Koch snowflake fractal | Perimeter, area, and volume | Geometry | Khan Academy

[@PHILANTHROPIST1988]

i cannot imagine what my life would be without khanacademy thank u very much khan -_-

[@jetpaq]

i disagree that the "area is "infinite" but the bounds is finite. Because if u zoom in..theres an area inside of each of the triangles then the "area" expands.. What do you think sal! : ) still this is freaking awesome and so are you!

[@chessandmathguy]

But the video doesn't say the area is infinite. It says the area is finite.

[@BurkeyAcademy]

@scottycatman No, it really is Infinite. The shape keeps getting more and more complex at smaller scales, so that you keep needing to add more "string" to outline all of the infinite facets, and you can never keep up. When we say that something is infinite, we never mean "trillions". Trillions are quite small. :)

[@robsonomasold]

Great sounding voice, very clear and easy to follow. This guys is good.

[@joechamm]

@jazzbuckeye - If each of the sides is length s, then the original area can be written as A = (1/2)s^2. After the first iteration, we are adding the areas of three triangles, each with area (1/2)(1/3s)^2 = (1/9)A, which gives us A + (1/3)A. You'll notice that after this you add 4 times the number of triangles on each successive iteration, each with area (1/9) the area of the last iteration's triangle(s)... ie you are adding (4/9)^k area on each iteration, giving you a geometric series.

[@metaldave08096]

Great Video, i read about the Koch snowflake in a book that i have and didn't quite understand it but this cleared everything up perfectly for me!!!

[@Ajaykumar-ij9ym]

One day before exam, its hard to know about the topic throw single video. But you really different. Now i understand the topic and also its become interesting

[@Soulsphere001]

I've been thinking about programming a fractal image, but I'm not quite sure when to start adding new triangles. I'll have to figure it out.

[@amoeba909]

Hey Sal, An interesting conjecture from an amateur physicist (0.0001 on a scale from 0.0001 to 10000000000.000). Can we bind the universe in n+1 space dimensions? It is quite interesting that time dimension is irrelevant in this observation!

[@Manodragon]

Can you do more biology videos? They are incredibly interesting :)

[@secretunknown2782]

Biology is always interesting 😂😂

[@518schenectady]

an infinite perimeter with a finite area thats nuts i saw 2 of ur videos and i had to subscribe to this channel

[@roflmao0702]

This is definitely interesting... but I'm just not sure when i'm ever going to use this.

[@kinghassy334]

It's a practical way to apply infinite series to find things like the area and perimeter

[@Aelea]

I'm here to figure out how some things manifest naturally.

[@fatelis]

If you become a biochemist you will

[@That.little.dinosaur]

@@Aelea John 3:16, God bless ❤

[@yuyo1948]

Es interesante esta posibilidad de aprender sin prisas. ¿Hay en castellano estos videos o subtitulados?

[@Jeed92]

Interesting sidenote: Nikola Tesla said if you want to understand the mysteries of the universe and the nature of excistance, you will have to work with the numbers 3 6 and 9. Fractals made is possible for humanity to have proper internet connection and information exchange without loss over cables (without it we wouldnt have internet or computer communication as we know it today). And here comes the 3 6 and 9 into play. Tesla wasnt just the inventor of modern day electricity and power contribution, he also was right about his 3 numbers. The first step of the snowflake is to make a triangle to the next shape is to make out of 3 sides, 6 cuts in order to get 9 lines of the same size. This single act is the entry into divine geometry, it is the entry into fractals. Fractals arent just geometry, they gave us understanding over how life was created and how life took its various shapes and forms. They gave us power over flawless communication techniques and they made us understand that there arent only 3 or 4 dimensions in space.

[@onumavirga]

Wow, that’s it’s really mind blowing!!!

[@scottycatman]

How do I explain this... Well, the area is finite because it's bound in that hexagon. The perimeter is only explained as "infinite" because it's ever-increasing. However, don't think for a second that by "infinite" he means that its perimeter can be in the trillions.

[@benschneider2710]

going to copy this into my math ia :)

[@dolnodjob]

There's something that bothers me. For finding the total length, it is assumed that the length of the limit curve equals the limit of the lengths of the individual curves. However this is not always the case, as it's demonstrated by the Pi=4 "proof".

