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FractalsUnit 6
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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
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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
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Fractal Generation
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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
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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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