The Fourth Dimension - Hypersphere

Video was created for submission to Summer of Math Exposition 2 (SoMe2)
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CREDITS
Music : • 🔥 Copyright Free Ambie...
Animation at the start : • Big Bang Explosion
The animations in this video were mostly made with manim

[@francescodoenz]

Best hypersphere demonstration I could find! Amazingly easy to understand! Thank you!!

[@angeldude101]

I'm not normally fond of trying to use cross sections to visualize 4 spatial dimensions, but here I don't mind it since 1) you were honest in saying that you really were giving a 4 dimensional image by making the 4th dimension time, and 2) it gives a clear choice for what to make the variable of integration. Integrating a function over time isn't uncommon, so what's being done here shouldn't be too confusing. I also like generalizing the formula to Lₙ₊₁ = ∫Lₙdt. This could theoretically extend the to even higher dimensions. Side note, I personally despise trigonometric integrals, but when I saw you go through the steps to simplify using expansions of powers, I might be able to find a way to work out the integral myself since, even if I don't memorize those expansions, I have an idea of how to derive them thanks to euler's formula... Holy cow, I actually managed to derive a power reduction formula using complex exponentials allowing me to complete an integral. (Yes, sin(x) = (e^ix - e^-ix)/2i is easier to remember for me than sin^2(x) = (1 - cos(2x))/2.)

[@Killer_Kovacs]

The double spiral symbol is probably the best way to show a hypersphere but it would be a four sided spiral.

[@Mathinity]

Nice! I liked how many different concepts of mathematics came out in this video! For next time I would suggest adding chapters, which I find useful in such explanatory videos. #peer_review

[@charlievane]

I'm surprised this doesn't have more views

[@luciazazel2683]

MOST minds cannot visualize 4 perpendicular spatial dimensions. SOME can, but they are very rarified.

[@Manny99999]

Great video! Actually learned alot

[@christophertinklerart2020]

Do more, you are clear in your explanations

[@AtomicFusionProductions]

1:35 did you say flatlandia???? That’s the same as flatlandland!!!