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Koch Island (Fractals)

\(\color{red}{\text{Fractals}}\) can be defined as "objects or figures which on magnification shows the original object or figure"

The nature has many fractals such as a fern leaf, path of a river etc.

A fern leaf

A fern leaf

A river's path

A river's path

Geometrical fractals includes a straight line, today's talk \(\color{green}{\text{Koch Island}}\) and much more!

Koch Island

Koch Island

The Koch Island is drawn by "copies of copies" method.

The base of Koch Island is a square as shown:

Base of Koch Island

Base of Koch Island

The motif of Koch Island is

Motif of Koch Island

Motif of Koch Island

The steps of making a Koch Island are as follows:

  • Draw the base.

Drawing a base

Drawing a base

  • Replace the sides of base by the motif (of appropriate size)

Replacing sides

Replacing sides

  • Repeat the previous step again and again (By "again and again", I mean nothing less than ∞)

The figure after

  • one iteration

After one iteration

After one iteration

  • two iterations

After two iterations

After two iterations

\(\color{blue}{Secret:} \color{orange}{\text{Beauty} \propto \text{Number of iterations}}\)

So repeat it as much as you can!!!!

One exciting fact about the Koch Island is that it can tesselate a plane.

Tesselating a plane means that a plane can be covered up by infinite Koch Islands without overlapping or leaving spaces.

Tesselations

Tesselations

Question 1: Can anyone guess the area and perimeter of the Koch Island after n iterations?

Question 2: What is the minimun number of colors that you could use to color in your tessellation if no two adjoining Koch Islands are allowed to be the same color (no islands can touch even at 1 point)?

Question 3: What is the self-similarity dimension of Koch Island?

Question 4: Can you offer a MSW-Logo program to draw a Koch Island?

Image credit:infohost.nmt.edu
Image credit:miqel.com

Note by Pranjal Jain
2 years, 7 months ago

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  1. Area is constant; perimeter doubles per iteration.

  2. Four colours (I think).

Jake Lai · 2 years, 7 months ago

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@Jake Lai Both are correct! Try \(3^{rd}\) and \(4^{th}\) too! Pranjal Jain · 2 years, 7 months ago

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@Pranjal Jain what is a self-similarity dimension? Curtis Clement · 2 years, 6 months ago

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@Curtis Clement Self similarity Dimension is the property of a fractal which is defined as "index for characterizing fractal patterns or sets by quantifying their complexity as a ratio of the change in detail to the change in scale".

You might like to read more about them here Pranjal Jain · 2 years, 6 months ago

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@Curtis Clement Self similarity Dimension is the property of a fractal which is defined as "index for characterizing fractal patterns or sets by quantifying their complexity as a ratio of the change in detail to the change in scale".

.You might like to read more about them here Pranjal Jain · 2 years, 6 months ago

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self- similarity dimension is 2? Abhinav Raichur · 2 years, 7 months ago

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@Abhinav Raichur I dont have much idea about self-similarity dimensions. Just saw a glimpse at National Science Camp, Kolkata. Can you prove it? Pranjal Jain · 2 years, 7 months ago

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@Pranjal Jain usually the self similarity dimension is given by d=log(n)/log(k) . ...... where 'd' is the dimension, 'k' is the scaling factor(actually scaled as '1/k' but taken as k) and n is the number of pieces of fractals in the figure given, resembling the original one.... It is simply like someone askin ya "hey! how many pieces of the original should i have if i scale the fractal by a factor '1/3' given its self similarity dimension is 2" ........ answer is 9 pieces. ( * agreed that fractals are the most beautiful objects ...... but theyre really too complex and scary .... have you heard of the non integral dimensions for fractals !! * )

NOTE : scaling is taken as equivalent to performing iterations. Abhinav Raichur · 2 years, 7 months ago

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@Abhinav Raichur What would be "n" and "k" in the given fractal? Non integral dimensions?? Well no! Something new for me! Curious to know about that! Pranjal Jain · 2 years, 7 months ago

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@Pranjal Jain well there are many other beautiful facts .. lets take koch island as an example .... after infinite iterations it has finite area but infinite perimeter!! ..... I mean you can have different perimeter if you scale the fractal at different factors ....... to exclude this contradiction dimensions for fractals were redefined as the formula above read this for an elegant detail about non integral dimensions. Abhinav Raichur · 2 years, 7 months ago

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@Abhinav Raichur Thanks for the info! Pranjal Jain · 2 years, 7 months ago

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@Pranjal Jain I think it must be n=16, k=4

So \(D=\dfrac{log\ 16}{log\ 4}=\boxed{2}\) Pranjal Jain · 2 years, 7 months ago

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@Pranjal Jain yes ure right! Abhinav Raichur · 2 years, 7 months ago

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@Abhinav Raichur and yes! ..... i really loved your post, no one really speaks much about fractals. CAN YOU KEEP GOIN AND POST SOME MORE (thanks) :) Abhinav Raichur · 2 years, 7 months ago

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@Abhinav Raichur Thanks! :) I would keep on adding as I get some more info about fractals and Koch island!! Pranjal Jain · 2 years, 7 months ago

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