Three Geometric Series in an Equilateral Triangle (visual proof without words)

Рет қаралды 12,344

Ай бұрын

This is a short, animated visual proof demonstrating the sum of three infinite geometric series using dissection proofs in an equilateral triangle. In particular, we show how to find the sum of powers of 1/2, of powers of 1/3 and of powers of 1/7 in the equilateral triangle. Geometric series are important for many results in calculus, discrete mathematics, and combinatorics.
If you like this video, consider subscribing to the channel or consider buying me a coffee: www.buymeacoffee.com/VisualPr.... Thanks!
Also, check out my playlist on geometric sums/series: • Geometric Sums
This animation is based on a proof by Stephan Berendonk (2020) from the November 2020 issue of The College Mathematics Journal, (doi.org/10.1080/07468342.2020... p. 385)
#mathshorts #mathvideo #math #calculus #mtbos #manim #animation #theorem #pww #proofwithoutwords #visualproof #proof #iteachmath #geometricsums #series #infinitesums #infiniteseries #geometric #geometricseries #equilateraltriangle
To learn more about animating with manim, check out:
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Music in this video:
Reaching The Sky (Long Version) by Alexander Nakarada (CreatorChords) | creatorchords.com
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Creative Commons / Attribution 4.0 International (CC BY 4.0)
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Пікірлер: 68

@BadlyOrganisedGenius Ай бұрын

The 1/7 construction is gorgeous

@dutchyoshi611 Ай бұрын

I noticed something here: The denominator is always one more than the numerator, and so i thought that the infinite sum from one to infinity of x divided by (x+1)^y should always equal one. And sure enough, when i plugged the function into wolframalpha, it did say that it does indeed converge to one. These proofs are a beautiful way of showing the beauty of complex mathematical equations, like infinite sums as shown here

@deananderson7714 Ай бұрын

Indeed if we use the formula sum = a/(1-r) for first term a and ratio r we get sum = (x/(x+1))/(1-1/(x+1)) multiply top and bottom by x+1 sum = x/(x+1-1) = x/x = 1 another observation we can make from the video is if we do the sum of first term 1/x with ratio 1/x we get 1/(x-1) as sum = (1/x)/(1-1/x) multiply top and bottom by x sum = 1/(x-1)

@eonasjohn Ай бұрын

1 - 1/2^n

@megachonker4173 23 күн бұрын

Infinite sums are not complex.

@ruilopes6638 Ай бұрын

I looked at the last construction and wasn’t convinced that it should be a seventh. Tried to prove it myself easily. Couldn’t. Brute force it with analytic geometry. 2 seconds later it rearranged itself and it was so obvious it had to be a seventh. Such a gorgeous construction

@douglaswolfen7820 Ай бұрын

I did something similar, but even after the rearrangement it wasn't completely obvious to me. Took some thinking, but I __think__ I could prove it rigourously now

@ruilopes6638 Ай бұрын

@@douglaswolfen7820 I could see that all the angles on the intersections were 60 degrees( the central triangle is clearly equilateral. The rest follow by alternating and opposing angles). After the rearranging all those new triangles have to be equilateral

@user-cd9dd1mx4n Ай бұрын

Amazing as usual! Actually very enjoyable ❤ Keep uploading 👏👍

@MathVisualProofs Ай бұрын

Thank you so much 😁

@Smartas599 Ай бұрын

Thanks! Keep up the good work

@MathVisualProofs Ай бұрын

Thank you too!

@bmx666bmx666 Ай бұрын

Amazing visualization, I love it, thanks! 🔥🔥🔥

@MathVisualProofs Ай бұрын

Thanks!

@leif1075 Ай бұрын

@@MathVisualProofsit's very nice thanks for sharing but zi don't think k the triangle proof at 1:30 is very clear .wjat is 2/3 and how os the denominator being multiplied by a factor of 3..I'd be surprised of anyone actually understood that one..how can they right? I.think something is missing?

