Beautiful Geometry behind Geometric Series (8 dissection visual proofs without words)

Рет қаралды 124,055

Жыл бұрын

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
___________________________________
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/...

Пікірлер: 328

@S.G.Wallner Жыл бұрын

The 1/9 breakdown was so clever.

@axion986 Жыл бұрын

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.

@MathVisualProofs Жыл бұрын

Thanks! Glad you liked this. And glad to hear you’re back to math!

@pauldokter2725 Жыл бұрын

Oh, sweet. Gave me chills. Makes me wish I were a kid again starting out fresh on math explorations. Thank you.

@MathVisualProofs Жыл бұрын

Thanks for checking it out. Glad you liked it!

@marca.f.3569 Жыл бұрын

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

@sleha4106 Жыл бұрын

If math is an art, you are no less than Vinc. These were so peaceful, elegant and true pieces of beauty

@MathVisualProofs Жыл бұрын

Glad you liked it!

@tomdekler9280 Жыл бұрын

Never call van Gogh that again.

@Devo491 Жыл бұрын

@@tomdekler9280 'Ear, 'ear!

@pritamyadav17 Жыл бұрын

@@MathVisualProofs please upload more and more...

@MathVisualProofs Жыл бұрын

@@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).

@corentinz6657 Жыл бұрын

is this a patern ? like sum (1 -> infinity) 1/k^i = 1/(k-1) ?

@MathVisualProofs Жыл бұрын

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.

@corentinz6657 Жыл бұрын

@@MathVisualProofs i will try to prove it just to see if it's that difficult or not. But thx for all your content !

@MathVisualProofs Жыл бұрын

@@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...

@Goaw2551 Жыл бұрын

@@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

@milanvasic1931 Жыл бұрын

@@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

@kornelviktor6985 Жыл бұрын

By far the most beautiful and relaxing video on youtube thanks :)

@MathVisualProofs Жыл бұрын

Thanks!

@louigomes154 Жыл бұрын

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.

@MathVisualProofs Жыл бұрын

@WomenCallYouMoid Жыл бұрын

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

@MathVisualProofs Жыл бұрын

Yes I have seen “folding proofs” as well. Cool way to think about them also :)

@theoriginaldrpizza Жыл бұрын

Awesome job putting this together! I had never seen many of those before.

@MathVisualProofs Жыл бұрын

Thanks! Check the playlist in the description for more too :)

@yuddhveermahindrakar6864 Жыл бұрын

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

@MathVisualProofs Жыл бұрын

👍

@sciencetechnician8787 Жыл бұрын

Amazing geometrical proof on GP, I am really happy that I had learnt something new..

@MathVisualProofs Жыл бұрын

Glad it was helpful!

@lapis.lareza Жыл бұрын

Thank you, thank you very much for the beautiful works !

@MathVisualProofs Жыл бұрын

Thanks for checking it out.

@78Mathius Жыл бұрын

I love the concept of algorithmic art and math as art. This is wonderful.

@MathVisualProofs Жыл бұрын

Thanks!

@vyacc.friend3798 Жыл бұрын

OMG, it is so beautiful! I have learned something about applied math in art and also some number mathematics! Thank you!

@MathVisualProofs Жыл бұрын

Glad it was helpful!

@ChannelDefault Жыл бұрын

Your channel is underrated. This is really beautiful and artistic.

@MathVisualProofs Жыл бұрын

Thanks! I appreciate the comment 👍😀

@user-ikono Жыл бұрын

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。

@MathVisualProofs Жыл бұрын

Yes! It’s fun to have both algebraic and geometric explanations.

@renesperb Жыл бұрын

Beautiful illustration!

@MathVisualProofs Жыл бұрын

Thank you so much 😀

@slrawana 4 ай бұрын

No Words. Excellent Work.

@MathVisualProofs 4 ай бұрын

Wow, thank you! I appreciate your comment :)

@eduardzakharian9274 Жыл бұрын

Thank you very much!)

@MathVisualProofs Жыл бұрын

Welcome!!

