The Road to Reality Study Notes: Difference between revisions

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===2.6 Historical aspects of hyperbolic geometry===

A historical backdrop is painted for the discovery of hyperbolic geometry as well as the importance of the proof by contradiction. Girolamo Saccheri’s work in 1733 used the proof by contradiction in attempt to prove Euclid’s fifth, but failed to show a contradiction. He did however discover the nature of hyperbolic geometry.

Following Saccheri, Heinrich Lambert derived many results by using this same method, including the hyperbolic angle/area relation that was mentioned in section 2.4. His tentative reasoning for belief in a consistent geometry without the fifth postulate was thinking about geometry on a ‘sphere of imaginary radius’. To illustrate this, Thomas Hariot’s angle/area relation for a Euclidean spherical triangle is given and compared to Lamberts hyperbolic relation. Lambert’s formula is recovered by replacing R2 with -1/R2 , showing that the ‘pseudo-radius’ is in fact imaginary: -C-1/2 .

Penrose makes the point that there are many instances in mathematics where the name attached to a concept is not that of the original discoverer. Some examples given:

* It is typical to attribute the discovery and first full acceptance of this geometry differing from Euclid’s via the absence of the fifth postulate to Carl Friedrich Gauss, but because Gauss did not publish his work, Janos Bolyai and Nicolai Ivanoivich Lobachevsky are also named as having independently rediscovered this geometry a few decades later. Hyperbolic geometry is frequently referred to as Lobachevskian geometry.

* Eugenio Beltrami was mentioned in section 2.5 for the discovery of the relations between the projective and conformal, via the hemispherical realizations. The conformal representation is commonly referred to as the Poincare model and the projective representation as the ‘Klein’ representation after their rediscoveries later in the 19th century. Beltrami is however best known for his [https://en.wikipedia.org/wiki/Pseudosphere#:~:text=In%20geometry%2C%20a%20pseudosphere%20is,immersed%20into%20three%2Ddimensional%20space. pseudo-sphere] representation involving Newton’s [https://en.wikipedia.org/wiki/Tractrix tractrix] curve.

===2.7 Relation to physical space===