Timeline for Trigonometry / Euclidean Geometry for natural numbers?
Current License: CC BY-SA 4.0
8 events
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| Oct 18, 2019 at 9:24 | history | edited | Anixx | CC BY-SA 4.0 |
change 1-sphere to unit sphere so to remove ambiguity
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| Oct 15, 2019 at 11:13 | comment | added | user142929 | Many thanks I add these comments but I do not realize if these have a good mathematical content, the feedback of previous user if I understand well is that it is required try to do specializations and applications of your distances for pairwise coprime integers, because in other case if your specializations are only for prime numbers there is a risk of loss of the arithmetic of the number theoretic functions with which you are working @orgesleka | |
| Oct 15, 2019 at 10:43 | comment | added | user6671 | @user142929: I was thinking the in the same direction. But I don't think that 3 points are enough, as one will get inequalities which are true for every 3 point metric space. To make use of embeddability, one has to take at least four points which could build for example a tetrahedron and then maybe use the isoperimetric inequality for this tetrahedron in $\mathbb{R}^3$. | |
| Oct 15, 2019 at 10:13 | comment | added | user142929 | Just as aside comment (isn't required a response, additionally I don't know if my comment has a good mathematical content) is that I was thinking in different applications, then for three of such points/integers that can be embedded in $\mathbb{R}^2$ one has the remarkable isoperimetric inequality (the article Isoperimetric inequality from Wikipedia shows the section 2, the section 6 and the section 8. Isoperimetric inequality for triangles) where the area of a triangle can be calculated using Heron's formula. I think that the interesting ones are formulas for pairwise coprime integers. | |
| Oct 15, 2019 at 9:01 | comment | added | user6671 | @yaakovbaruch: Yes, it holds for every three point metric space, as it can be embedded in $\mathbb{R}^2$. See math.stackexchange.com/questions/3393140/… | |
| Oct 15, 2019 at 8:57 | comment | added | Yaakov Baruch | This last formula, and all similar ones, though seem to hold for all reals $p,q,r$ (with obvious restrictions on the ranges)... Is it any easier to get them through these arithmetic constructions than simple trigonometric algebra? | |
| Oct 14, 2019 at 7:42 | history | edited | user6671 | CC BY-SA 4.0 |
added 314 characters in body
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| Oct 13, 2019 at 16:16 | history | answered | user6671 | CC BY-SA 4.0 |