Maths Problem: Connect the towns solution (Motorway Problem)
Why Mathematics Stack Exchange recommends it
Mathematics Stack Exchange users link it as a vivid demonstration of soap films finding minimal-length networks, whenever Steiner trees, road layouts between cities or minimal surfaces come up.
Recommended by 2 people across 2 different months, 2012 to 2023.
What it is
A short singingbanana demonstration of the Motorway (Steiner tree) Problem, asking for the least road needed to link four towns. It uses soap films stretched between pegs as an experimental way to find the minimal network, with help from University of East Anglia staff.
What people said
Recommends Jared · 2013-05 · Mathematics Stack Exchange · on “Minimizing the length of a pipeline between cities”
You are looking for the Fermat point [LINK] , $M(x)$, of the triangle whose vertices are $A=(6,3)$, $B=(0,4)$, and $C=(x,0)$, and trying to minimize the value $$d(x)=|AM|+|BM|+|CM|$$ According to this calculation [LINK] , the acute angles between $\overline{AM},\overline{BM},$ and $\overline{CM}$ are all $\frac{2\pi}{3}$. Appealing to intuition, we see that $d(x)$ will be minimized when $M(x)=(x,y)$, that is, when $M$ sits directly above $C=(x,0)$. Using these two facts, we see that the Fermat point $M$ minimizing $d$ will be the intersection of the following two lines: $$y=-\frac{x}{\sqrt{3}}+4\\y=\frac{x-6}{\sqrt{3}}+3$$ This is because these lines pass through the known points $A$ and $B$
Recommends Suzu Hirose · 2015-01 · Mathematics Stack Exchange · on “Is it possible to use physics or other form of non-canonical reasoning to study functions?”
Soap films create a surface with minimal surface area naturally. There is some more at this site [LINK] . Here is a video [this video] . Note that these are local rather than global minima.
Cites utdiscant · 2012-05 · Mathematics Stack Exchange · on “What would be a good outdoor maths puzzle for children?”
It is not much of an activity, but I remembered this old video of solving the Steiner Tree Problem using soap bubbles [this video]
Cites Shuhao Cao · 2013-07 · Mathematics Stack Exchange · on “Calculus of variations: find $y(a/2)$ if $y(x)$ maximizes the volume of rotation”
Summary of my answer: Some references for the classical isoperimetric problem of maximizing the area under a curve given a fixed arc length. Reformulating the revoluted volume problem into a non-constraint optimization problem. Example of why maximizing the revolution volume is not equivalent to maximizing the area under a curve given a fixed arc length. Numerical approach and the approximation to $y(a/2)$ is $0.74a$ with the approximated maximized revoluted volume $1.212108 a^3$. Let's say that the curve $y$ that maximizes the revolution volume subject to length $l$: $$\text{Maximize }V[y] = \pi\int^a_0 y(x)^2\,dx\quad \text{subject to}\quad \int^a_0 \sqrt{1+\big(y'(x)\big)^2}\,dx = l.\tag{
Cites JMoravitz · 2020-04 · Mathematics Stack Exchange · on “How many ideal ways are there to lay roads between cities?”
To be clear, your roads may only travel directly from one city to another in a straight bee-line path? You mention a centroid there... If you don't require this it can be more efficient [this video] .
Cites Belowabrightday · 2023-01 · Mathematics Stack Exchange · on “How to find the largest possible value when moving values between N nodes”
The solution to N nodes, is their maximum path. These paths all obey a couple logical rules. A. The only edges followed, are those whose origin was the previous edge's destination. B. The path utilizes bouncing as often as possible, which is to travel to a node and return from it immediately. These bounce edges, all connected by a return edge, make an outstretched-hand shape. C. All the return edges are connected to a some unifying node. This works until 6 nodes, where an important distinction arises. Two solutions exist for N nodes, depending on whether the nodes begin with equal populations or with unequal populations. D. The first moves of the solution, is to reduce itself from an N graph