Timeline for coordinate free foundations of trigonometry
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
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| Apr 12, 2020 at 9:03 | comment | added | Kugutsu-o | @Yemon Choi we are some thousands of years past archimedes, regardless, evidently now, one has to define angle either using arclengt or arch area of a circle, I just want to show that both are proportional to pi as given by some infinite series one calculates it with | |
| Apr 12, 2020 at 8:23 | comment | added | Kugutsu-o | In the answer it's stated that trig functions are defined as matrices of the homomorphism. So instead of matrix algebras you use Clifford algebras, certain spinors as their elements.. | |
| Apr 11, 2020 at 13:47 | comment | added | Yemon Choi | @Ezio I fail to see what Clifford algebras have to do with the question you actually asked. It is true that you can define rotations and reflections without coordinates, Alexandre himself says that the group SO(2) comes from geometry. However, to do actual measurements you need a frame of reference, which seems to be the "coordinates" that you are so keen to avoid | |
| Apr 11, 2020 at 13:43 | comment | added | Yemon Choi | @Ezio How do you propose to define the "length" of a circle? There is a reason why Archimedes used the principle of exhaustion | |
| Apr 11, 2020 at 13:42 | comment | added | Yemon Choi | @ogogmad What Wildberger has and hasn't "shown" seems rather debatable, once one strips away all the ultrafinitist polemic | |
| Apr 10, 2020 at 23:52 | history | edited | Alexandre Eremenko | CC BY-SA 4.0 |
added 119 characters in body
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| Apr 10, 2020 at 15:36 | comment | added | Kugutsu-o | @Alexandre Eremenko Also there are ways to have a rotation group representation without matrices or hyper complex numbers. Every rotation is a composition of two reflections, and in some systems, variations of Clifford algebra where this ratation groups are equivalent to spinors in a completely coordinate free way. | |
| Apr 10, 2020 at 15:31 | comment | added | Kugutsu-o | I agree that distinction is blurry, but does that mean that in the theory of pure geometry the ratio of circle length and radius is an undecidable question? | |
| Apr 10, 2020 at 8:45 | comment | added | wlad | Norman Wildberger has shown that trigonometry can be developed without using analysis, albeit by avoiding the notion of an angle | |
| Apr 10, 2020 at 0:58 | history | edited | kodlu | CC BY-SA 4.0 |
Spelling and grammar. Great answer!
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| Apr 9, 2020 at 23:31 | history | edited | Alexandre Eremenko | CC BY-SA 4.0 |
added 506 characters in body
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| Apr 9, 2020 at 12:08 | history | answered | Alexandre Eremenko | CC BY-SA 4.0 |