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Two dimensional Euclidean geometry works on a plane and corresponds to a space having constant zero curvature. However, it is not the only consistent geometry possible in two dimensions. The geometries with constant negative and positive curvatures are hyperbolic and spherical geometries, respectively. Spherical geometry leads to the construction of the Platonic solids via tessellations of a sphere and has triangles whose angles add up to more than 180 degrees. Hyperbolic geometry violates Euclid’s 5th axiom, it allows parallel lines to meet, and was so shocking to Gauss he didn’t publish it until after his death. In it triangles can have a total of 0 degrees!

The Cambridge undergraduate mathematics degre includes a short course in Geometry, 16 lectures rather than 24. The linked notes, from a lecture course by Dr. N.I. Shepherd-Barron, don’t include some of the graphical hand outs given in the lectures but still have a lot of illustrations. It covers the following;

  • Spherical geometry, tessellations and Platonic solids
  • Hyperbolic geometry and tessellations
  • Poincare disk, the upper half plane and Mobius transforms

This is a 2nd year lecture course.

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