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Title:Curvature of a one-parameter family of geometries
Author(s):Romney, Matthew
Abstract:The curvature of a geometric space measures its deviation from regular (or "Euclidean") space. For example, physicists often speak of the "curvature of space-time" in the context of general relativity. As a doctoral student in mathematics, my research deals with how the underlying geometry of a space things like volume, angles, and length is influenced by curvature. For this image, I began with two-dimensional Euclidean space but then weighted the distance function according to its distance from the unit circle, raised to the power α. The vertical axis represents the resulting curvature at each point; this varied greatly depending on the parameter α. The red graph (α=0) shows a constant curvature of zero, meaning that the underlying geometry is Euclidean. The blue graph (α=2) shows a constant negative curvature, which corresponds to hyperbolic geometry. The green graph (α=1) shows an intermediate case relatively flat away from the unit circle but high curvature near it. Finally, the orange graph (α=7) shows an extreme super-hyperbolic case, which has the opposite behavior. It's fascinating that this construction yields two of the three "model geometries" (namely, Euclidean, hyperbolic, and spherical) in such an elegant manner. This image was created with Mathematica.
Issue Date:2015-04
Rights Information:Copyright 2015 Matthew Romney
Date Available in IDEALS:2015-04-27

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