Video Infoblog - Another way to find the Fractal Dimension of a Coastline: Box Counting

Transcript

Intro

0:00

hey everyone in this video I'm gonna be

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showing you guys how to use the box

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counting technique and the reason why

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I'm doing this is partially because a

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few people have asked me about how they

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can use this technique because in my ia

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I really messed up the number so it

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doesn't make sense and it's not clear

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what to do oh this is a quick video just

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to show you guys how to do it it

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actually really isn't that difficult in

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this I am gonna be using Photoshop but

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that's basically all you need in order

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to do this technique yes let's just get

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started so firstly I'm gonna go over a

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little bit of the theory if you want to

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read more about it you can do so with

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the links below I'm not gonna go into

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too much detail about it I'm just gonna

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explain the equations part so I'm just

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gonna read out what the Box counting

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method is about so the procedure

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involves superimposing grids of

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different caliber size over an image

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then recording the number of blocks the

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coastline passes through each caliber

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with a finer grid more details supposed

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to be picked up therefore allowing you

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to better find the factor dimension the

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method can be multi using this equation

Box Counting Method

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and the number of boxes is equal to see

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a proportionality constant multiplied by

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s the scale factor ^ D which is the

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vertical dimension we can rearrange this

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formula to get it into the form y equals

1:14

MX plus C photograph and then find the

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gradient as the factor of dimension so

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let's do that so we take a log of both

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sides then placed off wall plus we can

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split the two so we get log N equals log

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C plus log s to the power of D then of

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course we can pull the D down so we get

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log N equals log C plus D log s and then

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let's just rearrange that put the log N

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equals the log s plus log C as you can

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see here now if you've done linear

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regression this is in the format of a y

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equals MX plus C equation so in this

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case Y would be log n m would be d X

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would be both s and C would be log C so

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this means we can ignore the

1:58

proportionality constant and if we plot

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a bunch of wise against a bunch of X's

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and we should be able to solve for D so

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let's briefly talk about scale factors

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because this is kind

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a difficult concept to talk about but

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I'm gonna simplify it so if you're

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confused it's me because I'm simplifying

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the math but this is a way to understand

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it just well enough in order to actually

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perform this method and get the fractal

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dimension so essentially every time you

The Scale Factor

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perform a scale you make it so that four

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boxes can fit inside one box let's call

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this one scale when you first start out

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you have a scale factor of zero then

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that scale factor becomes two to the

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power of the number of times you scaled

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the box this is a scale factor of two

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because it's two to the one then if we

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do it again like this because this is

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the second time we scaled that means the

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scale factor is four then if we do it

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again and scale for a third time that

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means the scale factor is eight and you

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really only need four values to find the

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practical dimension once you go beyond

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four it becomes difficult to perform now

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there's a reason why it's two to the

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power of the number of times that you

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scale down the box for but that

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mathematics is a bit too complicated and

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I'm not really qualified to explain that

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to you I'll lick resources down in the

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description that may be able to explain

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to you but anyways that's what the scale

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factor is okay so now I'm gonna actually

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go on my computer and show you guys how

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I actually do this technique so

Finding the Grid

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basically you're gonna start by going on

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Google Images and you're just gonna pick

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a random great you're gonna want to go

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look look for one that has no white in

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the background so it's just a grid it's

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got a tools color transparent this one

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should be good it doesn't really matter

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so we just we just need something that

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has a lot of boxes and then after you've

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got a grid you're gonna open up

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Photoshop then you are going to look for

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an image of the island that you are

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going to use to find the fractal

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dimension however what an important

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thing to do is that you cannot get a

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black image because

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you'll have issues with the grid because

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we will be able to see the boxes because

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the grid is the same color as your

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island and so this one will work because

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it's a different color to my grid my

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grid is black if that color was black

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then you will be able to see the grid

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then you're going to go to photoshop and

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then you're going to open the image of

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the Great Britain

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I want a white layer on the background

Box Counting

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so but before I do that I'm gonna get

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rid of all of the useless stuff around

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Great Britain okay now that we're done

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now we can put the grid on top and this

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grid is way too small so when you start

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out the boxes need to be really big and

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so this is gonna be for scale factor

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zero so it doesn't really matter how you

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place a grid you just got a place at

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somewhere and then once you've got that

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sorted what you're gonna do is you want

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to go into Excel and you want to set up

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a table so that you can record your data

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so the data I've got here is let me just

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make it bigger I've got the scale factor

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on one side and this is where I'm gonna

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write those numbers 0 2 4 8 here and

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then I've got log s I'm gonna log that

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later and then I'm gonna have the number

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of boxes what that is it's simply the

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amount of boxes that touch the coastline

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for each scale factor and then we're

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just gonna walk that so let's start off

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with the first one which is gonna be 0

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I'm going to see how many boxes are

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they're gonna touch what we're gonna do

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is we're just gonna count the number of

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boxes that touch the coastline and what

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you're gonna do is you're going to touch

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every box which touches the coastline

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and for every 10 bucks it you're gonna

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double click so you don't have to count

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all the way to that number in your head

