What Is the Shortest Road Connecting 4 Cities?

Say that you have been tasked with figuring out the shortest road connecting 4 cities. These four cities are located at the vertices of a square. The side length of this square is 1 Kilometre (Km).

Just to elaborate on the goal here: your task is to lay the least number of kilometres of road to connect all 4 cities. Although this is a fictional example, you can already imagine similar real-world problems that can benefit from this knowledge. It would be a useful skill if you could quickly come up with solutions for problems like these.

In this article, I’ll illustrate the intuition and phenomenon behind the solution for this problem. It will give you a brief introduction to a world where nature and mathematics come together.

This essay is supported by Generatebg

Playing With Intuitive Shapes

Let us start with the easiest connection imaginable to warm things up.

When we connect all 4 cities with straight lines to each other, we end up with a square and two diagonals. Each side of the square is 1Km long. And each diagonal of such a square is √2(square root of 2) Km long. In total, this road setup would be (4+(2*√2)) Km long or approximately 6.83 Km long.

This is a start, but you can sense some room for improvement. The question now becomes: Can we lower the total road length further? What if we eliminate the diagonals?

Sure, now the travel between the diagonal cities is a bit inconvenient, but we have reduced the total road length to just 4Km. Can we get away with more of this? What if we chop off one of the sides of the square?

We see immediately that the cities are still connected and we have managed to reduce the road length by a whole kilometre. The total road length now comes to 3 Kms. But if we were to travel from city ‘C’ to city ‘D’ or the other way around, this road connection is not the most efficient one.

Besides, is there a way to reduce the road length further? What about a circle?

Since we know that the diagonal of the square is √2 km long, we know that the diameter of the circle is √2 Km long. The corresponding circle would be approximately 4.44 Km long. This is certainly not an improvement compared to our previous attempt. So, what next?

---

Intuition at its Limit

The next logical idea that comes to mind is to just connect the diagonals alone. This way, the distance between any city is the same.

The resulting road network would be approximately 2.83 Km long. All 4 cities are connected, and we seem to have achieved the minimum possible road length between all 4 cities.

This is unfortunately not the shortest possible road. I wouldn’t blame you if you are surprised. Our human intuition only takes us this far. Beyond this point, we need to either turn to nature or embrace mathematics (or do both) to arrive at the best possible solution.

What is The Shortest Road Connecting 4 Cities?

I’ll just go ahead and reveal the solution to you directly:

The roads in this network meet each other at an angle of 120°. The total road length with this network is (1+√3) Km which is approximately equal to 2.73 Km. This is the shortest road connecting all 4 cities.

This solution is not so intuitive. How did we then arrive at this solution? Well, there are two ways of solving problems like these. One way is significantly easier than the other.

The significantly harder way is by using mathematics. The field associated with such problems is known as the calculus of variations (a field that is supposed to be one of my specializations). This is actually advanced calculus, and it doesn’t make sense to go into this before understanding calculus first. Since I intend for this article to be accessible to anyone, I will not go into the technical details here (perhaps in another article).

But hang on a minute? If there is an easier way to solve such a problem, why not discuss that? That is precisely what we will do next.

Nature

The easier way to solve this is by using analogies and leveraging nature. If you look closely, the shape of the final solution we arrived at would resemble a honeycomb. For ease of comparison, check out the picture of a honeycomb below.

The similarity is striking, isn’t it? It turns out that nature is very good at minimizing or maximizing stuff. For bees, producing wax is very effort-intensive. So, the bees implicitly ask the question: what is the best shape structure that allows for minimum wastage of wax while storing the maximum amount of honey? And the honeycomb turns out to be nature’s answer.

In fact, this is an age-old problem that mathematicians were trying to solve. They did eventually solve it. For more technical details, refer to the honeycomb conjecture.

On a related note, researchers have used soap bubbles to find the minimum area surfaces that stably connect multiple points in a plane.

This is essentially a higher dimensional extrapolation of the same problem we were trying to solve with the road connecting the cities.

The mathematics behind this phenomenon is complex, but nature has such elegant ways. Phenomena like these really make you wonder and question the so-called human intelligence.

---

I hope you found this article interesting and useful. If you’d like to get notified when interesting content gets published here, consider subscribing.

Further reading that might interest you:  The Thrilling Story of Calculus and How To Make Working With Squares Fun In Math?