Koch snowflake fractal
Area of Koch Snowflake (part 1) - Advanced Starting to figure out the area of a Koch Snowflake (which has an infinite perimeter)
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- We know how to find the area of an equilateral triangle
- What I want to do in this video is attempt to find the area of a a
- I know I'm mispronouncing it Koch snowflake
- And the way you construct one
- is you start one with an equilateral triangle
- And on each of the sides you split them into thirds
- And then the middle third you put
- another smaller equilateral triangle
- And that's after one pass
- And on the next pass you do that for all of the sides here,
- So little one here here here here here here here
- I think you get the general idea that's a nice pass
- So the next pass you do it for all these sides
- And what's really neat about this and
- we showed this on the previous video is that
- you have a figure here that has an infinite perimeter
- but we're about to see in this video it actually has a finite area
- Which is kinda of interesting to think about
- So let's start with a clean equilateral triangle right over here
- We're gonna assume that each of the sides have length S,
- So, it's going to be a clean equilateral triangle each of the sides
- I'll draw that a little bit neater, each of the sides have length S
- And so, what I wanna do that is keep to track of two things
- I wanna keep track of the sides of this triangle
- as we are or as it as it turns into a snowflake
- I gonna keep track of the number sides
- and I gonna keep track of the area
- after each pass of adding more smaller triangles
- So this is gonna be our count of the area
- Actually, let me give myself a little bit more real estate
- because I have a feeling I might need to use it
- So might keep this is the sides
- I'm gonna write up here
- and then this is our running count of the area down here
- So right when we start,
- We have three sides
- And our area we are to figure out in the previous video is going to be
- we assume that each of the sides are length S
- is going to be square root of three square root
- of three S squared over four
- Fair enough that's just a simple equilateral triangle
- Now we're going to take each of the sides divide them into thirds
- So we're gonna take each sides divid it into thirds
- And then, in that middle third we're going
- to add another smaller equilateral triangle
- So it will look like that on that side right over there
- I want you to think about what we're doing each side right here
- So before I did this, this is just one side
- Then I split it into thirds and that middle third
- I put I said you put two sides in there
- I put an equilateral triangle
- So one side has now turned into one two three four sides
- So every time we do a pass
- of the making the snowflake more intricate
- Each side will turn into four sides
- So you can imagine if we do this on all three sides,
- we have four times three which is now 12 sides
- So if you multiply these times four,
- so times four this gets us to 12 sides now
- We can count them out just
- to make sure we're on logic this correct,
- One, two, three, four, five, six, seven,
- Eight, nine, ten, eleven, twelve sides
- And now what is the area now?
- What's going to be the area
- of which of all yellow equilateral triangles
- plus the area of each of these smaller ones?
- And what's the area of each of these smaller ones?
- Well, first of all we have three of them
- There's three of these each of these smaller ones
- And then we use the formula for an equilateral triangle again
- So it's going to be the square root of three times S squared
- For now the length of each of these sides
- Each these sides are equal to these smaller equilateral triangles
- They aren't S anymore
- they are S over three
- remember this length right over here is S over three
- so this is going to be S over three as well
- Every pass,
- the sides of the equilateral triangle
- become one third of the previous pass
- So this is not going to be S squared anymore
- It's gonna be S over three squared
- and then all of that over four
- Then let's do another pass
- So, I'm gonna add these triangles right over here
- Then add these right over there
- And this is the last pass where I actually attempt
- to draw all of the triangles over there
- So how many how many sides am I going to have,
- first of all, after I do another pass?
- Well the previous pass I have 12 sides,
- each of those 12 sides are now going turn into four new sides
- When I add these little orange bumps there
- So when I multiply it times four again
- 'm gonna multiply it times four,
- so now I'm going to have 48 sides
- I'm gonna have 48 sides
- And how many new triangles,
- so what's the area what's gonna be the yellow area
- plus the blue area plus the orange area
- So how many new orange triangles do I have?
- Well I'm adding a new orange triangle
- to each of the sides for the previous pass
- And the previous pass I have 12 sides,
- so now I'm going to add 12 orange triangles
- I'm going to add 12 orange triangles
- And actually let me write that
- or I'll just write 12 orange triangles but it's really,
- I just multiply it times four
- And then I'm going to have times the square root of three
- And now this is not going to be S over three anymore
- This is going to be s over nine
- These have one third dimensions of these blue triangles
- So this is going to be S over nine squared,
- S over nine squared over four
- And so, I think you might start to see the pattern building
- if we do another pass after this one
- Move to the right a little bit what will that look like
- Let gonna do this in a different color that I haven't used yet
- Let me see if haven't use this pink yet
- So now we're going to have,
- I'm going to have the previous number sides
- that's my number of new triangles
- 48 times the square root of three times S over --
- I'm going to, now these are gonna be one third of these
- S over 27 to the second power all of that over four
- And I'm going to keep adding an infinite number of terms of this
- to get the area of a true Koch snowflake
- So I'm just gonna keep doing this over and over again
- So that trick really is is finding this infinite sum
- and see if we get a finite number over here
- So the first things I wanna do just to simplify,
- Well let me just rewrite it a little bit?
