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Generally geometry and trigonometry are the visual areas of mathematics. So, it is readily used in $1,2$ and $3$ dimensions.

However generalization in mathematics has its on beauty. Can we generalize circles, squares, etc. or the trigonometric functions to $n$ dimensional Euclidean space?

Sanji Vinsmoke
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  • Sure. For example, the $n$-sphere: https://en.wikipedia.org/wiki/N-sphere. Also, this might be of interest: https://en.wikipedia.org/wiki/Generalized_trigonometry – 1729 Dec 17 '17 at 13:09
  • Thats nice, but doesnt that defeat the purpose of geometry being visual? – Sanji Vinsmoke Dec 17 '17 at 13:11
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    Not everyone would agree that the purpose of geometry is visual, – Malcolm Dec 17 '17 at 13:14
  • But doesn't geometrical questions need critical visualization to be solved? Consider this - https://www.youtube.com/watch?v=OkmNXy7er84&t=332s – Sanji Vinsmoke Dec 17 '17 at 13:22
  • @SanjiVinsmoke Not always. Moreover, with some training one develops ability to visualize in higher dimensions (even infinite dimensions). – Moishe Kohan Dec 17 '17 at 13:37

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Yes. For example, the generalisation of a triangle is an $n$-simplex in $\Bbb{R}^n$. See https://en.wikipedia.org/wiki/Simplex.

Results from trigonometry such as the Law of Sines and the Law of Cosines can be extended to $n$-simplices.

A. Goodier
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  • Is the Pythagoras theorem also valid in higher dimensions? If so, is there a proof, for higher dimensions? – Sanji Vinsmoke Dec 17 '17 at 13:25
  • Yes, it is. See https://math.stackexchange.com/questions/1588798/how-would-pythagoreans-theorem-work-in-higher-dimensions-general-question and for the proof, see http://www.cs.bc.edu/~alvarez/NDPyt.pdf. – A. Goodier Dec 17 '17 at 13:57