Beautiful Geometry behind Geometric Series (8 dissection visual proofs without words)
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- Опубліковано 22 лют 2023
- This video is a compilation of eight shorter videos I have created showing dissection proofs for infinite geometric series with ratio of the form 1/n and first term 1/n. To see the original videos (in shorter form and typically with more dramatic music), check the links below. When you have seen enough of these dissections, you should be able to guess the general formula for such a geometric series (and perhaps the more general form, which can be found in other videos on my channel).
Here are the original series videos (along with attribution; for more detailed attribution, see the original videos):
r = 1/2: • Geometric series: sum ... (attribution: Roger B. Nelsen)
r = 1/3: • Geometric series: sum ... (attribution: Rick Mabry)
r = 1/4: • Geometric series: sum ... (attribution: Rick Mabry)
r = 1/5: • Geometric series: sum ... (attribution: Rick Mabry)
r = 1/6: • Geometric Series: sum ... (attribution: James Tanton)
r = 1/7: • Geometric series: sum ... (attribution: James Tanton)
r = 1/8: • Geometric Series: sum ... (attribution: Roger B. Nelsen)
r = 1/9: • Geometric Series: sum ...
If you like series dissections, check out my playlists:
• Geometric Sums
• Infinite Series
#manim #math #mathshorts #mathvideo #mtbos #manim #animation #theorem #pww #proofwithoutwords #visualproof #proof #iteachmath #calculus #series #geometricseries #infiniteseries #dissection #dissectionproof #geometricsum #sums #calculus2
To learn more about animating with manim, check out:
manim.community
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Music:
Adrift Among Infinite Stars by Scott Buckley | / scottbuckley
Music promoted by www.free-stock-music.com
Attribution 4.0 International (CC BY 4.0)
creativecommons.org/licenses/...
The 1/9 breakdown was so clever.
I never really liked math as a teen and I'm just now starting my journey into mathematics and things like this make me really appreciate beauty in it. Excellent job.
Thanks! Glad you liked this. And glad to hear you’re back to math!
Oh, sweet. Gave me chills. Makes me wish I were a kid again starting out fresh on math explorations. Thank you.
Thanks for checking it out. Glad you liked it!
just came to say the same, I really miss the evenings where I went to the internet and started reading maths texts, first divulgation ones, then more complex (but still in the divulgation/not really formal field) and finally more formal ones (which i didn’t fully understand until i went to university).
I like to think that math is like art, first the mathmatician fights to find the proof, then the rest of us admire the piece and finally we get to understand it. Still amazes me how even after that many years learning math it stills shows me there is more beauty hidden out there
If math is an art, you are no less than Vinc. These were so peaceful, elegant and true pieces of beauty
Glad you liked it!
Never call van Gogh that again.
@@tomdekler9280 'Ear, 'ear!
@@MathVisualProofs please upload more and more...
@@pritamyadav17 I am doing my best. I do have about 200+ videos already up if you want to check the back catalog (but the older ones were when I was first learning so they could be updated perhaps).
is this a patern ? like sum (1 -> infinity) 1/k^i = 1/(k-1) ?
For sure it is! Notice that the dissections for k=6 and k=7 can be generalized for any integer k. You can also prove the formula you have in a variety of ways for any k>1.
@@MathVisualProofs i will try to prove it just to see if it's that difficult or not. But thx for all your content !
@@corentinz6657 The key to the proof is to think about partial sums, S_n = sum (1-> n) 1/k^i. Think about how this sum is related to (1/k)*S_n...
@@MathVisualProofs it also work with k=1 too right? 1/1+1/1²+1/1³+...+1/1^n approaches inf, 1/0 also approaches inf
@@Goaw25511/0 does not approach anything. The limit of s to 0 for 1/s approaches infinity from above. From below for example it approaches negative infinity.
If you take s=((-1)^n)/n you can take the limit of n to infinity and see that it doesnt converge at all. Thus we mathematicians dont like to talk about 1/0
By far the most beautiful and relaxing video on youtube thanks :)
Thanks!
I have seen the majority of them on olympics or challenges, and i finally discovering that it has some logix behind, the point that it isn't just uses to be in a random question, but the beauty of geometry.
:)
I've seen this explained in an analogy of folding infinity, somehow making it a finite number. As in, fold S, get S+1. Like a weird rule
Yes I have seen “folding proofs” as well. Cool way to think about them also :)
Awesome job putting this together! I had never seen many of those before.
