Editing Chapter 2: An ancient theorem and a modern question
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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 \triange </math> | ||
where <math>C</math> is just some constant determined by the ''units'' by which we measure a give length or area. The ''units'' we use can always be chosen such that <math>C=1</math>. | where <math>C</math> is just some constant determined by the ''units'' by which we measure a give length or area. The ''units'' we use can always be chosen such that <math>C=1</math>. |