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MAADHYAM Nurturing Gifted Minds

Printed Under Gifted Education Abhiyaan

An Initiative By The Office Of Principal Scientific Advisor To The

Government Of India

MONTH: OCTOBER ISSUE NO:2016(14)

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Yes! I see patterns in

architecture, flowers,

rangoli, henna designs,

computer graphics, thunder

lightning etc.

But I am more fascinated

by the branching structure

of trees found in nature.

Have you noticed

any such instance

from your

surroundings?

Do you know the form of these trees resulted from experimenting with

FRACTAL branching patterns.

MONTH: OCTOBER ISSUE NO:2016(14)

INTRODUCTION TO FRACTALS

Do you ever want to

explore the process of

how it is made — who

decides to put it in a

place, who dreams it up,

what are the steps in

constructing it?

When you see a work of

art at a school or a

park or a public building,

you might appreciate its

forms or enjoy its

relationship to the

pattern found in nature.

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MONTH: OCTOBER ISSUE NO:2016(14)

A fractal is a never

ending pattern that

repeats itself at

different scales. This

property is called

“Self-Similarity.”

Fractals are

extremely complex,

sometimes infinitely

complex - meaning you

can zoom in and find

the same shapes

forever

Fractals are

extremely simple to

make

A fractal is made by

repeating a simple

figure again and again

FRACTALS???

WHAT ARE THEY?

WHERE DO WE

FIND FRACTALS?

Fractals are found in

all over nature

We follow the same

pattern again and again,

from tiny branching of

our blood vessels and

neurons to the

branching of trees,

lightning bolts and

river networks.

Purely geometric

fractals can be

made by repeating

a simple process.

INTRODUCTION TO FRACTALS

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MONTH: OCTOBER ISSUE NO:2016(14)

INTRODUCTION TO FRACTALS

Many people are fascinated by the beautiful images. Extending beyond the typical perception of mathematics as a body of complicated, boring formulas, fractal geometry mixes art with mathematics to demonstrate that equations which is more than just a collection of numbers. What makes fractals even more interesting is that they are the best existing mathematical descriptions of many natural forms, such as coastlines, mountains or parts of living organisms. Although fractal geometry is closely connected with computer techniques, some people had worked on fractals long before the invention of computers. Those people were British cartographers, who encountered the problem in measuring the length of Britain coast. The coastline measured on a large scale map was approximately half the length of coastline measured on a detailed map. The closer they looked, the more detailed and longer the coastline became. They did not realize that they had discovered one of the main properties of fractals. A fractal is a geometric pattern that repeats at every level of magnification. Another way to explain it might be to use Mandelbrot's own definition that "a fractal is a geometric shape that can be separated into parts, each of which is a reduced-scale version of the whole." Think of Russian nesting dolls. Fractals are common in nature and are found nearly everywhere. An example is broccoli. Every branch of broccoli looks just like its parent stalk. The surface of the lining of your lungs has a fractal pattern that allows for more oxygen to be absorbed. Such complex real-world processes can be expressed in equations through fractal geometry. Even to the everyday person, fractals are generally neat to look at even if you don't understand what a fractal is. But to a mathematician, it is a neat, neat subject area.

Why are fractals important? Fractals help us study and understand important scientific concepts, such as the way bacteria grow, patterns in freezing water (snowflakes) and brain waves. Their formulas have made possible many scientific breakthroughs. Wireless cell phone antennas use a fractal pattern to pick up the signals better, and pick up a wider range of signals, rather than a simple antenna. Anything with a rhythm or pattern has a chance of being very fractal-like. Read more at: http://phys.org/news/2011-10-beautiful-math-fractals.html#jCp

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MONTH: OCTOBER ISSUE NO:2016(14)

Regardless of scale, these patterns are all formed by repeating a simple branching process. A

fractal is a picture that tells the story of the process that created it.

Neurons from the human

cortex.

The similarity to a tree is

significant, as lungs and trees

both use their large surface

areas to exchange oxygen and

CO2.

“Lightning” formed by

rapidly discharging

electrons

The four images above are successive zooms into a detail. It starts with Two-fold symmetry branches

and becomes into 4-fold, which doubles into 8-fold, and then 16-fold. The branching process continues

forever, and the number of arms at any level is always a power of 2.

A spiral galaxy is the largest

natural spiral comprising

hundreds of billions of stars.

Oak tree, formed by a sprout

branching and then each of the

branches branching again and

again.

The plant kingdom is full of

spirals. An agave cactus forms

its spiral by growing new pieces

rotated by a fixed angle.

