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This is for my underground dwelling Kepler Bb people. I was thinking of having underground foot highways as a startup once the civilization gets big enough.

Now what do I mean, as a startup, you might be wondering.

Well what I mean is that later on these highways will be expanded and lane lines changed when these people start using 100% renewable energy. But right now, if they want to establish another civilization somewhere else or if they want to expand their civilization by miles, highways would be needed.

Maximum distance currently

If you were to go in a straight line like you were flying from 1 place to another, the current maximum distance is 1 mile.

The Pythagorean theorem gives a more accurate, walking through tunnels, maximum distance. But even this only considers the horizontal distance. And while you could do it again for the vertical distance, and make a total distance vector, you would have to do the pythagoream theorem once again to figure out that vector's magnitude and thus a good approximation of the total distance.

But even this still ignores the shape of the tunnels and there is only 1 way that you could get the exact maximum distance no matter the shape of the tunnels and that is calculus.

Calculus will answer the question "what is the total length of this curve when fully stretched out?" and the answer to that question is the exact maximum distance. Now yes you would again have to separate into horizontal and vertical curves and then apply calculus to both and then after doing that, make a distance vector from these components and do the Pythagorean theorem but doing this, you get the total exact maximum distance. Note that this question is basically the same thing as "what is the limit as you do the Pythagorean theorem and all dimensions approach 0(thus lining up with the curve perfectly)". I think I would have to use a multidimensional derivative to find this limit and while a 1 dimensional derivative, I can do(such as f'(x)=2x if f(x)=ax^2), I don't know how to apply the derivative to more than 1 dimension.

Anyway, let's go with the simplest of these and say that the maximum distance is 1 mile. I might go as far as the Pythagorean theorem later but not right now.

So we are basically saying right now that hypotenuse = distance.

Foot highway

I have never seen a foot highway except on maps(and even then only Indian Foot Highways(referring to the American Indians)). But yeah a foot highway would have high foot traffic.

My foot highways I plan to take 1 step further. I plan on my foot highways being like long distance running(high speed and high foot traffic).

Now I would of course need plenty of rest stops on these foot highways because some runners will be out of breath quickly whereas others will have high endurance and can run for much longer without being out of breath.

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So how can I have enough rest stops so that the people with the least endurance and the people with the highest endurance and the rest of the people on that endurance spectrum all have rest stops without having 1 long rest stop the same length as the highway?

I mean look at it this way. If I just take the person with the least endurance and measure the distance they can run before they are out of breath and put rest stops just at that distance from any given rest stops or endpoints(where the highway starts and ends), there are bound to be runners with higher endurance that are out of breath and not at a rest stop. But the only way that I can think of to satisfy all runners with a rest stop in case they are out of breath is to have 1 rest stop the same length as the highway.

So how can I avoid having out of breath runners without a rest stop close to them but at the same time avoid having 1 long rest stop the same length as the highway?

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    $\begingroup$ look at the road network Ancient Rome built. $\endgroup$ – Marshall Tigerus Dec 28 '16 at 20:53
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    $\begingroup$ What, exactly, are those pictures adding to your question? $\endgroup$ – kingledion Dec 28 '16 at 21:07
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    $\begingroup$ You can't. Any spacing is going to preference one runner over another. The way to solve it would probably be something along the lines of high-endurance runners simply stopping early. $\endgroup$ – ifly6 Dec 28 '16 at 21:11
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    $\begingroup$ Caters, we love all contributions but a significant number of your questions have irrelevant information. Discussing the Pythagorean theorem and calculus, inserting stock photos of people and signs, further elaborating on measurement, etc. don't contribute here. $\endgroup$ – Zxyrra Dec 28 '16 at 21:46
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    $\begingroup$ In other words you could get what you need by saying "My people are running though 1 mile tunnels. How do I keep them rested / not out of breath?" As opposed to straying into discussions on irrelevant aspects of your world. $\endgroup$ – Zxyrra Dec 28 '16 at 21:48
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It's somewhat unrealistic to have a runners-only foot highway. If it's good enough to run, it's good enough to walk and drive carts.

Lanes for different speeds would be a good idea - but no one thought of that in ancient times. A law saying, say, "runners at center", well-known by the populace, should be enough.

Into the question itself: put rest stops separated by half a day's walking, say, 10 to 15 km. Runners will just skip one or two of them, while the rest will use each one.

With some luck, the rest stops will develop into inns, which will eventually grow into markets and villages.

About the derivatives of functions of more than one variable: derive the function separately on each variable. Deriving along a vector is a linear combination of the derivatives on the variables. More at

https://en.wikipedia.org/wiki/Derivative#Derivatives_in_higher_dimensions

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Very small alcoves along the side of the tunnel spaced maybe 500 meters with larger rest stops with amenities spaced perhaps at intervals of 5 km. This would serve the twin purpose of roman milestones and a spot for the less fit to get out of the way. The alcoves could also house ventilation, to make them truly multipurpose.

You sometimes see examples like this in the southwester US. A highway will have larger rest stops with amenities, bathrooms, etc. every 100 miles, but may have 2 smaller stopping points in between with maybe a shaded picnic table.

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