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

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:
manim.community
______________________________
Music in this video:
Reaching The Sky (Long Version) by Alexander Nakarada (CreatorChords) | creatorchords.com
Music promoted by www.free-stock-music.com
Creative Commons / Attribution 4.0 International (CC BY 4.0)
creativecommons.org/licenses/...

[@BadlyOrganisedGenius]

The 1/7 construction is gorgeous

[@ruilopes6638]

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]

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]

@@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

[@dutchyoshi611]

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]

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]

1 - 1/2^n

[@megachonker4173]

Infinite sums are not complex.

[@muse0622]

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

[@MathVisualProofs]

Yep!

[@alanthayer8797]

Da VISUALS Visuals visuals = complete Individuals !

[@MathVisualProofs]

👍

[@user-cd9dd1mx4n]

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

[@MathVisualProofs]

Thank you so much 😁

[@cookiehead4759]

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

[@MathVisualProofs]

Glad you liked it!

[@Smartas599]

Thanks! Keep up the good work

[@MathVisualProofs]

Thank you too!

[@adw1z]

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]

I have it on the channel recently: ruclips.net/video/Mi_Uo4eRWcE/видео.htmlsi=4e0Vhp8KNF0iOab6

[@bmx666bmx666]

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

[@MathVisualProofs]

Thanks!

[@leif1075]

​​@@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]

@@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.

[@youoyouoyou]

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]

👍😎

[@anadiacostadeoliveira4]

Triangle fractals!!!

[@matematicasantiagofiore]

Excellent!

[@MathVisualProofs]

Thanks!!😀

[@tomjones6777]

Cool !

[@MathVisualProofs]

👍😀

[@puzzleticky8427]

Chill math
I like you cutchi

[@astropeter31415]

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

[@astropeter31415]

THAT IS ACTUALLY GORGEOUS

[@astropeter31415]

❤❤❤❤❤❤❤

[@user255]

Nice!

[@MathVisualProofs]

Thanks!

[@mysyntax1311]

could you post the manim code

[@ishtaraletheia9804]

Quite literally breathtaking! :O

[@MathVisualProofs]

👍😎

[@KaliFissure]

Corny but the classical and plane geometry are just perfect together

[@MathVisualProofs]

👍

[@user-kn6sw2jl2p]

WoW

[@MathVisualProofs]

:)

[@user-pq8qi6mn8n]

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

[@MathVisualProofs]

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]

@@MathVisualProofs thanks! will do

[@vennstudios9885]

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

[@user_08410]

wow

[@MathVisualProofs]

😀

[@happystoat99]

I don't get where the * 1/3 and *1/6 come from for 2/3 * 1/3 and 1/6^2?

[@Kokice5]

Because the smaller shapes are 1/3 and 1/6 of the size of tthe original.

[@happystoat99]

@@Kokice5 Ha yes, got it, thanks :)

[@Bruh_80575]

with that we can make a formula that every fraction that goes like 1/x^i equals 1/x-1

[@duckyoutube6318]

What do you do when x=1? Or 3^0?

[@Bruh_80575]

@@duckyoutube6318 when x=1 we get that this is equal to 1/0, but is also equal to 1+1+1+1+..., therefore we could say that 1/0 is infinity

[@Bruh_80575]

But there are some other proofs that say that 1/0 can not be equal infinity so its a really complicated problem

[@Bruh_80575]

Maybe i’ll do a video solving this problem sometime soon

[@duckyoutube6318]

@@Bruh_80575 ahh that makes sense. Ty for the reply

[@ESeth-xb5cu]

lim X -> inf x sig n=1 ((y-1)/(y^n)=1

[@stevehines7520]

"All" from Divine Be-ginning non-material.

[@_.1_teja]

Initially there was no infinity in the triangle...

[@DriftinVr]

There always has been, just not discovered or thought of

[@Babychesssalmon]

hi first

[@IgnDolphin]

hi second

[@abdo01386]

Mathematician hate v proof and like more abstract math

[@learnenglishwithash5383]

But Math is also art