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.
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.