FractalsUnit 6

Agenda/Topics to Be Covered

• Introduction

•Classification

• Fractal Dimension

• Fractal generation•Snowflake•Triadic curve•Hilbert curve

•Applications

• Summarize

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What are Fractals?

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• A Fractal is an geometrical object whose shape is irregular at all scales

• Fractals are formed by iterations

• Fractals are used to create complex objects

• Defined in terms of self similarity

• Conventional geometry concerns mostly with regular shapes and whole number dimensions•Example : Line, Cones

• Fractal geometry on the other hand deals with shapes found in nature that have non-integer, or fractal dimensions•Example : Cloud, River, Terrain ,Mountains etc

Examples of Fractals

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Classification

• Self Similar• fractals have parts that are scaled down versions of

the entire object, we construct the object subparts by applying a scaling parameter s to the overall shape. • Exact self similar: identical at different scales

• Quasi- self similar: approximately (but not exactly) identical at different scales.

• Statistical self similar: numerical or statistical measures which are preserved across scales.

• Self Affine• formed with different scaling parameters. -terrain,

water, clouds are typically modelled using this.

• Invariant Fractal Sets• this class of fractals includes self squaring fractals.

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Fractal Dimension

• Fractal has infinite detail and fractal dimension.

• A fractal imbedded in n-dimensional space could have any fractional dimension between O and N.

• The Fractal Dimension D= Log N / Log S • Where N is the number of smaller self-similar figures needed to create a larger

figure

• S is the Scaling Factor.

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Dimension of Square Dimension of CubeDimension of Line

Fractal Generation

Koch (Triadic) Curve

• Discovered in 1904 by Helge von Koch

• Start with straight line of length 1

• Recursively: Divide line into 3 equal parts

• Replace middle section with triangular bump, sides of length 1/3

• S = scale factor = 3

• N = number of copies of original = 4

• Dimension = log N/log S=log 4/log 3=1.261

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Snowflake• Can form Koch snowflake by joining

three Koch curves

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Hilbert’s Curve

• Discovered by German Scientist, David Hilbert in late 1900s

• Space filling curve

• Drawn by connecting centers of 4 sub‐squares, make up larger square.

• Iteration 0: 3 segments connect 4 centers in upside‐down U

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• Each of 4 squares divided into 4 more squares

• U shape shrunk to half its original size, copied into 4 sectors

• In top left, simply copied, top right: it's flipped vertically

• In the bottom left, rotated 90 degrees clockwise,

• Bottom right, rotated 90 degrees counter‐clockwise.

• pieces connected with 3 segments, each of which is same size as the shrunken pieces of the U shape (in red)

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• Each of the 16 squares from iteration 1 divided into 4 squares

• Shape from iteration 1 shrunk and copied.

• 3 connecting segments (shown in red) are added to complete the curve.

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At each subdivision the scale changes by 2 but length changes by 4 . Therefor the dimension is 2

Applications

• Fractals are used in many areas such as •Astronomy − For analyzing galaxies, rings of Saturn, etc.

•Biology/Chemistry − For depicting bacteria cultures, Chemical reactions, human anatomy, molecules, plants,

•Others − For depicting clouds, coastline and borderlines, data compression, diffusion, economy, fractal art, fractal music, landscapes, special effect, etc.

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Summarize

• Fractals are very complex pictures generated by a computer from a single formula. They are created using iterations.

• This means one formula is repeated with slightly different values over and over again, taking into account the results from the previous iteration.

• Questions• Write Short notes on Fractals

• Explain Fractal Dimension

• What are the different types of Fractals?

• Explain the following : Koch Curve(Triadic Curve), Snowflake, Hilbert’s Curve

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