Non Euclidean Geometry & Hyperbolic Social Networks 
Non Euclidean Geometry & Hyperbolic Social Networks
by Richard Southwell
Video Lecture 16 of 16
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Date Added: March 5, 2015

Lecture Description

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I begin by introducing non-euclidean geometry by discussing Euclid's postulates. I then discuss spherical geometry, great circles, warped triangles and map making. Then I discuss Beltrami's hyperboloid model, and its relationship to the Poincare disk. Then I discuss regular tiling of the plane, platonic solids and regular tilings of the hyperbolic plane. Then I discuss how to make hyperbolic paper, how triangles on saddles are distorted, and how to model hyperbolic space using the pseudosphere. The next topics are the Euler number and genus of a graph, and combinatorial curvature. I also discuss graph scaling dimension and the notion of scaled Gromov hyperboic graphs. Using this idea I discuss how real complex networks such as the LiveJournal can have negative curvature on a large scale.

Wolfram Demonstrations Used:

Lines through Points in the Poincaré Disk

Tiling the Hyperbolic Plane with Regular Polygons

Hyperbolic paper from `The Shape of Space' by Jeffrey R. Weeks. Original credit for the Hyperbolic paper idea is given to Bill Thurston.

I reccommend Wildberger's course:

Universal Hyperbolic Geometry 0: Introduction

Course Index

Course Description

Protective geometry is deeper and more fundamental than standard euclidean geometry, and has many applications in fundamental physics, biology and perspective drawing. We shall introduce it visually, without relying upon equations. The hope is make this beautiful subject accessible to anybody, without requiring prior knowledge of mathematics. At the same time, there are some very deep, rarely discussed ideas in this subject which could also benefit experts.


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