@AxelinickRapGirl Жыл бұрын

Thank you, that was beautiful beyond words

@MathVisualProofs Жыл бұрын

Glad you enjoyed it!

@chunkiatlim406 Жыл бұрын

wow, this is such an enjoyable video to watch

@MathVisualProofs Жыл бұрын

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!

@keinKlarname Жыл бұрын

Indeed: beautiful! Thanks a lot for this.

@MathVisualProofs Жыл бұрын

Thank you too!

@kmjohnny 4 ай бұрын

I'm so glad I found this channel.

@MathVisualProofs 4 ай бұрын

Glad you're here!

@dylanparker130 Жыл бұрын

Beautiful video!

@MathVisualProofs Жыл бұрын

Thank you very much!

@-ZH Жыл бұрын

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.

@-ZH Жыл бұрын

I suppose that also applies to the 3:05 method

@-ZH Жыл бұрын

Just realised the 2:13 proof is just a fancy way of drawing the 3:05 proof.

@hydrogenbond7303 Жыл бұрын

This is really beautiful.

@MathVisualProofs Жыл бұрын

Thanks!

@rohitsk6068 Жыл бұрын

Great work .

@MathVisualProofs Жыл бұрын

Thank you!

@algorithminc.8850 Жыл бұрын

Fun channel. Thanks. Cheers.

@MathVisualProofs Жыл бұрын

Glad you enjoy it!

@DoxxTheMathGeek Жыл бұрын

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.

@MathVisualProofs Жыл бұрын

Agree!

@shahin07140 10 ай бұрын

Such a nice way to present the mathematical expression.. Awesome experience with ur background music🎶 nice choice of background music..

@MathVisualProofs 10 ай бұрын

Thank you so much 🙂

@pizzarickk333 Жыл бұрын

Surprised I could understand all of it. Thanks for the video

@MathVisualProofs Жыл бұрын

Glad to hear that!

@Scrolte6174 Жыл бұрын

Great videos!

@MathVisualProofs Жыл бұрын

Thanks!

@Scrolte6174 Жыл бұрын

How welcome😁

@TRZG246 Жыл бұрын

Please don't stop making these videos they are helpful

@MathVisualProofs Жыл бұрын

I'll try! :)

@TRZG246 Жыл бұрын

@@MathVisualProofs thanks

@Zangoose_ Жыл бұрын

Math Degree here. This makes me feel like "All that challenging work don't seem so hard no more."

@MathVisualProofs Жыл бұрын

👍😀

@danielsantrikaphundo4517 Жыл бұрын

Beautiful

@MathVisualProofs Жыл бұрын

Thank you!

@aleksanderorzechowski5580 Жыл бұрын

This is beautiful 😮

@MathVisualProofs Жыл бұрын

Thanks!

@jordkris Жыл бұрын

Proudly, I can guess any result of infinity sum just with see portions of shape

@vishalramadoss668 Жыл бұрын

This was amazing. Geometry forever

@noble2834 Ай бұрын

Wow, please keep it up

@MathVisualProofs Ай бұрын

Doing what I can. Thanks!

@KaliFissure Жыл бұрын

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.....❤

@MathVisualProofs Жыл бұрын

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!

@KaliFissure Жыл бұрын

@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.....

@l.v.6715 9 ай бұрын

Wonderful!!!!

@MathVisualProofs 9 ай бұрын

Glad you like it!

@DidarOrazaly 8 ай бұрын

Amazing geometry

@ccona2020 Жыл бұрын

Maravilloso. Muchas gracias.

@MathVisualProofs Жыл бұрын

Thanks for checking it out!

@anadiacostadeoliveira4 Ай бұрын

Really like fractals! 😊

@hontema Жыл бұрын

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?

@MathVisualProofs Жыл бұрын

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: kzfaq.info/get/bejne/mN6Ddq2Dm7i2qKc.html

@nonameee0729 Жыл бұрын

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

@MathVisualProofs Жыл бұрын

@@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.

@johnchessant3012 Жыл бұрын

very neat!

@MathVisualProofs Жыл бұрын

Thanks!