@MathVisualProofs Ай бұрын

@@leif1075 The first part cuts the triangle into three equal area pieces. Then only two are left shaded. In the next step, we divide the unshaded 1/3 into 3 equal area pieces and shade two of them. So we have just shaded 2/3 of the unshaded 1/3. That means we shaded 2/3^2. After that, we repeat on the remaining unshaded 1/3 of 1/3 and shade 2/3 of that, etc.

@muse0622 Ай бұрын

These fractals are the visualization of 0.nnnn(base n+1).

@MathVisualProofs Ай бұрын

Yep!

@adw1z Ай бұрын

So beautiful as always, thank u for sharing! I have a video suggestion, on a very underrated fact I feel everyone should know: can u show that sin(54*) = phi/2, where phi is the golden ratio?

@MathVisualProofs Ай бұрын

I have it on the channel recently: kzfaq.info/get/bejne/g8-PiNJlyri7k3k.htmlsi=4e0Vhp8KNF0iOab6

@matematicasantiagofiore Ай бұрын

Excellent!

@MathVisualProofs Ай бұрын

Thanks!!😀

@alanthayer8797 Ай бұрын

Da VISUALS Visuals visuals = complete Individuals !

@MathVisualProofs Ай бұрын

👍

@cookiehead4759 Ай бұрын

Beautiful and smart way to make you love geometry and understand the link with algebra. Thank you for sharing

@MathVisualProofs Ай бұрын

Glad you liked it!

@youoyouoyou Ай бұрын

Fun! You take an equilateral triangle and remove area such that you leave one or more smaller equilateral triangles. Then you repeat. Simple. Beautiful.

@MathVisualProofs Ай бұрын

👍😎

@tomjones6777 Ай бұрын

Cool !

@MathVisualProofs Ай бұрын

👍😀

@user255 Ай бұрын

Nice!

@MathVisualProofs Ай бұрын

Thanks!

@anadiacostadeoliveira4 Ай бұрын

Triangle fractals!!!

@puzzleticky8427 Ай бұрын

Chill math I like you cutchi

@ishtaraletheia9804 Ай бұрын

Quite literally breathtaking! :O

@MathVisualProofs Ай бұрын

👍😎

@astropeter31415 20 күн бұрын

The infinite sum of half reminds me of me making a spiral in a rectangle only using half, quarter, eighths, sixteenths,...

@astropeter31415 20 күн бұрын

THAT IS ACTUALLY GORGEOUS

@astropeter31415 20 күн бұрын

❤❤❤❤❤❤❤

@KaliFissure Ай бұрын

Corny but the classical and plane geometry are just perfect together

@MathVisualProofs Ай бұрын

👍

@mysyntax1311 Ай бұрын

could you post the manim code

@user-kn6sw2jl2p Ай бұрын

WoW

@MathVisualProofs Ай бұрын

@user-pq8qi6mn8n Ай бұрын

did you use the manim library if so how did you learn it i want to learn it too

@MathVisualProofs Ай бұрын

Yes. This is in manim. If you know python, then I would just pick something you want to animate and start playing around. The documentation on the site will get you started and then you want to maybe check out a view tutorials online (something like Benjamin hackl, Brian amedee, theorem of Beethoven, or Varniex). Join the manim discord. I didn’t do these things - I just started playing around (over three years ago). Slowly you will pick things up.

@user-pq8qi6mn8n Ай бұрын

@@MathVisualProofs thanks! will do

@user_08410 Ай бұрын

wow

@MathVisualProofs Ай бұрын

😀

@vennstudios9885 Ай бұрын

wait so let me get this straight the sum of all n^-x where x is an integer is basically just (n-1)^-1 right? we already know that right so if we were to do something like (n-1)×SUM ALL(n^-x) is basically just 1 or maybe even maybe if we make (n-1) be any number it can now be solved as Ω Where Ω is any number other than 0 Ω/(n-1) where n is greater than 1