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because trust me is hearing into the

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hundreds you need a way to keep track

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and so then you can just count tens when

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you go back so for example 1 2 3 4 5 6 7

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8 9 10 1 2 3 4 5 6 7 8 9 1 2 3 okay so I

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had to double tap so that's 20 + 6 26 so

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over here I'm gonna write 26 then what

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I'm gonna do is I'm gonna do what I said

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before is that I'm going to scale the

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boxes

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four boxes could fit into one original

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box I'm actually going to there's a few

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ways to do it but I'm just gonna draw a

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box here and I'm going to give it an

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outline and I'm gonna make that one fit

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I'm gonna make sure that doesn't have an

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outline okay so I don't know how to get

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those dots off so I'm gonna do is I'm

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gonna make a new grid donate the old

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grid just bring a new grid in and then

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scale this grid so that four squares

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gonna fit inside that black box okay

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there we go so now we've got a great and

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I'm just gonna do the same thing again

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with this scale factor so I'm gonna take

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a paintbrush

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okay so now we've scaled it down we buy

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a scale factor of two so now what we

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simply have to do is we need to count up

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to the boxes again so we do the same

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thing and in this case you didn't know

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why we DoubleTap because it's gonna get

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like a big heart so okay and then once

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we reach the end basically you know do

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you remember the last number baby

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founded which oh my god was like seven

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okay let's just say seven and then we're

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gonna count all the dark squares so one

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cuz I'm here one two three four five six

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seven six seven so 70 so 77 so then

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we're gonna note that down here so 77

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then the next scale factor because it's

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to the tonic - that means it's four so

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the next scale factor before and then

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what we're gonna do is we're gonna scale

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the grids down again in the similar way

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in which four boxes are now gonna be

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squished into one more box well I'm

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gonna do is going to take my rectangle

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one Square and now we've got us trying

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to fit four squares into that one and

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now I'm gonna try and squeeze that a lid

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into that little dot

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let's try okay there we go

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so there we go four boxes contained into

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that one grid so that we can use this

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bit but as you can see you may run into

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the problem there is no later that up

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big enough that's okay we just pop the

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command C command V and then we just

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drag that grid up here you know it's

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already the right scale factor now you

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just got to try and align it so there

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you go so I guess not I'm gonna make

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mine our brush size a bit smaller

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[Music]

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[Music]

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okay now I've done that I'm gonna count

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up all my dark circles one two three

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four five six seven eight nine ten

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eleven twelve thirteen then my last

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number I remember was four so 13 times

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ten hundred thirty so the number of

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boxes is 134 now I'm not going to do the

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last one because three is enough but you

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definitely should you can even do five

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if you want but this is enough to get

Graphing

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some good data okay so now that we've

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got our data what you could do is you

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could go and do another one so for

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example this one would be eight because

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it's two to the power of three cuz it's

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the third time you've done it and the

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next one would be 2 to the power 4 which

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would be 16 but I'm just gonna use 3 so

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I'm gonna do here is I'm gonna log as

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all of these numbers now you may be

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wondering oh my god wait what why do you

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can't log Sara right because log 0 like

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doesn't actually give you a solution so

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what the answer for this one is it's

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just 0 nothing too complicated about it

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you know log base 10 of 0 is not 0 for

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this case you just put it as 0 and then

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over here - we're gonna equals and all

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right log doesn't matter really what

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base it is as long as there as long as

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you're consistent log to 10 and then

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equals 4 and then you do the same over

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here

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you can just do this quickly with the

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shortcut but I'll just do each one so we

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log 77 okay

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so now we've got our numbers we've got

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our data now we can plot our graph and

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find the fractal dimension so what we're

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gonna do is we're gonna go to a chart

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we're gonna click scatter I'm gonna mark

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scatter and then I've got this chart

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here I'm gonna select data and then

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click add and then for the x value and

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gonna be the scaling factor so these

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ones and then for the Y values those of

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you these ones and once we've selected

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our data we're gonna press ok and we're

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gonna get a graph like this and that's

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exactly what we want so what we're gonna

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do now is we're going to put a while

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invest it in and then the gradient is

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actually gonna be the fractal dimension

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so we're gonna right-click or a double

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click add trendline and then click

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options display our value on chart why

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not display equation on chart and then

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you can see here we've got the equation

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y equals 120 180 - 8 X plus 1 point 145

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and based off our previous equations you

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should know that the gradient of this

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graph 1.18 is the fractal dimension and

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that makes sense because the fact of

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dimension is supposed to be between 1

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and 2 this is actually somewhat

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different to the actual value the actual

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value is 1.25 but they didn't do it so

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accurately and the box counting method

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isn't as accurate as other methods that

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they are so I hope that helps anyone

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who's trying to figure out how to do the

12:25

Box counting method to find the fractal

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dimension that's how I did it there's

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actually lots of different ways that you

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can do it that's just kind of a method

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that I made up on my own so if you have

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another method please feel free to share

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in the comments because also people

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always asking me how to do it and yeah

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this is just the method that I used

12:43

thanks for watching I'll see you later

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