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

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Von der Zerlegung der Zahlen in Teile,uch a surface would resemble ℝ. when extended indefinitely, even if small parts of it matched small parts of ℝ. with absolute precision. Indeed, we may never know enough about the large-scale structure of the universe to say what an unbounded flat surface would really be like. What we can do, however, is find which flat surfaces are . possible.

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https://doi.org/10.1007/978-3-662-25901-6 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 self-contained structure. This intrinsic structure makes the sphere the first example of a non-euclidean geometry.

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,Die Größenordnung der Kardinalzahlen,d-for-word (provided “line”, “distance” etc., are understood in the hyperbolic sense), showing that any complete, connected hyperbolic surface is of the form ℍ./Γ, where Γ is a discontinuous, fixed point free group of ℍ.-isometries.

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Planar and Spherical Tessellations,ges). 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 symmetry group contains enough elements to map any tile onto any other tile.