file, Theora/Vorbis, length 3 min 31 s, 638 × 360 pixels, 1.47 Mbps overall, file size: 36.83 MB) This is a file from the Wikimedia Commons . Information from its description page there is shown below.
One has a hyperboloid of revolution if and only if =. Otherwise, the axes are uniquely defined (up to the exchange of the x-axis and the y-axis). There are two kinds of hyperboloids. In the first case (+1 in the right-hand side of the equation): a one-sheet hyperboloid, also called a hyperbolic hyperboloid.
This page is a list of hyperboloid structures. These were first applied in architecture by Russian engineer Vladimir Shukhov (1853–1939). Shukhov built his first example as a water tower ( hyperbolic shell ) for the 1896 All-Russian Exposition .
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 ...
Three RCA connectors - yellow for composite video, and white and red for stereo audio RCA connector: Widely used in consumer electronics for audio and video. A single connector must be used for each signal. SCART: Consumer electronics, mostly in Europe. Carries analog stereo sound, along with composite video and/or RGB video.
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 ...