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A while ago I saw an image like the one below in a lecture, which was supposed to represent a rail network in a (square) city:

Rail network in a city

The circles represent trains that are moving either North/South or East/West along the vertical and horizontal tracks. The purpose of this picture was to show that, even though the grid is unevenly spaced, it is always possible to schedule the trains so that connections were perfectly timed: that is, any time a train arrives at an intersection, there is another train also arriving at that intersection, in a perpendicular direction.

The math behind this is very simple, so my question is: what is the area of study in which one would encounter an image like this?

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    $\begingroup$ It seems very likely that someone fluent in the theory of configuration spaces will be able to formalize this. Please note that I am not saying that the connection of this problem to configuration spaces is clear and canonical, but I think that 'configuration space' is the 'direction in knowledge space' in which you should go, possibly asking someone working in this direction. $\endgroup$
    – Peter Heinig
    Commented Sep 25, 2017 at 11:46
  • $\begingroup$ For what's it's worth, let me remark that I think I have seen many mathematical pictures and mathematical exhibitions, have never seen this before, and find it a pleasure to see. I think that (copies of) this should be 'inducted' into, simultaneously, the Mathematikum, the MoMath, and the ix-quadrat. Seriously. $\endgroup$
    – Peter Heinig
    Commented Sep 25, 2017 at 15:31
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    $\begingroup$ Not an answer, but isn't the creation of such a situation trivial? Place horizontal and vertical lines randomly in $[0, 1] \times [0,1]$ and place "trains" on the lines at positions $(a_i, a_i)$ and set them running with constant speed. $\endgroup$
    – David G. Stork
    Commented Sep 25, 2017 at 16:34
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    $\begingroup$ @Brendan, his solution is actually exactly what's being shown in my picture. Notice that the cars all hit the diagonal of the square at the same time. $\endgroup$
    – Tom Solberg
    Commented Sep 26, 2017 at 12:34
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    $\begingroup$ @BrendanMcKay: Consider: a train at $(x_1, x_1)$ will meet a train at $(x_2,x_2)$ at point $(x_1, x_2)$ after both have traveled the same distance, $d = |x_1 - x_2|$, at the same speed... and hence at the same time. True for all pairs of trains. $\endgroup$
    – David G. Stork
    Commented Sep 26, 2017 at 16:16

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Not an answer, but two related, almost inverse topics suggesting in which areas of mathematics they fall:

(1) The Lonely Runners Conjecture. See, e.g., Terry Tao, "Some remarks on the lonely runner conjecture." Perhaps this could be classified as a mixture of number theory and combinatorics.

Lonely
Guillem Perarnau, Oriol Serra, "Correlation Among Runners and Some Results on the Lonely Runner Conjecture." Electr. J. Comb. 2016.

(2) "At Museum of Mathematics, Meet 2 Beavers That’ll Never Meet." This construction uses Truchet Tiling. Perhaps this could be classified as a mixture of topology and combinatorics.

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