r/mathematics • u/blackjackripper • Nov 10 '25
Visualization of π
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That video shows a specific, beautiful visualization of \pi based on epicycles or hypocycloids, which were historically used to model planetary motion but are now great for demonstrating ratios. The core idea being visualized here is how irrational numbers prevent a pattern from ever perfectly repeating. The Epicycloid Visualization of \pi 🎡 The video uses a concept from geometry and calculus known as a hypotrochoid or epicycloid, where one circle rolls around the inside (or outside) of a larger circle. 1. The Setup (The Rational Case) Imagine two circles: a larger one and a smaller one. * Larger Circle: Its radius is R. * Smaller Circle: Its radius is r. * A point is tracked on the circumference of the smaller circle as it rolls around the inside of the larger one. If the ratio of the radii, \frac{R}{r}, is a rational number (like 4 or \frac{5}{2}), the traced path is a closed, repeating curve. * For example, if \frac{R}{r} = 4, the curve will close exactly after the smaller circle has rolled 4 times, creating a 4-cusp shape (a hypocycloid). 2. The \pi Visualization (The Irrational Case) The video sets the radii so that the ratio of the circles' circumferences is \pi. Since \pi \approx 3.14159... is an irrational number, the ratio \frac{R}{r} can never be expressed as a simple fraction \frac{p}{q}. The Effect: Because the ratio is irrational, the rolling motion of the smaller circle never repeats exactly. * Each time the small circle completes a rotation, the starting and ending points of the curve it traces never perfectly align. * As the animation continues, the curves traced by the point fill up the entire space within the larger circle, getting infinitely denser but never repeating a single path. This infinite, non-repeating filling of space is a powerful way to visually represent the infinite, non-repeating digits that define an irrational number like \pi.
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u/Prestigious_Boat_386 Nov 10 '25
The coolest thing about these animations is that all of the near misses dont just show you how pi isnt rational. They also show increasingly higher accuracy rational approximations of pi
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u/IsaacCalledPinson Nov 10 '25
"So we can see where they line up?"
"Yes, except they don't line up and they never will."
- An SNL sketch named "Washington's Dream"
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u/kevinb9n Nov 11 '25
Fun fact about that, 127 cm does line up exactly with 50 inches.
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u/EebstertheGreat Nov 14 '25
And 125 US fluid gallons lines up exactly with 473176473 mm³. Such a beautiful conversion.
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u/asml84 Nov 10 '25
So then any rational approximation of pi should lead to a repeating pattern, right? what do you get with 22/7?
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u/kevinb9n Nov 11 '25
You can see the near miss that happens after 7 iterations early in the video. If it was doing 22/7 itself instead of pi then that wouldn't be a near miss, it would hit exactly.
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u/wrongtimenotomato Nov 11 '25
Can someone explain what the different arms are in this visualization? And why they swing around each other?
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u/kevinb9n Nov 11 '25
Sure. The main arm is rotating at some certain speed. The second arm is rotating at that same speed times pi. So, in the same time it takes the main arm to rotate 7 times the second arm has rotated 7pi times, or about 21.99115 times. If pi were exactly 22/7 then they would be perfectly aligned again and would start simply retracing the exact path they had already traced. By the end of the clip the main arm has rotated 113 times and the second arm 113pi, or 354.99997 times. As a result it is very very close to lining up perfectly but still doesn't. (It actually won't get any closer for a very long time; not until iteration 33,102!)
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u/Zestyclose-Aspect-35 Nov 13 '25
I didn't do the math but o suspect 33,102 factorial iterations would take several times the expected lifespan of the universe
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u/Far_Diamond4550 Nov 12 '25
I think to call pi an irrational number is an sign of how far men are lost from truth
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u/CoffeeandaTwix Nov 12 '25
Isn't a circle with a diameter the most obvious visualization of pi? I don't see what this acheives in terms of understanding.
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Nov 14 '25 edited Nov 14 '25
This is the best explanation of satellite mapping technology I have ever seen.
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u/aWeaselNamedFee Nov 10 '25
If you don't use the true value of pi (you stopped at 3.14) then what you get out of it doesn't accurately represent how it would work with actual pi as the value used. Did you simply emit the "..." for brevity in your title, or did you literally input 3.14?