[@MetalMilitia072583]

its a little after 4am and i was falling asleep... i started watching this and now im wide awake!! go figure.... P.S. i credit khan academy for my A in calculus and preach it to all of my friends

[@gupta-pw5xb]

3:08 after each *generation*

[@charlenelariosa5378]

Thankkk yoooou

[@silencedidgood]

Sal talking about fractals....infinite perimeter....finite area. What better way to start the morning in a fractious world.

[@lmcdowall]

This is why I love math! Great video!

[@dakshsaxena660]

no

[@A010795]

lol I love how you repeat the things you say while writing it out :P

[@alextate1525]

That’s exactly what I was saying. It’s kinda satisfying

[@elswamidor]

MAGENTA!!!

[@someonetoogoodforyou]

The total area is A = 2*sqrt(3) * S^2 / 5

[@jacobskarby1389]

So proud Koch went to my high school! :)

[@atheno3061]

so you can go infinitely smaller but once you have that initial snowflake , theres nothing going on in the other direction ? meaning there’s not a bigger one ? or not one that looks like the initial triangle with the 3 coming off

[@chessandmathguy]

The area does get larger and larger forever, but doesn't diverge to infinity.

[@BigDoodles]

Can you give us the formal proof for the finite area of the Koch Snowflake? For those of us who are far past the basics.

[@charlottespears9977]

I would argue its not an infinite area or infinite perimeter. rather the perimeter and area as you zoom in increase by smaller and smaller decimal places making it more precise and the decimal places of these measurements extend to infinity. therefore, the values, as you zoom in closer and closer to infinity, approach a constant.

[@chessandmathguy]

I get where you're coming from, but that's incorrect. The perimeter truly is infinite but the area isn't.

[@jazzbuckeye]

Could you express this in sigma notation as an infinite series?

[@pnbllwzrd95]

I hope by the island of England you mean Great Britain - (excluding outlying islands such as the Orkney Islands, Shetland islands etc. as they would make it more than one Island) note I use the term Great Britain and not the United Kingdom as that would include Northern Ireland. :)

[@Revelationxxx]

It is not an infinite area but an area that is impossible to measure exactly as it is getting infinitely more precise? :o

[@tylergerald]

Would it be correct to say if the second step the perimeter is 4/3Po, would the third step be 4squared/3squared Po? Rather than 4/3P1?

[@chessandmathguy]

Same thing.

[@porkypine1888]

Yes fractals!!!!!!

[@superdau]

@KiloSierraAlpha Far from it. But I think it's hard for english speakers since you don't really have the sound of "ch". It's like the "loch" in "Loch Ness" or like the "ch" in "Bach" (the composer).

[@scottycatman]

@scottycatman Y'know what? I forgot to factor in that there are more triangles in each 'tier'... I was going off of the idea that each tier had the same amount of triangles. Thanks for replying, otherwise I wouldn't have second guessed myself.

[@fjordland5646]

@KarmaProstitute It's a german surname. The retroflex R you speak of is not correct, it is a gutteral sound similar to how you would pronounce Johann Bach.

[@darrellawlor]

How was this video made?

[@TheLlamachupacabra]

when we use the limits of the universe n physics, planck time/ length, speed light etc we dissolve infinite paradoxes ( like the fact that u could never draw a koch triangle since it would require infinite time) but from what i understand mathmaticians freak out when u say that, why... why not undo zenos paradox w planck length?

[@yuyo1948]

Me gustan y quiero ampliar, pero....hay estos videos es castellano o subtitulados.

[@igorkrupitsky]

Assuming that initial triangle is 1 meter, it would take about 30 iterations to go to down to atomic level. In Excel type: =POWER(2,-30). Hydrogen Atom size is about POWER(10,−10) m.

[@QueenFondue]

The star of the star of the star of the star of the star of the star of David.

[@Electronieks]

how long is a piece of string -theory ! :)

[@alcesmir]

I looked into his family name a bit more and found that the name is pronounced like "kokk" would be pronounced in Swedish (my native language). So, the English pronounciation would be more or less: "cock". Fun facts: Helge von Coch was born into a family belonging to the Swedish nobility. He studied at Stockholm University under Gösta Mittag-Leffler, another famous mathematician.

[@hypotherima1]

I wonder what a vi hart video on this would be like

[@AureliusProject]

To be a bit more specific. It's spoken as: Coke But the k sound you make is like when you hack up stuff in your throat

[@Revelationxxx]

Don't the bumps have an area? If so then yes you know the area the object can not exceed a certain area but you still don't know the exact area of that object and wouldn't in looking for that you'd find if you can keep adding bumps you can keep adding area at some scale that is measurable? I'm confused how it can have an infinite perimeter but not an infinite area... can someone explain?