Thanks! Check the playlist in the description for more too :)
विविध भौमितिक आकृत्या,त्यांचे अनंत विभाग करून त्यांची बेरीज ,प्रात्यक्षिकासह दर्शवल्यामुळे अनेक घटकांची माहिती मिळाली, यामुळे विविध कल्पना सुचतात, भूमिती मध्ये लपलेल्या सौंदर्याच दर्शन घडले
धन्यवाद सर
👍
Amazing geometrical proof on GP, I am really happy that I had learnt something new..
Glad it was helpful!
Thank you, thank you very much for the beautiful works !
Thanks for checking it out.
I love the concept of algorithmic art and math as art. This is wonderful.
Thanks!
OMG, it is so beautiful! I have learned something about applied math in art and also some number mathematics! Thank you!
Glad it was helpful!
Your channel is underrated. This is really beautiful and artistic.
Thanks! I appreciate the comment 👍😀
数式の導出自体は高校数学でもできてしまいますけど、こういう風に図にして視覚的に捉えられるというのは面白いし勉強になりますね。ありがとうございます。
Yes! It’s fun to have both algebraic and geometric explanations.
Beautiful illustration!
Thank you so much 😀
No Words. Excellent Work.
Wow, thank you! I appreciate your comment :)
Thank you very much!)
Welcome!!
Thank you, that was beautiful beyond words
Glad you enjoyed it!
wow, this is such an enjoyable video to watch
Oh excellent! I really like these dissections and put many up individually but I hoped people might also like to see a themed compilation. Thanks!
Indeed: beautiful! Thanks a lot for this.
Thank you too!
I'm so glad I found this channel.
Glad you're here!
Beautiful video!
Thank you very much!
5:00
Just realised this proof can be used for any of the sums, as long as you find a way to evenly divide the area of the triangle.
I suppose that also applies to the 3:05 method
Just realised the 2:13 proof is just a fancy way of drawing the 3:05 proof.
This is really beautiful.
Thanks!
Great work .
Thank you!
Fun channel. Thanks. Cheers.
Glad you enjoy it!
With math you can create really nice looking things like fractals, geometric series, etc.
I still can't understand how most people don't like math.
Agree!
Such a nice way to present the mathematical expression.. Awesome experience with ur background music🎶 nice choice of background music..
Thank you so much 🙂
Surprised I could understand all of it. Thanks for the video
Glad to hear that!
Great videos!
Thanks!
How welcome😁
Please don't stop making these videos they are helpful
I'll try! :)
@@MathVisualProofs thanks
Math Degree here. This makes me feel like "All that challenging work don't seem so hard no more."
👍😀
Beautiful
Thank you!
This is beautiful 😮
Thanks!
Proudly, I can guess any result of infinity sum just with see portions of shape
This was amazing. Geometry forever
Wow, please keep it up
Doing what I can. Thanks!
I had no idea you had then all in one place.
How beautiful and perfect that the infinite sum is the previous fraction. It can't help but be such and yet still.....❤
Thanks! I made them one at a time, but then I tried making a compilation video here (and I turned them into shorts). The compilation video did better than all my other long-form videos, so maybe I'll have to create more compilations.... Appreciate you watching them and commenting!
@MathVisualProofs I'm trying to model a generalization of this in my head.
There should be one as the descending series of triangles in a curl.....
Wonderful!!!!
Glad you like it!
Amazing geometry
Maravilloso. Muchas gracias.
Thanks for checking it out!
Really like fractals! 😊
for the circle one, cant you prove the same thing with the hexagon one previously? can't you split it into an infinitely large number of segments and prove infinitely many geometric series?
For sure. Both n=6 and n=7 in this video can be generalized to any geometric series of the form 1/n where n is a positive integer. I even have another old video showing how you can use the circle idea (and so the polygon idea) to get some different series: ua-cam.com/video/bxSCJR6RRxs/v-deo.html
I think actually from (1/3)^i you can do all proofs on circle and perhaps there is general solution for 1/n+(1/n)²+(1/n)³+...=1/(n-1)
I think so cuz you cut inside circle smaller one so that you can cut a bigger piece into n-1 parts and n-th part is circle where you repeat process for eg if you have series with 1/4^i you make 3 pieces on circle and there must be a smaller circle so that small circle is equal to each of one pieces from bigger one co you taking 1/4 of circle and going into small one and repeat process
@@nonameee0729 for sure! You can even use the circle for sums of 1/2, but it's a bit strange because you get an inner circle of 1/2 area and an annulus of 1/2 area... so the annuli just shrink in at various powers of 1/2.
very neat!
Thanks!
3:08 BLUE LOCK LESSGOOOOOOO!!!
So calming :)
👍
I am so dumb. The second one surprised me 🤣
4:05 I think this circle method can be used to prove the general situation since any circle can be dissected such that there are r-1 sections surrounding a central circle. So this is the general proof that the sum of any geometric sequence of Σ(1/r)^n is equal to 1/(r+1).
