\(\color{red}{\text{Fractals}}\) 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.

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

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

The base of Koch Island is a square as shown:

The motif of Koch Island is

The steps of making a Koch Island are as follows:

- Draw the base.

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

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

The figure after

- one iteration

- 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.

**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?

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## Comments

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TopNewestFour colours (I think).

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Both are correct! Try \(3^{rd}\) and \(4^{th}\) too!

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what is a self-similarity dimension?

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"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

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"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

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self- similarity dimension is 2?

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I dont have much idea about self-similarity dimensions. Just saw a glimpse at National Science Camp, Kolkata. Can you prove it?

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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.

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read this for an elegant detail about non integral dimensions.

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 aboveLog in to reply

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So \(D=\dfrac{log\ 16}{log\ 4}=\boxed{2}\)

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