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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 21:42, 16 May 2020
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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'': | 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 \ | <math> \pi - (\alpha + \beta + \gamma) = C \triangle </math> | ||
where <math> C </math> is jsut some constant. | where <math> C </math> is jsut some constant. | ||