Editing The Road to Reality Study Notes (section)

===2.4 Hyperbolic geometry: conformal picture===

[[File:Escher_circle_limit_1.png|thumb|M. C. Escher’s woodcut Circle Limit I, illustrating the conformal representation of the hyperbolic plane.]]

The topic of questioning fundamental assumptions is taken to a level deeper with the example of Euclid’s fifth postulate and hyperbolic geometry, illustrated with [https://mathstat.slu.edu/escher/index.php/Circle_Limit_Exploration M.C. Escher’s Circle limit I].  The notion that all the black and white fish near the boundary are equal in ‘size’ to the fish near the center starts the section with an interesting point of confusion between our visual perception of Euclidean geometry and the hyperbolic representation.

Within the hyperbolic plane, Euclid’s first four postulates hold true, however, the fifth parallel postulate is false.  Penrose notes some interesting consequences of this are that the interior angles of a triangle do not add to π, the Pythagorean theorem fails to hold, and a given shape does not have a [https://en.wikipedia.org/wiki/Similarity_(geometry) similar] shape of a larger size.  The idea of conformal representations is then introduced with the example that angular relations between lines in the Euclidean and this representation of the hyperbolic plane are precisely the same.

Penrose then gives us the relation between the interior angles of the hyperbolic triangle to its area and says that while there is a sense of ‘unpleasantness’ with the fact that the sum of the interior angles of a triangle within the hyperbolic frame do not add to π, there is something ‘elegant and remarkable’ in direct relation of these angles to the area of the triangle, which is impossible to show within the Euclidean framework.  He then leaves us with the equation for hyperbolic distance and notes that with this, hyperbolic geometry has all the properties of Euclidean geometry apart from those that need the parallel postulate, and this allows us to form [https://en.wikipedia.org/wiki/Congruence_(geometry)#:~:text=In%20geometry%2C%20two%20figures%20or,mirror%20image%20of%20the%20other. congruent] shapes.