So through the work of Plateau (as I understand it), we know that soap tries to find the shortest connection between points. At least, that's what I was taught. With this in mind, I had to solve the following problem:
In a cube frame with sides of length 20cm, soap forms a square, at half height in the center of the cube (Don't know how to properly translate this into English). Calculate the area of the square in $cm^2$.
So, first of all to give you an idea of what is meant here:
So this is a sketch I made to solve this problem:
So we want the distance between the vertices of the square and the vertices of the cube to be minimal. So I thought that the total distance would equal:
$$8\sqrt{x^2 + 100} + 4(20-2x)$$
So we want this to be as small as possible, take the derivative and make it equal zero: $$ \dfrac{8x}{\sqrt{x^2+100}} - 8 = 0$$
But this doesn't have a solution. What did I do wrong?
All you need to know for this is that, when three soap films meet along an edge, the mutual angles are $120^\circ.$
Among many, I can recommend this book and, probably, this video by the author.
EDIT: he does your soap film at about minute 23. It is not a real square! I will leave this here... Alright, this requires that the faces that resemble flat trapezoids are not really planar, either. So the whole thing is not as it first appears.