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Chapter 2: An ancient theorem and a modern question: Difference between revisions
Chapter 2: An ancient theorem and a modern question (edit)
Revision as of 21:39, 16 May 2020
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So just remember, <math> \pi = 180^\circ </math>. Further explanations are given in the [[Preliminaries| preliminaries]] section. | So just remember, <math> \pi = 180^\circ </math>. Further explanations are given in the [[Preliminaries| preliminaries]] section. | ||
=== Hyperbolic Geometry === | |||
A type of geometry which can emerge when the fifth postulate is no longer taken to be true. Objects like triangles obey different rules in this type of geometry. For instance, [https://en.wikipedia.org/wiki/Hyperbolic_triangle hyperbolic triangles] have angles which sum to '''less''' than <math> \pi </math> radians. In fact, we have we have a triangle with an area represented by <math> \triangle </math> and three angles represented by <math> \alpha, \beta, \gamma </math> then by the ''Johann Heinrich Lambert formula'': | |||
<math> \pi - (\alpha + \beta + \gamma) = C \triange </math> | |||
where <math> C </math> is jsut some constant. | |||
In contrast to euclidean geometry where the angels of a triangle alone don’t tell you anything about its size - in hyperbolic geometry if you know the sum of the angels of a triangle, you can calculate its area using the formula above. | |||
== Preliminaries == | == Preliminaries == | ||