Timeline for Two integer sided equilateral triangles with integer distances
Current License: CC BY-SA 4.0
9 events
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| Feb 11, 2021 at 22:07 | comment | added | WhatsUp | @HerbertKociemba This is not surprising, as the sum of $\cos \theta$ and $\cos (\theta \pm \frac{2\pi}3)$ is zero, for any $\theta$. | |
| Feb 11, 2021 at 21:06 | comment | added | Zomulgustar | Phrased another way, the sum of the squares of your three red lines must equal the sum of the squares of the edge lengths of the two triangles. | |
| Feb 11, 2021 at 19:37 | comment | added | Herbert Kociemba | Btw., from your equations (*) we get the remarkable relation u^2 + v^2 + w^2 = a^2 + 1, but I do not see in the moment how this could be exploited. | |
| Feb 11, 2021 at 2:28 | comment | added | Laska | @WhatsUp thanks sorry I deleted my comment about a before I saw you'd swiftly replied. I had realized we don't gain any more solutions by looking for only the 3 inter-triangle lengths to be rational. Using any two of the quadratics in a we can eliminate a^2 to see a is rational. | |
| Feb 11, 2021 at 2:09 | comment | added | WhatsUp | $a$ is, by definition, the side length of one of the given triangles, hence it belongs to "distance between any two of the $6$ vertices". | |
| Feb 10, 2021 at 14:00 | comment | added | Herbert Kociemba | The solution I found using integers and not rationals nevertheless has distances small enough and can be found just by brute force in an appropriate way. | |
| Feb 10, 2021 at 2:14 | comment | added | Herbert Kociemba | Thank you very much for this analysis which gives me new insights to the problem. I can confirm your set of diophantine equations and have transformed my solution to a rational tuple (a,t) which indeed then gives rational u, v and w, which are the lengths of the red lines above if the outer triangle is scaled to edge length 1. I agree that it would be very interesting to know something more about the solution space in general. My knowledge about elliptic curves is more or less zero. | |
| Feb 9, 2021 at 22:17 | history | edited | WhatsUp | CC BY-SA 4.0 |
added 96 characters in body
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| Feb 9, 2021 at 22:12 | history | answered | WhatsUp | CC BY-SA 4.0 |