Titlebook: Geometry of Surfaces; John Stillwell Textbook 1992 Springer Science+Business Media New York 1992 Area.Fractal.curvature.differential geome

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Textbook 1992lcome the opportunity to make a fresh start. Classical geometry is no longer an adequate basis for mathematics or physics-both of which are becoming increasingly geometric-and geometry can no longer be divorced from algebra, topology, and analysis. Students need a geometry of greater scope, and the

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The Euclidean Plane,d circles, and proceed from “self-evident” properties of these figures (axioms) to deduce the less obvious properties as theorems. This was the classical approach to geometry, also known as .. It was based on the conviction that geometry describes actual space and, in particular, that the theory of

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The Sphere,e, however, is of interest . in relation to the plane. Its intrinsic structure is locally the same as the line because we have the map θ→.θ which is a local isometry between the line and the unit circle. The sphere, on the other hand, is . locally isometric to the plane, hence it is of interest as a

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The Hyperbolic Plane,t . ∉ ., more than one line through . which does not meet . Such a surface departs from the euclidean plane in the opposite way to the sphere, and the hyperbolic plane, in fact, emerged from the study of surfaces which “curve” in the opposite way to the sphere. The train of thought, in brief, was th

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Hyperbolic Surfaces, function . such that each . ∈ . has an ε-neighborhood isometric to a disc of ℍ.. The proof of the Killing-Hopf theorem (Section 2.9) carries over word-for-word (provided “line”, “distance” etc., are understood in the hyperbolic sense), showing that any complete, connected hyperbolic surface is of t

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Paths and Geodesics, problem of classifying groups Γ. In the spherical and euclidean cases this problem is easy to solve, as we have seen in Chapters 2 and 3, because there are only a small number of possibilities. However, in the hyperbolic case the number of possibilities is infinite, and the problem is best clarifie

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Planar and Spherical Tessellations, tile”, i.e., if any tile II. can be mapped onto any tile II. by an isometry which maps the whole of . onto itself (faces onto faces and edges onto edges). The isometries of . onto itself are called . of ., and they form a group called the . of . Thus, we are defining . to be symmetric if its symmet

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