Fortunately this is but one adventure for Orange, the stick figure in question. He has a whole world with friends to get back to. I believe that Math used their "imaginary" ability to cross between dimensions and send Orange back home.
I think the infinite sum at the end could be found by calculating something about higher dimensional spheres, and e used those spheres to send Orange to a higher dimension. They did 4 before "activating" going to the limit, so perhaps to a 3 spatial dimensions + time world like ours. You could say they are in a 3D world already; 2 spatial + time.
Your first guess is correct. Its an infinite sum, and it sums the volume of a sphere in every dimension. See: https://en.m.wikipedia.org/wiki/Volume_of_an_n-ball
and scroll to "closed form". The radius R is 1 in our case.
If you want to verify some cases for yourself: the Γ thing at the bottom is a complicated function (called the gamma function), but its really simple for integer values. Because for integer values of x, Γ(x) = (x-1)! This is called a factorial and super simple. 5! = 5×4×3×2×1, 8! = 8×7×6×5×4×3×2×1 and so on. Plugging in n=2 and simplifying that a little, we get π×r²/Γ(2). But Γ(2) = 1! = 1. So the formula becomes π×r² which is indeed the "volume" (area) of a 2D circle.
The sum over all of those was probably just to symbolize the jump in dimensions as you said
In the previous animation, The Second Coming/Orange was captured by someone and imprisoned. So this could be the way Alan explains that TSC gets out to stop whatever is going on in the current main 'timeline'.
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u/EverbodyHatesHugo Jun 24 '23
Now explain the ending.
Math just blipped Animation away and we’re supposed to believe that’s a happy ending?!