Euclid's orchard

Once upon a time Isaac was lounging under a tree in Euclid's apple orchard, when something struck his foresight.

“In the future,” he imagined, “the layout of these trees could help understand how to relieve an urban congestion problem.”

Our hero roughed out the idea on his mobile iTypewriter as you can see. What was he thinking?

________|        :     :      :                    .'
________      red:   g:een   :red    green    red.'  green    red    green  .'
|       :    .'     :                  .'                       ..''
|       :    :     :                 .'                      ..'
________|      :    :     :                .'                     .''        .
________     gr:en .' red:   green    red.'  green    red    g..''    red..''
|      :   :    :              .'                  ..'       ..''
|     :   :    :             .'                 .''      ..''
________|     :  .'   :            .'               ..''     ..''
________     :red:   :reen    red.'  green    red..' gree..'' red    green
|    :  :   :          .'             .''    ..''                  ...
|    : .'  :         .'           ..''   ..''                ...'''
________|   :  :  :        .'          ..'   ..''              ...'''
________    :g:ee:    red.'  green  .'red..''green    red...'''een    red
|  : .' :      .'       ..'' ..''          ...'''
|  : : :     .'      ..' ..''        ...'''                  .....''''
________|  :: :    .'     .''..''      ...'''              .....'''''
________  :.':red.'  g..''.'' red...'''een    red.....'''''   red.......''''''
| :::  .'  ..:.''  ...'''      .....'''''  .......'''''''.......::::::
|::: .' .::''...'''  .....'''''.....'''''''...:::::::::.::::::::::::::
________|::.:.:::.:::...:::::.::::::::::::::::::::::::::::::::::::::::::::::::
red:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::


(Those dotty diagonals are meant to be solid, straight and thin. The lower portion is meant to show an increasing density of diagonals.)

Hint:

   time  ----------------------->

So, the situation appears to be this. We've got a single street with regularly spaced lights on it. The lights alternate between red and green, all with the same period and with adjacent lights exactly out of phase. (Perhaps there's an amber phase in between that isn't shown -- perhaps, e.g., it lies between the words "red" and "green" on each horizontal line.)

Each of the straight lines in the graph corresponds to a vehicle going along the street with constant velocity. Gradient of line = speed. The vehicle will have to stop if and when it reaches a red light.

These lines all emanate from the same spot at bottom-left, which I take to be just after the end of a red light. (Before the start of the word "green", but never mind that.) And they all appear to (in some cases only barely) miss the word "red", which I take to indicate that the lines correspond to trajectories that completely avoid red lights.

So presumably the idea is (something like)

that diagrams of this kind would enable sufficiently accurately-controlled cars, with sufficiently accurate knowledge of the lights ahead of them, to drive at a nice constant speed while never being stopped by red lights. It would probably require computer-driven vehicles.

I confess myself not entirely convinced

that this could actually work with realistic light positions and timings

but never mind.

• How do you like it now? – Gareth McCaughan Jun 16 '16 at 17:59
• Only one of the lines (more or less 45 degrees on my screen) is meant for use in reality, and it does allow for some human sloppiness. The other lines indeed represent robotic-precision scofflaws and slowpokes. Your skepticism might be on target, alas, as my favorite example of this seems to have been dismantled recently. – humn Jun 16 '16 at 23:37
• Sorry I gave feedback as an edit, and unattributed at that – humn Jun 17 '16 at 0:23
• That's OK! It was useful feedback... – Gareth McCaughan Jun 17 '16 at 7:29

Was he thinking that

Alternating red and green traffic lights facilitates traffic to move efficiently* both N-S and E-W?

where

"Efficiently" means reliably travelling Euclidean distances as opposed to Manhattan distances.

My reasoning:

Say we want to get from one corner to the other with minimal stops. Consider the major and minor diagonals and the four diagonals around them. If the lights along any diagonal are always the same color, and the diagonals alternate color, we have the following scenario (WLOG) for both the major and minor diagonal: We start travelling N-S at one of the "corners," we have a green light. As we're between streets, the lights change, and we have a green light at the next intersection, so we can turn. Change the lights again, and once more we'll have a green light at the next intersection. Continue in this fashion until we've traveled the entire diagonal without ever stopping at a red light.

That said, we also consider other possibilities:

Other desired paths at different "angles" will be less optimal if we travel along their gradient (on second thought, will they?), but in any case, travelling at a "45 degree angle" and then traveling N-S or E-W directly to our desired destination will have us always at a green light. We can (in theory - yellow lights** and possible delays/blockages be damned) get anywhere we want without ever stopping.

In practice though

Having yellow lights at every intersection in both directions every so often is an utterly frightening idea - I suppose it's no coincidence there are no Golden Delicious trees in this orchard, though.

Disclaimer: I'm tired and there's probably a flaw in my reasoning, and I don't think it accounts for all the lines he drew, but I don't think it's a bad attempt.

• Nice, @Will! Your answer made Isaac think so hard that he had to fill out more of the diagram in order to remind himself of the original idea's asymmetry. I hope you enjoy it at a more rested moment as much as I enjoy your take. Good point about not having Golden Delicious apples in this orchard. – humn Jan 3 '16 at 10:21