 Hello friends, myself Darshan Pundit, Assistant Professor, Department of Computer Science and Engineering from Walchand Institute of Technology, Solapur. So, today we are going to see about Fractal Lines and Surfaces. So, at the end of this session student will be able to identify and distinguish various types of fractals. So, here first we are going to see fractal and fractal types that is exactly self-similarity fractal, Cauchy self-similarity fractal and statistically self-similarity fractal. So, fractal is discovered by mathematician Dr. Benoit Mandelbrot. The word fractal is derived from Latin word that is fractals which means broken. A fractal is a mathematical object that is both self-similar and chaotic. Self-similar is nothing but as you magnify you will see the object over and over again in its parts and chaotic is nothing but these fractals are infinitely complex. Fractals are very complex picture generated by a computer from a single formula and they are created using iterations. One formula is repeated with slight different values over and over again taking into account the result from previous iteration. Fractal is defined in mathematical limit of infinitely many iterations. So, the example of fractal you can see, so here see whenever you zoom this object you can see the same object is repeated. So you can see this part, so whenever you zoom this thing again you are going to see the same part. So fractal types the level of details remains same as we zoom in. So as we have seen in example surface roughness and profile will be same as we zoom in. So types of fractals that is exactly self-similarity, causes self-similarity and statistical self-similarity fractals. So let us see these types one by one. Exactly self-similar fractal is strongest type of fractal which appear identical at different scales. So these fractals are defined by iterated function system. So in this we are having initiator and generator. That is nothing but start with a given geometric shape and generator is a pattern which replaces the sub part of initiator. So as example we are having Serpinski triangle, coach snowflake and fernleaf etc. So Serpinski triangle is a fractal described in 1915 by Serpinski, Serpinski triangle starts as a shaded triangle of equal length and we split the triangle into four equal triangle by connecting the center of each side together and remove the central triangle. So we can then repeat this process on newly generated triangle that is three newly created smaller triangle. So let us see that, so this process is repeated several time on newly created smaller triangle to arrive the picture displayed. The Serpinski triangle is created infinitely repeating this construction process. So Serpinski triangle example, so this is initiator and generator and this is the iteration 1 of Serpinski triangle where we divide this triangle in equal halves and we remove the middle triangle. So this will create three new triangle and again on this part, again this part is divided in three parts, again this triangle is divided in three parts. So this process is repeated. So this is iteration 2 of Serpinski triangle again when we divide again this part we get iteration 3 of Serpinski triangle. So whenever we zoom this part, so we see the original triangle, see in this way we get the Serpinski triangle. So whenever you zoom this part, so whenever you zoom this part you get exactly same structure is repeated over the fractal. So this is exactly self similarity fractal. Remember that we are having coach snowflake triangle, so it is discovered by one coach in 1904. It start with straight line of length 1 and recursively divide it into three equal parts and replace middle section with triangular bump with sides of length 1 by 3. So now new length will be 4 by 3. So let us see the example. So this is the initiate on in coach snowflake after that generate on and each side is replaced by generator. So that we will get the first iteration. So stage 0 after that each side is replaced by generator. So this will be the stage 1 after that again each side is replaced by generator. So that you will get stage 2 that is coach star in stage 2 after that stage 3 and this process will be repeated and whenever you zoom this part you can find the initial stage. So last is fern leaf. So where you can see, so whenever you zoom this part, so you can find identical similar structure is repeated over a wide range of length scale. And these are the examples of exactly self similarity fractals. Next type is Cauchy self similarity. So this is a looser form of self similarity fractal which appears approximately identical at different scale. So in this we would not get exactly self similar but approximately it is identical at different scale. So this contains small copies of entire fractal in a distorted and degenerate form. So these are defined by recurrence relation example, Mandelbrot set, satellite approximation. So this is the Mandelbrot set. So in this we can see the Mandelbrot set. So whenever you zoom this thing you will find the approximate identical fractal. Next type is Statistically self similarity fractal. So this is weakest type of self similarity as fractal has numerical or statistical major which are preserved across scales. Self similarity fractals are example of fractals which are statistically self similar but neither exactly nor Cauchy self similar. So example you can have coastline, microscopic or root fractals. So this is the example of coastline. So where you can see statistical self similarity fractal, so in this fractal you cannot find exactly similar or approximate. So it is statistically self similar that is it is in a distorted form. So it is the microscopic self similarity fractal. So here we can pause the video and answer this question, distinguish between exactly self similarity, Cauchy self similarity and statistical self similarity fractals. So form of exactly self similarity fractal it is the strongest form of self similarity. Cauchy is loose form of self similarity and statistical is weakest form of self similarity. So scales, so in exactly self similarity fractals appear identical at different scale. So fractals appear approximately identical at different scale and fractals are measured numerical and statistical which preserved across the scale. Examples of self similarity is Serbinsky triangle, Cauchy-Noflec and Fernleaf. Statistically self similarity examples are Mandelbrotz and Satellite approximation and statistically self similarity fractals are microscopic coastline etc. So these are the references which I have used to create this video, thank you.