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

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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.
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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/10.108... p. 385)
#mathshorts #mathvideo #math #calculus #mtbos #manim #animation #theorem #pww #proofwithoutwords #visualproof #proof #iteachmath #geometricsums #series #infinitesums #infiniteseries #geometric #geometricseries #equilateraltriangle
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14 окт 2024

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Комментарии : 69

@BadlyOrganisedGenius 4 месяца назад

The 1/7 construction is gorgeous

@dutchyoshi611 4 месяца назад

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 4 месяца назад

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 4 месяца назад

1 - 1/2^n

@megachonker4173 3 месяца назад

Infinite sums are not complex.

@ruilopes6638 4 месяца назад

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 4 месяца назад

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 4 месяца назад

@@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

@bmx666bmx666 4 месяца назад

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

@MathVisualProofs 4 месяца назад

Thanks!

@leif1075 4 месяца назад

@@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 4 месяца назад

@@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.

@adw1z 4 месяца назад

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 4 месяца назад

I have it on the channel recently: ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-Mi_Uo4eRWcE.htmlsi=4e0Vhp8KNF0iOab6

@حسينالقطري-ب8ص 4 месяца назад

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

@MathVisualProofs 4 месяца назад

Thank you so much 😁

@alanthayer8797 4 месяца назад

Da VISUALS Visuals visuals = complete Individuals !

@MathVisualProofs 4 месяца назад

👍

@cookiehead4759 4 месяца назад

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

@MathVisualProofs 4 месяца назад

Glad you liked it!

@Smartas599 4 месяца назад

Thanks! Keep up the good work

@MathVisualProofs 4 месяца назад

Thank you too!

@youoyouoyou 4 месяца назад

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 4 месяца назад

👍😎

@matematicasantiagofiore 4 месяца назад

Excellent!

@MathVisualProofs 4 месяца назад

Thanks!!😀

@muse0622 4 месяца назад

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

@MathVisualProofs 4 месяца назад

Yep!

@anadiacostadeoliveira4 4 месяца назад

Triangle fractals!!!

@tomjones6777 4 месяца назад

Cool !

@MathVisualProofs 4 месяца назад

👍😀

@ishtaraletheia9804 4 месяца назад

Quite literally breathtaking! :O

@MathVisualProofs 4 месяца назад

👍😎

@puzzleticky8427 4 месяца назад

Chill math I like you cutchi

@user255 4 месяца назад

Nice!

@MathVisualProofs 4 месяца назад

Thanks!

@محمدالسباعي-ك1ب 4 месяца назад

WoW

@MathVisualProofs 4 месяца назад

@KaliFissure 4 месяца назад

Corny but the classical and plane geometry are just perfect together

@MathVisualProofs 4 месяца назад

👍

@astropeter31415 3 месяца назад

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

@astropeter31415 3 месяца назад

THAT IS ACTUALLY GORGEOUS

@astropeter31415 3 месяца назад

❤❤❤❤❤❤❤

@user_08410 4 месяца назад

wow

@MathVisualProofs 4 месяца назад

😀

@HamzaAsif-o9v 4 месяца назад

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

@MathVisualProofs 4 месяца назад

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.

@HamzaAsif-o9v 4 месяца назад

@@MathVisualProofs thanks! will do

@mysyntax1311 4 месяца назад

could you post the manim code

@happystoat99 4 месяца назад

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

@Kokice5 4 месяца назад

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

@happystoat99 4 месяца назад

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

@Bruh_80575 4 месяца назад

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

@duckyoutube6318 4 месяца назад

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

@Bruh_80575 4 месяца назад

@@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 4 месяца назад

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

@Bruh_80575 4 месяца назад

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

@duckyoutube6318 4 месяца назад

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

@ESeth-xb5cu 3 месяца назад

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

@stevehines7520 4 месяца назад

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

@vennstudios9885 4 месяца назад

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