@Mo-uq4ix Жыл бұрын

3:08 BLUE LOCK LESSGOOOOOOO!!!

@WiecznyWem Жыл бұрын

So calming :)

@MathVisualProofs Жыл бұрын

👍

@peterwolf8092 Жыл бұрын

I am so dumb. The second one surprised me 🤣

@SocratesAlexander Жыл бұрын

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).

@MathVisualProofs Жыл бұрын

Both circle and polygon methods generalize. But the result is 1/(r-1)

@williamribeiro4622 4 күн бұрын

beautiful

@MathVisualProofs 3 күн бұрын

Thank you! 😊

@GamerzInfinite Жыл бұрын

3:06 ayo blue lock

@바나나는최고의과일 Жыл бұрын

so beautiful video

@MathVisualProofs Жыл бұрын

Thanks!

@yashmithmadhushan888 Жыл бұрын

Good job

@MathVisualProofs Жыл бұрын

Thank you!

@user-qh4wc5zz6m Жыл бұрын

Awesome😮😮😮😮

@ricekuo853 Жыл бұрын

So cool！

@MathVisualProofs Жыл бұрын

Glad you liked it!!

@utsavmitra Жыл бұрын

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

@MathVisualProofs Жыл бұрын

Thank you!

@calicoesblue4703 Жыл бұрын

What specifically is this song called? It’s very relaxing & beautiful.

@MathVisualProofs Жыл бұрын

It’s linked in the description - check it out!

@SridharGajendran Жыл бұрын

Mesmerising...

@MathVisualProofs Жыл бұрын

@ruilopes00 Жыл бұрын

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.

@MathVisualProofs Жыл бұрын

Glad you enjoyed it!

@MaJetiGizzle Жыл бұрын

Bwaaahhh it’s just one more in the denominator gotta love that geometry tho!!!

@Unknown-kc8xz Жыл бұрын

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

@minakadri2221 Жыл бұрын

so relaxing

@MathVisualProofs Жыл бұрын

😀👍

@robertingliskennedy Жыл бұрын

superb

@MathVisualProofs Жыл бұрын

Thank you!

@mangus8759 9 ай бұрын

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

@MathVisualProofs 9 ай бұрын

I have a couple general approaches for any ratio between -1 and 1 on my channel too.

@natanytzhaki8665 Жыл бұрын

(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)

@MathVisualProofs Жыл бұрын

sounds about right :)

@tamirerez2547 Жыл бұрын

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.

@MathVisualProofs Жыл бұрын

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!

@tamirerez2547 Жыл бұрын

@@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.❤️👍

@MathVisualProofs Жыл бұрын

@@tamirerez2547 Thanks! :)

@Patrik-bc2ih Жыл бұрын

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.

@GrifGrey Жыл бұрын

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.

@GrifGrey Жыл бұрын

ope, now realised how many people came to the same conclusion sorry for the filler

@MathVisualProofs Жыл бұрын

Definitely cool to see it right?

@MathVisualProofs Жыл бұрын

@@GrifGrey No worries! I am glad you noticed it and commented on it. That's the fun of it!

@GreenPower713 Жыл бұрын

Wow! Just... wow!

@MathVisualProofs Жыл бұрын

😀👍

@minhperry Жыл бұрын

Is it possible to prove this visually for each (1/n)^k series with a (n-1)-gon instead?

@MathVisualProofs Жыл бұрын

Yes! The circle proof generalizes as well

@mrsillytacos Жыл бұрын

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..."

@benjaminbertincourt5259 Жыл бұрын

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.

@MathVisualProofs Жыл бұрын

Yes! The circle is nice for sure. But I love the different ones here too :)

@prarthananeema9774 Жыл бұрын

Thank you God for recommending this 🙏❣️

@MathVisualProofs Жыл бұрын

Glad you liked it!

@kadirjaelani8112 Жыл бұрын

What application did you use to make the video animation?

@MathVisualProofs Жыл бұрын

I use manim (manimgl currently) for all the videos on my channel. But manimce will be better to use I think.