[@SethShivang]

this is the part of class 6th syllabus in India

[@TasmanianTigerGrrr]

I think it would have infinite area because you are always adding onto its size, but the areas added are so microscopic that it appears to be finite . The area is growing infinitely smaller with each iteration approaching a size scale we cannot comprehend.

[@lloydkim3990]

it wouldnt be infinite

[@chessandmathguy]

It's possible to add an infinite number of positive numbers and still not diverge to infinity. Many infinite series do not diverge to infinity.

[@TasmanianTigerGrrr]

Would that not make the area infinite then? The area keeps growing with no end in sight@@chessandmathguy

[@chessandmathguy]

@@TasmanianTigerGrrr area keeps growing but stays finite despite growing forever. As an example of such a phenomenon, the series 1+1/2+1/4+1/8+1/16... keeps growing forever but always stays below 2. Formally, it is said that this series converges to 2. But this shows that it is indeed possible for something to grow forever without diverging to infinity.

[@TasmanianTigerGrrr]

Is it possible to measure the area of the snowflake? It seems like it would be asking the question "how long is a piece of string"@@chessandmathguy

[@somnys]

@personkid20 do you know what "described" means?

[@jamesjosephclarke]

I always like liked the Menger Sponge, infinite surface area, zero volume.

[@That.little.dinosaur]

John 3:16, God bless ❤

[@Pirate44444]

oh yeah, that coast line between England and Wales

[@FloraSlimesOfficalYT]

dont pretend this is just interesting geometry - its Your life. So do changes in time and place

[@artmonger78]

This guy sounds like an older version of Chris (seth green) from family guy... Great vids though.. love me some khan academy

[@xmyccymx]

@eksman187 are you partly German?

[@TheLlamachupacabra]

how big is the wall? ( mass area perimeter doesnt matter) all infinite. one one of the wall, an infinite plain, other side another infinite plain. three infinities in one, none could be called smaller than the other even the original undivided plain. so i would argue u could "increase resolution" or "zoom in" on a koch triangle and reveal its infinite area. u can hide its infinitness, but never get rid of it. why does math disagree w my seemingly (to me) solid logic.

[@mastertrey4683]

How does this have an perimeter length of infinity... makes no sense its like a repeating decimal its not infinite

[@MaciejProsowski]

I thought that not P(infinity) is equal to infinity bu the sum of all P's.

[@Buoy2]

@ICarnag3 everything

[@jewljuju]

this was helpful thanks! Although i wont use this in my everyday life, i will use this as part of my math essay

[@TheLlamachupacabra]

infinity is a process of time(actually lack thereof) not a number. if an object grows an inch a year, or a mile a year, for all eternity i would argue no difference. at any point in TIME i could say one bigger than other but both have infinite expansion. to attach a numerical value implies time which ceases to exist at infinity. no infinity is bigger than another. only one infinity... the infinite complex. sum of all infinities. infinite plain.. build a wall dividing it in two..

[@nicolathing]

TheLlamachupacabra not true. Mathematicians have proven how some infinities are larger than others. This is true for the real number set, which is proven to be larger than the whole number set

[@learnfromstudent]

it takes a little khan magic.

[@alcesmir]

Correction: Coch -> Koch

[@somnys]

@Freshman000000 if you're really far past the basics, then you should be able to prove it yourself.

[@Hiandbye95]

Koch is German and it's pronounced like the Scottish loch. Like in Loch Ness, you know. ;)

[@vaibhavpatil2611]

so it's called snowflake because it looks like snowflake or actual snowflake follows this pattern ?

[@nejlaakyuz4025]

vaibhav patil this looks like snowflake

[@SanctumZero]

The correct pronunciation would sound similar to "cock", I believe 8D

[@Jeed92]

No because cock sounds like Kok in German. There is not a single word in english that has the pronunciation of the german ch after an o. If you try to say Loch Ness, the Lok will be similar to the proper Koch Pronounciation. Now try to make the "k" in the Lok Ness sound like the 2nd half of the k is missing. Or just google Koch and listen to the pronounciation. Theres no elegant way to solve this with written language. you have to hear it.

