Geometry: from Isometries to Special Relativity by Nam-Hoon Lee

Author:Nam-Hoon Lee
Language: eng
Format: epub, pdf
ISBN: 9783030421014
Publisher: Springer International Publishing

The sum of the interior angles in every triangle is exactly π.

There exists a pair of similar, but not congruent, triangles.

The Pythagorean Theorem.

There is no upper limit to the area of a triangle.

The circumference of any circle of radius r is 2πr.

There is a quadrilateral whose interior angles are all .

Now, it is easy to see that all the postulates except the fifth one hold for the hyperbolic plane. Hence, the existence of hyperbolic geometry can be regarded as the evidence that Euclid’s fifth postulate is independent of the other four postulates. For a more detailed history of the development of hyperbolic geometry and non-Euclidean geometry, we refer to [15].

There are many differences between the geometries of the Euclidean plane, the sphere, and the hyperbolic plane. The most intrinsic one is the difference between their Gaussian curvatures, as shown in Theorem 4.35 (see also Exercise 4.18). One of the other differences is the tessellation on them. A tessellation is an arrangement of flat shapes, called tiles, without overlaps or gaps. Some tessellations involve many types of tiles; however, the most interesting tessellations use only one or a few different types of tiles. A regular tessellation is a pattern made by repeating a regular polygon. A regular tessellation is described by two positive integers, [s, t], where s is the number of sides on the regular tiling polygon and t is the number of these polygons that meet at a vertex.

In the Euclidean plane, the vertex angle of a regular s-sided polygon is equal to

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