@guigazalu Жыл бұрын

Cool use of manim.

@MathVisualProofs Жыл бұрын

Thanks!

@ludmilavokareva719 Жыл бұрын

Математика завораживающее зрелище! Спасибо!👍🏻👏👏👏

@MathVisualProofs Жыл бұрын

Thanks for checking it out!

@maximosavogin50 Жыл бұрын

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

@parimalpandya9645 2 ай бұрын

What about Basel problem of sum of reciprocal of the squares and other problems of reciprocal of cubes

@MathVisualProofs 2 ай бұрын

Hard to get nice dissection proofs because of the values those produce

@Gunslinger-us1ek 5 ай бұрын

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)

@MathVisualProofs 5 ай бұрын

Check out my playlists on finite and infinite sums. There are others besides geometric sums (though geometric are my faves)

@bizon1271 Жыл бұрын

This amazong work. May God bless you and your loved ones!

@MathVisualProofs Жыл бұрын

Thank you so much!

@antoniocampos9721 Жыл бұрын

This is absolutely wonderfull....for the ones who love math...

@MathVisualProofs Жыл бұрын

I hope that includes you :)

@antoniocampos9721 Жыл бұрын

Of course I'm included...

@MathVisualProofs Жыл бұрын

@@antoniocampos9721 👍😀

@MisterSnail1234 Жыл бұрын

But does that mean that if you make the sum i root of 1/k as i -> infinity = 1/(k + 1)??

@noah-tl1gv Жыл бұрын

wait so does it converge to a number, infinitely close but never reaching it, or does it actually eventually equal it?

@MathVisualProofs Жыл бұрын

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.

@adipy8912 Жыл бұрын

Geometry is my favorite thing in math(s)

@MathVisualProofs Жыл бұрын

👍😀

@aeoliaxd Жыл бұрын

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...

@MathVisualProofs Жыл бұрын

@limenlemon3116 Жыл бұрын

I tried 1/(n^n) with the sigma function, and it was ~1.291285. I named the constant after myself.

@kitten6317 Жыл бұрын

Ok but what does the square root of -1 have to do with geometric series /j

@jpopelish Жыл бұрын

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.

@MathVisualProofs Жыл бұрын

Here’s one way to do it in general : kzfaq.info/get/bejne/jJ18aNGkt9Cbc3k.html . I have a playlist that contains others too : kzfaq.info/sun/PLZh9gzIvXQUsgw8W5TUVDtF0q4jEJ3iaw

@neon9334 4 ай бұрын

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

@MathVisualProofs 4 ай бұрын

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.

@neon9334 4 ай бұрын

oh thanks man @@MathVisualProofs

@villaratanaphom-sg3hg Жыл бұрын

do you think you can do a proof on why the infinite series of (1/a) = (1/ (a-1) ) ??

@MathVisualProofs Жыл бұрын

I have a few on the channel. Here’s a nice one : kzfaq.info/get/bejne/jJ18aNGkt9Cbc3k.html

@sunsetbyauraYT Жыл бұрын

For anyone who is still confused: ∞ ∞ ∑ 1/(nⁱ) = 1/(n-1) and ∑ n/((n+1)ⁱ) = 1 i=1 i=1

@tompeled6193 Жыл бұрын

Σ(i=1, ∞)1/n^i=1/(n-1)

@MathVisualProofs Жыл бұрын

👍

@k-senpai3203 Жыл бұрын

You can just use a circle for every example instead of different shapes. Just watch the 1/7 part and be creative.

@Amoro.369 Жыл бұрын

Mathematics is so unique ✨️

@MathVisualProofs Жыл бұрын

👍😀

@WilliamWizer Жыл бұрын

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?

@MathVisualProofs Жыл бұрын

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.

@WilliamWizer Жыл бұрын

@@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.

@Siya0000 Жыл бұрын

@WilliamWizer You just need the smaller pentagon to be 1/6 of the larger one. The radius should be 1/sqrt(6).

@Siya0000 Жыл бұрын

@WilliamWizer Which is approximately 40.8%.