Both circle and polygon methods generalize. But the result is 1/(r-1)
beautiful
Thank you! 😊
3:06 ayo blue lock
so beautiful video
Thanks!
Good job
Thank you!
Awesome😮😮😮😮
So cool！
Glad you liked it!!
you are wonderful, i think you are that kind of person who imaging number not only number but what it's acutely number is very beautiful work
Thank you!
What specifically is this song called? It’s very relaxing & beautiful.
It’s linked in the description - check it out!
Mesmerising...
:)
Beautiful video for a fascinating concept. I "sense" this has a deep meaning in our universe. I know their inversed, but it's like a sum of powers of an integer originates the next integer. Or maybe I'm just crazy. Probably the latter.
Glad you enjoyed it!
Bwaaahhh it’s just one more in the denominator gotta love that geometry tho!!!
I have developed a proof of sum upto infinite powers of 1/2 which includes bisection of angles by extending the hypotenuse further into a base forming an infinite length base
so relaxing
😀👍
superb
Thank you!
Something you will discover, by using the same method as the proof at 3:04, the proof goes like this:
For any number 'n' from 2 to infinity, the infinite sum of 1/n^i, where i = 1 to infinity, is equal to 1/(n-1). If n=1 then the resulting infinite sum is infinity.
The geometric proof works for all whole numbers greater that or equal to 4 but breaks down lower than that. There is probably a way to use different geometric proofs for n=[1, 4) but I don't know them off the top of my head.
Edit:
The infinite sum of 1/3^i can be visually proven by using the approach at 3:51
I have a couple general approaches for any ratio between -1 and 1 on my channel too.
(1/k)^i for i:0 => infinity and k integer grater then 1 is 1/(1 - 1/k) which is k/(k-1). Now we can subtract fisrt item which is 1/k^0=1 and we get k/(k-1) - 1 = 1/(k-1)
sounds about right :)
In the geometric series of ½+¼+⅛ you wrote 1/2^i which is fine and correct, but not acceptable.
The letter "i" is reserved for the root of -1.
And what if you were to solve a problem with a triangle blocked inside a circle, would you mark one of the angles in the triangle with the Greek letter π?
of course not. The spectators or students in the class will not understand what you mean when you say 2π.
Twice the angles π, or twice pi.
The same with the letter i.
From the day it was established that i is the root of -1, this letter (i) should not be used for any variable in an equation.
Besides of this, an amazing video, and teaches a lot. Graphics and illustration at a high level. BIG LIKE.
Thanks! In my experience, the letter "i" is often used as the index of a summation. So I think it is fairly standard, especially when complex numbers aren't involved. I agree that if I were using complex numbers in any way here, I wouldn't use i for the index . Thanks!
@@MathVisualProofs
Thanks for your response.
After 35 years as a math teacher, I can say that I have never called an angle the letter delta (even if there is no ∆x in the problem) nor pi (even if there is no π^2 or 2π in the problem)
And for similar reasons I didn't call the variable e and more...
For me these are "holy" letters or, as a student once told me, these are "married" letters... 😉 they are already taken.
In any case, I mentioned at the beginning that it is correct and okay to use the letter i but...
there is a "but" here
I love your videos and I admit that something in this visual illustration of yours is new and fascinating to me.
Thanks for the great videos.❤️👍
@@tamirerez2547 Thanks! :)
That is interesting! I usually use X for the root of -1. I mean someone can see X^2=-1. So (3x+2)^2=-5+6x. Jokes aside what I have written is Z[X]/(x^2+1) which is isomorph to C. I understand your point, but Math is not about symbols but rather the meaning behind them.
so what i am getting is, the sum of (1/n)^i as i goes from 1 to infinity is 1/(n-1)? That's very cool.
ope, now realised how many people came to the same conclusion
sorry for the filler
Definitely cool to see it right?
@@GrifGrey No worries! I am glad you noticed it and commented on it. That's the fun of it!
Wow! Just... wow!
😀👍
Is it possible to prove this visually for each (1/n)^k series with a (n-1)-gon instead?
Yes! The circle proof generalizes as well
Bro I swear your videos really give me 3Blue1Brown vibes, like I can hear him explaining "why this calculus equation is so beautiful yet elegant..."
Artistically, I can appreciate doing it with several different shapes but to demonstrate that it generalize to N partitions, I find it easier to show that you can do it all with just a circle.
Yes! The circle is nice for sure. But I love the different ones here too :)
Thank you God for recommending this 🙏❣️
Glad you liked it!
