Quote - So what is the math of these gaps? 

It's pretty simple. I'll just say what it is, but if somebody asks, I can make a trigonometry diagram to show how I derived this formula.

Let's let D represent the "density" of the tubes, expressed as a fraction...

Now we also need to consider the angle between the normal of the "surface" comprising these tiny tubes and the observer. Let's call that angle alpha.

Ready?

The opacity of the surface is

D / cos(alpha)
...

BB, I would like to see the trigonometry diagram showing how you derived the formula.