Whenever I would see Mandelbrot zoom videos pop up on my YouTube, I would always watch the entire thing. It always amazed me.
Recently I thought how "much" are we zooming in. How small of a point in the complex plain are we going. Each video gives a number for how far they are going, but that doesn't mean anything. I figured I could go and scale the set as if it were a real world object being zoomed in on, then I could get a better grasp by how small we are going.
I decided I would scale the area of The Mandelbrot Set to the area of the universe. Assuming they are both flat of course.
For the Sets area I used what is called the Escape Radius, which typically is 2.0 on the complex plain. A = 4.0pi
And for the universe I used a rough estimate of its diameter being 7trillion light years. U = upi (too many numbers to type out, the calculations don't matter tbh for my question)
The number that represents how far was zoomed I used Z.
The equation I got was
upi/x = 4.0pi/Z
upi*Z = x*4.0pi
(upi*Z)/4.0pi = x
uZ/4.0 = x (canceled pi)
So I have x equal to the zoom on the universe, but the issue I have is I am lacking a unit. Assuming my math/understanding of The Mandelbrot Set (if it is please let me know) isn't flawed, then all I need to put this into proportion is a unit for x. Basically, if I am looking at the entire universe at once, and I pick a random point in the universe, and I zoom there by x, what is the x's unit? This is probably super simple and I'm just missing it.
Sorry if tag is off
Most or all the units of physical length you are familiar with are arbitrarily defined. You can use meters, light-years, miles, furlongs, or what have you. The numerical value of the diameter of the universe will change depending on the unit you choose. There are length values that are defined by physical constants. The classical radius of the hydrogen atom, about $5.3\cdot 10^{-11}$ meter, is one, the $21$ centimeter line of hydrogen is another, and some will argue for the Planck length, about $1.6\cdot 10^{-35}$ meter. Take your pick. This gives a radius of the universe of about $4\cdot 10^{63}$ Planck lengths.