What application did you use to make the video animation?
I use manim (manimgl currently) for all the videos on my channel. But manimce will be better to use I think.
Cool use of manim.
Thanks!
Математика завораживающее зрелище! Спасибо!👍🏻👏👏👏
Thanks for checking it out!
so imagine applying this to one it will result with infinity being the sum of infinite one to the power of any value as 1^x is always 1, therefor approaching the limit value of having 0 as the divisor which is a neat idea
What about Basel problem of sum of reciprocal of the squares and other problems of reciprocal of cubes
Hard to get nice dissection proofs because of the values those produce
so you are practically done with geometric sums. I challenge you to try to give visual proves for certain harmonic progressions and arithmetic progressions. (I know I'm evil XD)
Check out my playlists on finite and infinite sums. There are others besides geometric sums (though geometric are my faves)
This amazong work. May God bless you and your loved ones!
Thank you so much!
This is absolutely wonderfull....for the ones who love math...
I hope that includes you :)
Of course I'm included...
@@antoniocampos9721 👍😀
But does that mean that if you make the sum i root of 1/k as i -> infinity = 1/(k + 1)??
wait so does it converge to a number, infinitely close but never reaching it, or does it actually eventually equal it?
To make sense of an infinite sum we let it be the limit of the partial sums with n terms as n goes to infinity. So the partial sums get infinitely close to the infinite sum but the infinite sum is the limit so the infinite sum is the fraction shown.
Geometry is my favorite thing in math(s)
👍😀
Idk if it's just a coincidence, but: The infinite sum of (1/n)^x, and for each sum x increases +1, equals 1/(n-1). This could be just a short and finite pattern, but it could be an infinite pattern too...
So... Idk...
:)
I tried 1/(n^n) with the sigma function, and it was ~1.291285. I named the constant after myself.
Ok but what does the square root of -1 have to do with geometric series /j
Looks like it works for denominators that are not integers, too, as long as they are more than 1. For example, sum of powers of 1/pi = 1/(pi-1). I don't know how to show that, geometrically, though. Perhaps you can help me.
Here’s one way to do it in general :
ua-cam.com/video/V7L5nsRj7CE/v-deo.html . I have a playlist that contains others too : ua-cam.com/play/PLZh9gzIvXQUsgw8W5TUVDtF0q4jEJ3iaw.html
in 3:07 you take all sides to be 1/6 so does that mean the pentagon inside the pentagon has the same area as the other part of pentagon(Bigger one) or is it something else plz explain brother, Thanks
All the trapezoids have area 1/6 because I take the central pentagon to have area 1/6. The other space is 5/6 of the area and is spread equally among 5 trapezoids so they are 1/5 of 5/6 or 1/6 area too.
oh thanks man
@@MathVisualProofs
do you think you can do a proof on why the infinite series of (1/a) = (1/ (a-1) ) ??
I have a few on the channel. Here’s a nice one : ua-cam.com/video/V7L5nsRj7CE/v-deo.html
For anyone who is still confused:
∞ ∞
∑ 1/(nⁱ) = 1/(n-1) and ∑ n/((n+1)ⁱ) = 1
i=1 i=1
Σ(i=1, ∞)1/n^i=1/(n-1)
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You can just use a circle for every example instead of different shapes. Just watch the 1/7 part and be creative.
Mathematics is so unique ✨️
👍😀
is there any proof that the parts of the pentagon are truly 1/6 each?
not that I say they aren't but... how do you find the size of the inner pentagon?
You have to construct the pentagon with the appropriate radius. This is possible with straightedge and compass (though not straightforward). So you scale the radius of the inscribed circle down by the right value and then the outer ring will be evenly divided into 5 equal pieces.
@@MathVisualProofs that's the think I'm asking. by what value you scale the radius? how do you find that value?
for the rest of the series it's easy to see how it's done but for pentagons it's a "scale to the right value" with no hint of how to find that value.
@WilliamWizer You just need the smaller pentagon to be 1/6 of the larger one.
The radius should be 1/sqrt(6).
@WilliamWizer Which is approximately 40.8%.
Just fascinating! And the natural world is expressed in these geometric mathematical truths! Just fascinating.
H.M. Sims
Citizen Mathematician
👍😃
can you name the music ?? ... so relaxing..
Is linked in the description.
Thank you ,it is a really good visualize, please check your video at the time 5:43/6:45 , the result equal 1 ==> 1/7
I’m not sure what I should check ?
Can we Prove that the sum of 1/x^y (if y = [1, 2, 3, ...]) is 1/x-1?
For sure!
What is the software he usees
Manim. It’s in every video description and my about page