To make it even simpler we'll start with some static theory.
Even simpler still--two-dimensional static theory.
We know that, in two dimensions, the most economical shape, as pertains to boundary versus area encapsulated, is a circle. The formula for the area of a circle is
πr2
If we let r = 3
Then, the area is 28.26
And the circumference is πD = 18.84
So, if we attempt to encapsulate an area of 28.26
in a square instead of a circle, we need a side of length 5.32 to do so [5.322 = 28.3]. But the boundary in this case is 4 x 5.32 = 21.28.
This is 13%
more costly than our circle encapsulating the same area [(21.28 - 18.84)/18.84].
If we choose a rectangular shape, it is even worse. For instance, if we choose a rectangle
1 x 28.26, the boundary becomes
1 + 1 + 28.26 + 28.26 = 58.52
, or 213% more costly than the circle and 175% more costly than the square.
Keep in mind, we have stayed with a two-dimensional static problem here. In reality, our machine hall is a three-dimensional building and the web has velocity. Think starting with a sphere (πr3) and thinking about rate of production.