[@svergurd3873]

It is pronounced "kock" He was Swedish and it is pronounced so in Swedish. The spelling Koch is not typical Swedish and the name probably originally comes from German, but it is not pronounced in the German way. However, we pronounce the "von" correctly, so it is "fon kock" in Swedish, that's what he called himself.

[@GermanTacos]

On the pronunciation of Koch; I'm not sure of the Swedish pronunciation, but odds are it has the same sound as the German pronunciation, which is pronounced as 'Coke' (As in Coca-Cola).

[@MrLaggy2000]

Exactly just like the Koch brothers. (Last in the family of Koch industries and two of the wealthiest people in America.)

[@orinsmith1004]

GermanTacos we learned this as “co” for pronunciation. Like co-dependent.

[@patinho5589]

Doesn’t the perimeter calculation converge though? I guess the answer is no if people have said so. Oh multiplying by 4/3 diverges.. but man.. this is nuts... infinite perimeter in a bounded area.. just by drawing.. I guess it’s all because a line has no width.. so you can ‘fit’ in a infinity of line length into a 2D space... you zoom in and in and can still draw a line with no width.. the lines don’t get wider as you zoom in.... therefore you find more and more ‘space’ to draw lines in... I will call it “‘zoom in’ space”.

[@borgholable]

we have that letter in arabic ..... and it is very weird that there are 8 letters in arabic that doesnt exists in english ,,, that is 9 letters ... damn

[@TheLlamachupacabra]

I have such a problem w this! If i had a sheet of paper n a pen cabable of infinite detail. And began drawing a koch triangle ignoring the quantum limitations of planck length, continuing to draw smaller triangles each triangle continues to have a 'white spot' inside, w an area greater than zero. The sum of an infinite series all greater than zero is infinite! No i dont run off the paper... but still an infinite amount of white triangles! Arghh infinite dimensionality... one is infinite, .999...

[@davetriplett4779]

Hurry! Someone do it an infinite number of times! ;p

[@patinho5589]

So the key is that perimeters don’t exist because lines don’t exist

[@jigsawpuzzle6340]

I'm sorry as I have never done fractals.. but why is perimeter of the line 4/3... what about the base s/3? why is 5/3 incorrect?

[@nataliess]

I'm not sure, but I think its because you dont count the line on the bottom of the triangle

[@LiteraryInsanity]

my teacher wuz here

[@theolevy]

he says infinite perimeter, and finite area

[@crysisgoty]

he said the area is finite; perimeter is infinite

[@Lojikish]

So in other words, England owns the universe.

[@Hiandbye95]

Oh? Seems, I falsely took it for granted that everyone would know how to pronounce loch? xD

[@scottycatman]

Now I can't sleep at night.

[@alaverga173]

use the flower of life to create a snowflake god does..lol but try it theres infinite possibilities

[@Sodapop04]

Came from the odd1sout

[@khroma441]

DuoMusic ???

[@TheLlamachupacabra]

Not a mathematician, obviously! Lol.. Mathematical proof didnt help me... as a thought experiment i get an object that never runs off the paper, but has infinite perimeter, infinite area, and by default, infinite mass. Thus swallowing the entire universe, all while never running off the page... if i draw triangles forever i would need infinite ink, w infinite mass. Arghhh!!!!

[@TheLlamachupacabra]

i could post my crazy comments on a vid w more views but i figure less trollers here anyone that finishes this vid is probably way more educated than me... lol maybe ill get an answer that doesnt involve whether or not god exists...lol stupid youtube arguements.

[@dharma6662013]

This is GREAT video. Unfortunately, you just annoyed all of Scotland and Wales. The shape you drew was Britain, which is made up of England, Scotland and Wales. It's like saying "here's a picture of Texas" and then drawing the whole mainland USA.

[@FishbirdDD]

dharma6662013 *great Britain

[@superCattaz]

Great Britain.*

[@kindykrei]

It sounds like he is talking to him self I am going to draw a equilateral triangle I am going to draw a equilateral triangle

[@LiteraryInsanity]

who ever clicks my profile will probrably know who i am im talking to my teacher

[@TheLlamachupacabra]

ok seriously though... the amount of ink it would take to draw a koch triangle would be infinite right? infinite mass on an 8x11 sheet of paper... hmm infinite perimeter, an infinite mass, an infinite amount of time to draw it, infinite amount of triangles, each w area greater than zero. infinite all round, even area... just cuz it doesnt leave the page doesnt break infinity. stupid math... not conforming to my logic!

[@somnys]

@personkid20 do you know what "described" means?