- Let me rewrite it a little bit different over here
- So the first thing that's kind of it, that might be obvious is that
- We can throughout the square root of three S squared over four
- So let me just factor that out
- So if we factor a square root of three S squared over four
- from all of the terms then this term right
- over here will become a one
- This term right over here is going to become a three,
- let see we factor a squared of three we factor out of four
- and we factored out the S squared
- We factored out all in the S squared,
- so now it's going to have plus three times
- one third three times one third squared
- That's all we have left here,
- We have the one third squared
- and then we have this three
- And I'm not simplifying this on purpose
- so that we see a pattern emerge
- And then this next term right over here plus
- so this 12 is still going to be there
- but I'm gonna write that as three times four
- And we're gonna lose, this that we're factoring out
- the square root of three
- we're factoring out the four
- we're factoring out the S squared
- And so we're gonna be left with three squared
- that's what this is down here, squared
- So this is one over one third squared and then that squared
- So that's what we're left with that orange term
- And then we're going to the pink term
- This pink term, this is 48 it is just three times four times four
- Three times four
- I'll write four squared here
- cause each time we're gonna multiply times four again
- So next is gonna be four to the third
- Because we're really each iteration,
- each sides turns into four sides
- that's where that came from
- Four squared we lose we're factoring out the squared of three
- we're factoring out the four we're, factoring out the S squared
- And all we're left is one over three to the third power squared
- So, times one over three to the third power squared
- and we'll just gonna keep going like that forever
- Keep going like that forever
- So, on each each step we're incrementing,
- we're multiply by four and we're also multiplying
- we're, I guess we say, the power of this four as incrementing
- So it goes from there's usually fourth of the zeroth power here
- We have a one here
- You can kind of imagine implicitly the fourth
- of the first power four squared and then it'll go four to the third
- And then we have this power is also increment
- Three to the first, three to the second, three to the third
- We see that this power is always one more than that
- And it'll be much easier to calculate this infinite
- which is gonna turn into infinite geometric series
- if those were actually the same power
- So, what I wanna do is I wanna
- increase the power four in all of those
- But I can't just willy-nilly multiply everything by four
- If I'm gonna multiply everything by four
- I also need to divide everything by four
- So, what I'm gonna do in this right over here
- is I'm gonna multiply and divide everything by four
- So, if we divide by four I can do that on the outside
- So, I'm gonna multiply it one fourth times this right over here
- And, so I'm dividing by four and here
- I'm going to multiply this by four
- So, I'm not going to be changing the value of the actual thing
- So, this is going to be four plus three times four
- plus three times four squared four to the third
- And so what was cool about this is now
- at the power of power and the power of this three down here
- are going to be the same power
- But it still seems a little weird because we're
- taking this one over three squared and we're squaring it,
- one over three to the third then we're squaring it
- And here we just have to realize,
- so this is always gonna be squared
- and this is the thing that is incrementing
- But in general,
- if I have one over three to the end
- and I'm squaring it
- This is equal to one over three to the 2N power,
- which is, so I'm just multiplying it by two, right?
- If I'm raising something to an exponent
- then raising that to an exponent,
- that's just multiplying it times the or raising
- to the or raising to the Nth times to exponent
- And this is the exact same thing
- as one over three squared raised to the Nth power
- So we could actually switch these two exponents
- in a very legitimate way
- And then let me rewrite everything
- 'cause I do wanna do too much on this one step right over here
- So, part right over her gives us square of three S squared
- over 16 and then that's going to be times,I'll open
- and close parenthesis
- So then we have four plus,
- then in blue,
- I'll write three times four to the first power
- And then I can write this,
- I can rewrite this as one times one third
- We could view this as one third squared
- or we could view this, we could view this
- as one over three to the first power square
- or we can view this as one over three squared to the first power
- And I'm gonna write it that way
- So time one ninth to the first power
- And then plus three times four squared
- And then these we can write as times one ninth to the second power
- And then these where we can write plus three times four to the third
- times
- and this is we could write
- this is on over 27 to the second power
- But we could also write this based on what we saw over here
- Let me make this clear
- One over three to the third to the second power
- This is the same thing as one over three squared to the third power
- That's what we showed right over here
- So, this is equivalent to one ninth to the third power
- Now we start to see the pattern
- and starting to clean up a little bit
- And let me just do one more step,
- and then we'll finish this to the next video
- So this is equal to square root of three S squared
- over 16 times four plus four plus three times,
- this is four ninths plus,
- the next term is three times four ninths squared
- And then we have plus times four over nine to the third power
- And we're just gonna keep on and on and on and on
- taking three time four ninths
- To the successively larger and larger power
- So, this is what we have to find the sum of
- to find our area and we're gonna do that on the next video
- We're gonna use some of the tools
- we've used to find the sums of infinite geometric series
- But we're kind of going to redo it in the next video
- just so you don't have to remember that formula or that proof
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At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
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