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The question is written like this:

Is it possible to find an infinite set of points in the plane, not all on the same straight line, such that the distance between EVERY pair of points is rational?

This would be so easy if these points could be on the same straight line, but I couldn't get any idea to solve the question above(not all points on the same straight line). I believe there must be a kind of concatenation between the points but I couldn't figure it out.

What I tried is totally mess. I tried to draw some triangles and to connect some points from one triangle to another, but in vain.

Note: I want to see a real example of such an infinite set of points in the plane that can be an answer for the question. A graph for these points would be helpful.

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You can even find infinitely many such points on the unit circle: Let $\mathscr S$ be the set of all points on the unit circle such that $\tan \left(\frac {\theta}4\right)\in \mathbb Q$. If $(\cos(\alpha),\sin(\alpha))$ and $(\cos(\beta),\sin(\beta))$ are two points on the circle then a little geometry tells us that the distance between them is (the absolute value of) $$2 \sin \left(\frac {\alpha}2\right)\cos \left(\frac {\beta}2\right)-2 \sin \left(\frac {\beta}2\right)\cos \left(\frac {\alpha}2\right)$$ and, if the points are both in $\mathscr S$ then this is rational.

Details: The distance formula is an immediate consequence of the fact that, if two points on the circle have an angle $\phi$ between them, then the distance between them is (the absolute value of) $2\sin \frac {\phi}2$. For the rationality note that $$z=\tan \frac {\phi}2 \implies \cos \phi= \frac {1-z^2}{1+z^2} \quad \& \quad \sin \phi= \frac {2z}{1+z^2}$$

Note: Of course $\mathscr S$ is dense on the circle. So far as I am aware, it is unknown whether you can find such a set which is dense on the entire plane.

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  • $\begingroup$ Can you please explain this in more details, especially the tan part. $\endgroup$ – Ahmed Amir Oct 21 '16 at 12:04
  • $\begingroup$ Which part in particular? Can you get the distance formula? $\endgroup$ – lulu Oct 21 '16 at 12:17
  • $\begingroup$ No, I tried to use Pythagoras, but couldn't get the same formula. $\endgroup$ – Ahmed Amir Oct 21 '16 at 12:26
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    $\begingroup$ Suppose two points on the circle have angle $\phi$ between them. Bisect the chord connecting them to get two right triangles, each with central angle $\frac {\phi}2$. The length of the side opposite that central angle is $\sin \frac {\phi}2$. I then use the addition formula for $\sin$ applied to the angle $\frac {\alpha - \beta}2$. $\endgroup$ – lulu Oct 21 '16 at 12:33
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    $\begingroup$ @J.Pak well, the $\tan x2$ transformation is a rationalization of the circle so my set is just a rational transformation of the Pythagorean set. To be precise: take the point $(-1,0)$ on the circle and extend the line connecting it to a point $(0,z)$ for $0<z<1$ until it meets the circle. That gives you the connection you seek. So far as I am aware, there are no known examples of a non-rational algebraic curve that admits a dense set of the form we seek. (N.B. I could be wrong there, but I don't think I am). $\endgroup$ – lulu Oct 22 '16 at 15:31
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Yes, it's possible. For instance, you could start with $(0,1)$ and $(0,0)$, and then put points along the $x$-axis, noting that there are infinitely many different right triangles with rational sides and one leg equal to $1$. For instance, $(3/4,0)$ will have distance $5/4$ to $(0,1)$.

This means that most if the points are on a single line (the $x$-axis), but one point, $(0,1)$, is not on that line.

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  • $\begingroup$ Thanks for your answer.. But how could you find the point (3/4, 0) and how can I find the other similar infinite points? $\endgroup$ – Ahmed Amir Oct 21 '16 at 11:48
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    $\begingroup$ Let $(a,b,c)$ be a Pythagorean triple. Then $\left(\frac ab\right)^2+1^2=\left(\frac cb\right)^2$, so the distance from $(a/b,0)$ to $(0,1)$ is rational (since it equals $c/b$). There are infinitely many Pythagorean triples, and each one gives you a point $(a/b,0)$. $\endgroup$ – Arthur Oct 21 '16 at 13:32
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    $\begingroup$ The Pythagorean triple makes it so simple.. It's totally clear now. $\endgroup$ – Ahmed Amir Oct 21 '16 at 17:02
  • $\begingroup$ @Arthur Apparently, the above solution of lulu could be also built from rational Pythagorean triples, as $(\frac {1-z^2}{1+z^2}, \frac {2z}{1+z^2},1)$ is a rational Pythagorean triples, too. $\endgroup$ – user370634 Oct 22 '16 at 14:02
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There is essentially only one known infinite rational-distance set, built from rational Pythagorean triples, and all other examples are derived from this by inversions (with rational radius and center one of the points in the set), isometries, dilation, and taking subsets.

There are no other examples on algebraic curves (Solymosi and de Zeeuw 2008, http://arxiv.org/abs/0806.3095).

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  • $\begingroup$ Yeah,, but what makes the unit circle example special is that there are no more than two points on the same straight line.. While in the essential example(built from Pythagorean triples) all the points are on the same line except one on top of them.. Anyway, thanks for your addition. $\endgroup$ – Ahmed Amir Oct 22 '16 at 7:53
  • $\begingroup$ @zyx Apparently, the above solution of lulu could be also built from rational Pythagorean triples, as $(\frac {1-z^2}{1+z^2}, \frac {2z}{1+z^2},1)$ is a rational Pythagorean triples, too. $\endgroup$ – user370634 Oct 22 '16 at 14:03
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    $\begingroup$ Yes, the words "built from Pythagorean triples" were written for the purpose of including both constructions and not treating either one as more basic than the other. The solution with a unit circle is to take 2S where S is the subset (and subgroup) of angles with rational sine and cosine, also known as rational Pythagorean triples. @J.Pak $\endgroup$ – zyx Oct 22 '16 at 16:56
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    $\begingroup$ @Ahmed: In inversive geometry, lines and circles are the "same thing" -- in particular, I believe for any specific line and circle, there is an inversion that gives a bijection between them. (if you include the point at infinity on the line) $\endgroup$ – user14972 Oct 23 '16 at 9:08
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    $\begingroup$ Another indication of the the inversive nature of the problem is that it is natural to allow $\infty$ as a rational distance, and add one point at infinity to the plane, that can be included (or not) in any rational distance set. In addition to making inversions bijective, this restores symmetry in some constructions. @Hurkyl $\endgroup$ – zyx Oct 24 '16 at 19:31

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