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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

Connor Smith
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  • The Mandelbrot set (please don't be rude - capitize proper names) is a mathematical set, not a physical object. It does not come with units of "meters" or "kilometers" or "miles" or "feet". It's size is a pure number, not a measure of physical distance. – Paul Sinclair Apr 04 '18 at 23:45
  • @PaulSinclair I understand that it is not a physical object, and I apologize for not capitalizing, I was using my phone and it kept making it lowercase. The Mandelbrot set however can be graphed, as that has been demonstrated in several instances. It is that graph (more specifically the area determined by the escape radius) that I am scaling to be the size of the universe. – Connor Smith Apr 04 '18 at 23:54
  • The area of the Mandelbrot set can be given in a set of units by deciding on what the area of the mathematical 1x1 square is in that set of units. Then the "physical" area is the mathematical area (dimensionless) times the physical area of a 1x1 square. – Ian Apr 05 '18 at 00:01
  • The Mandelbrot set has no "natural" relationship to the physical universe. You can equally accurately choose the unit square to have an area that dwarfs that of the universe, or choose to give it a side that is dwarfed by the Planck length. Any size comparison between the Mandelbrot set and the universe is an artificial one based on an arbitrary scale that you will have to decide on yourself. – Paul Sinclair Apr 05 '18 at 03:18

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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.

Ross Millikan
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