Box and String, Part 2

I'm going to carry on with the theme from yesterday (but the boxes are green for St. Patrick's Day).

The length of the string is different in each of these three cases. I think we could probably experiment with boxes (or a drawing) and eventually we could prove that the string is the same length in all 3 cases ONLY IF the box is a cube - all three sides are of equal length.

We would have to prove that:

(2D) + (2W) = (2H) + (2W) = (2H) + (2D)

First we would divide everything by 2 to simplify it the formula. A cube's side must then be consistent with this:

D + W = H + W = H + D

Let's try it with our existing box. If we have sides of 1, 2 and 3, the formula's results give us

2 + 3 = 1 + 3 = 1 + 2 

BUT when we solve it, 5 ≠ 4 ≠ 3  This box with unequal lengths doesn't fit the formula.

If we use a cube with each side 2 units long, then the results are like this

2 + 2 = 2 + 2 = 2 + 2 or 4 = 4 = 4 

That works.

Here's a flattened cube.  

 

And a rotating one.