# Edit Create an Image Page: Hyperbolic Geometry

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 Image Title*: Upload a Math Image This is an animation of a square rotating in hyperbolic geometry as represented by the Poincaré Disk Model. The geometry with which most people first learned to visual basic shapes such as lines, triangles, and squares is the traditional geometry that most of us are used to, formally called '''Euclidean geometry'''. In two-dimensions, Euclidean geometry is viewed in a flat, infinite plane. However, there also exists '''non-Euclidean geometry''', examples of which include the '''elliptic''' and '''hyperbolic''' geometries. [[Image:TriangleGeometry.gif|center|thumb|550px|left|Triangles in Different Geometries]] One way to describe the difference between these three geometries is by comparing each geometry's definition of a triangle. In Euclidean geometry, triangles must have three angles that total to 180 degrees. However, in elliptic geometry, the angles in a triangle must sum to greater than 180 degrees, and in hyperbolic geometry, the angles must sum to less than 180 degrees. ''Click here for a [[Summary of Geometries|summary]] of the Euclidean, Hyperbolic, and Elliptic Geometries.'' Euclidean geometry is governed by a list of axioms know as the Five Postulates of Euclid, who was a Greek mathematician who lived around 300 BC. These five rules are enough to describe an entire geometry, and if we take away only one of these postulates, we will emerge with completely new geometries. This is the case with hyperbolic (and elliptic) geometry. Hyperbolic (as well as elliptic) geometry follows all of the postulates of Euclid except the final one, known as the '''parallel postulate''', which defines parallel lines and will be addressed below. ==Poincaré Disk Model== {{{!}} border=0 cellpadding=10 cellspacing=10 {{!}} [[Image:HyGeometry_Octagon.gif|left|thumb|150px|Octagon as seen in the Poincaré Disk Model]] {{!}}{{!}} Since hyperbolic geometry is non-Euclidean, this geometry cannot be viewed in a traditional flat plane. There are various models that can be used to represent the hyperbolic plane, but the '''Poincaré Disk Model''' will be used in this discussion. The image featured at the top of this page shows a triangle rotating in the Poincaré Disk Model and the animation to the side is an octagon in the model. To learn more about the other models commonly used when dealing with hyperbolic geometry, please click [http://www.mi.sanu.ac.yu/vismath/sazdanovic/space/main.htm here]. {{!}}} ==Parallel Lines== Image:Poincare_Disk_Line.png Algebra Analysis Calculus Dynamic Systems Fractals Geometry Graph Theory Number Theory Polyhedra Probability Topology Other None Algebra Analysis Calculus Dynamic Systems Fractals Geometry Graph Theory Number Theory Polyhedra Probability Topology Other None Algebra Analysis Calculus Dynamic Systems Fractals Geometry Graph Theory Number Theory Polyhedra Probability Topology Other :Jos Leys' article about creating a futuristic hyperbolic chamber, http://www.josleys.com/article_show.php?id=83 :More basic information about hyperbolic geometry, http://euler.slu.edu/escher/index.php/Hyperbolic_Geometry :To see animations of shapes and lines represented in other models, http://www.mi.sanu.ac.yu/vismath/sazdanovic/space/main.htm :To download Cinderella (a free software) to draw geometry figures in Euclidean, Hyperbolic, and Elliptic geometries, Yes, it is.