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MONTH: OCTOBER ISSUE NO:2016(14)

FRACTALS AND DIMENSIONS

Fractals are some of the most beautiful and most bizarre geometric shapes. They look the same at various different scales – you can take a small extract of the shape and it looks the same as the entire shape. This curious property is called self-similarity. To create a fractal, you can start with a simple pattern and repeat it at smaller scales, again and again, forever. In real life, of course, it is impossible to draw fractals with “infinitely small” patterns. However we can draw shapes which look just like fractals. Using mathematics, we can think about the properties a real fractal would have – and these are very surprising. Here you can see, step by step, how to create two famous fractals: the Sierpinski Gasket and the von Koch Snowflake.

The name “fractals” is derived from the fact that fractals don’t have a whole number dimension – they have a fractional dimension. Initially this may seem impossible – what do you mean by a dimension like 2.5 – but it becomes clear when we compare fractals with other shapes.

Source: http://world.mathigon.org/Fractals

Source: http://world.mathigon.org/Fractals

To create the Sierpinski Gasket, start with a triangle and repeatedly cut out the centre of every segment. Notice how, after a while, every smaller triangle looks exactly the same as the whole.

Source: http://world.mathigon.org/Fractals

To create the von Koch Snowflake you also start with a triangle and repeatedly add a smaller triangle to every segment of its edge. After a while, the edge looks exactly the same at small and large scales.

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FRACTALS PROPERTIES

Fractals’ properties Two of the most important properties of fractals are self-similarity and non-integer dimension. What does self-similarity mean? If you look carefully at a fern leaf, you will notice that every little leaf - part of the bigger one - has the same shape as the whole fern leaf. You can say that the fern leaf is self-similar. The same is with fractals: you can magnify them many times and after every step you will see the same shape, which is characteristic of that particular fractal. The non-integer dimension is more difficult to explain. Classical geometry deals with objects of integer dimensions: zero dimensional points, one dimensional lines and curves, two dimensional plane figures such as squares and circles, and three dimensional solids such as cubes and spheres. However, many natural phenomena are better described using a dimension between two whole numbers. So while a straight line has a dimension of one, a fractal curve will have a dimension between one and two, depending on how much space it takes up as it twists and curves. The more the flat fractal fills a plane, the closer it approaches two dimensions. Likewise, a "hilly fractal scene" will reach a dimension somewhere between two and three. So a fractal landscape made up of a large hill covered with tiny mounds would be close to the second dimension, while a rough surface composed of many medium-sized hills would be close to the third dimension.

Fractal patterns can also be found in commercially available antennas, produced for applications such as cellphones and wifi systems by companies such as Fractenna in the US and Fractus in Europe. The self-similar structure of fractal antennas gives them the ability to receive and transmit over a range of frequencies, allowing powerful antennas to be made more compact.

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MONTH: OCTOBER ISSUE NO:2016(14)

FRACTAL APPLICATIONS

Cables and Bridges Modern engineers still use this same idea today in the construction of high strength cables that make possible such masterpieces as the giant suspension bridges like the Golden Gate Bridge. A steel cable is formed from a bundle of smaller cables which themselves are formed of smaller bundles, etc. Cable technology is essential for building suspension bridges. The Inca weavers and the modern steel cable makers both use a repetitive process to create strong, fractal cable patterns.

Fractal Devices Engineers are using the ideas of fractal geometry in a variety of applications. Often we are faced with a task that is similar to something that nature has already found a solution for. The idea of deriving inspiration for human designs from the natural world is called "biomimicry".

we will examine some engineered fractals that solve the challenge of fluid transport by copying the fractal patterns of our blood vessels and lungs.

A fractal heat exchanger designed by Deb Pence at Oregon State University, and etched in silicon. Photo courtesy of Tanner Labs.

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As computers get smaller and faster, they generally produce more heat, which needs to be dissipated or else the computers will overheat and break. The smaller they are, the more this becomes a problem. Engineers at Oregon State University have developed fractal pattern that can be etched into a silicon chip to allow a cooling fluid (such as liquid nitroge) to uniformly flow across the surface of the chip and keep it cool. The fractal pattern above derived from our blood vessels provides a simple low-pressure network to accomplish this task easily.

An excerpt from a publication of Amalgamated Research Inc, showing some manufactured fractals used for fluid mixing. Image courtesy of Amalgamated Research Inc.

FRACTAL APPLICATIONS

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Maadhyam News Corner

MONTH: OCTOBER ISSUE NO:2016(14)

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RESPONSE SHEET

MONTH: OCTOBER ISSUE NO:2016(14)

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YOUR FEEDBACK

My Name: _______________________________________________

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PROJECT TEAM

Dr. Jyoti Sharma , Prof. Pankaj Tyagi , Prof. Shobha Bagai , Prof. Bibhu Biswal

Email id: giftededucationcic@gmail.com

RESEARCH TEAM

Geetu Sehgal , Shilpi Bariar, Savita Bansal, Uzma Masood , Jyoti Batra , Ritu Verma

MONTH: OCTOBER ISSUE NO:2016(14)