Fractals

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Fractals

• Alan CHEE Ka Ho
• Justina LAI Siu Kwan
• Gloria WONG Wing Yan

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Outline

• What fractals are
• Properties of fractals
• How to create fractals
• Math concepts
• Appreciations and applications

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Observation
What are the similarities among these photos??
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Characteristics

• Scaling ( lt1)
• Self-similar

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Fractal is

• Latin fractus broken or fractured
• Mathematically, based on an equation that
  undergoes iteration(??)
• A fractal is "a rough or fragmented geometric
  shape that can be split into parts, each of which
  is (at least approximately) a reduced-size copy
  of the whole."

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Properties

• Reduced scale
• Self similar
• Area?
• Perimeter?
• http//fractalfoundation.org/resources/what-are-fr
  actals/

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How to create fractals?
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Fractals Line Segment Generator

• Koch curve
• The middle one-third of the line segment is
  replaced by another two line segment formed as an
  equilateral triangle.
• The triangles have length as the original
  one-third of the line segment.

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Fractals Line Segment Generator

• Minkowski Sausage

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Fractals Line Segment Generator

• Create your own Fractals
• http//www.dangries.com/Flash/FractalMaker.html

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Math concepts
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Calculation Time

• Sierpinski Gasket (Triangle)

http//www.csua.berkeley.edu/raytrace/java/sierpi
nski/gasket.html
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Rules Sierpinski Triangle

• 1) Recognize the midpoints on each side of the
  equilateral triangles.
• 2) Connect the midpoints internally to form the
  next level's fractal
• 3) Remove the middle triangle

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Rules Sierpinski Triangle

• Keep repeating this process, we will eventually
  have something like this
• http//math.bu.edu/DYSYS/applets/fractalina.html

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Change in Area

• Originally

 
 

S
OR
 
 
 


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Change in Area

• After 1st iteration,
• After 2nd iteration,

 
 
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Change in Area

• After nth iteration,
• As the number of iteration tends to infinity,

 
 
http//www.csua.berkeley.edu/raytrace/java/sierpi
nski/gasket.html
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Change in Perimeter

• Originally
• After 1st iteration,

 
S
 
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Change in Perimeter

• After 2nd iteration,
• After 3rd iteration,

 
 
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Change in Perimeter

• After nth iteration,
• As the number of iteration tends to infinity,

 
 
http//www.csua.berkeley.edu/raytrace/java/sierpi
nski/gasket.html
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More Examples

• KOCH Snowflakes

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More Examples

• Sierpinski Carpet

http//www.csua.berkeley.edu/raytrace/java/sierpi
nski/carpet.html
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Conclusion about Fractal Properties

• 1) Scaling
• 2)Self-similarity
• 3) Area usually converges to certain value as
  iteration level increases
• 4) Perimeter usually increases even up to
  infinity as iteration level increases

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Appreciation

• Human body

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AppreciationHuman Body

• Aorta -gt Arteries -gt Arterioles -gt Capillaries
• Capillaries are 1 cell thick.
• Humans have about 150,000 km of blood vessels -
  enough to go around the world several times!

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AppreciationHuman Body
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AppreciationHuman Body
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Appreciation

• Nature

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Appreciation--nature

• Nature is somehow complicated and irregular for
  us to understand.
• However, after learning fractals, it seems that
  the nature is following some rules.

For since the creation of the world Gods
invisible qualities his eternal power and divine
naturehave been clearly seen, being understood
from what has been made, so that people are
without excuse. (Romans 120)
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Application
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ApplicationArts

• http//hk.myblog.yahoo.com/jw!vjC_crWaHwNGi_.lEaAt
  KIIfZPg-/article?mid2180
• http//fractalfoundation.org/images/

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Application--Music

• Method of creating the self-similarity of
  fractals (L-Systems)
• starting with a short string of symbols
• replacing the symbols with corresponding rules
  (The symbols are then interpreted as notes,
  chords, and several other things.)

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Application--Music

• C D
• F E D C E F
• D E F D C E F F E D F D D E

• Example
• Start C D
• Rules
• C -gt F E D
• D -gt C E F
• F -gt D E
• E -gt F D

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Other Applications

• Find out yourself and share in the online
  discussion

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Extra and Homework
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Extra(fun)

• Fractal Maker
• http//www.dangries.com/Flash/FractalMaker.html
  
• http//fractalfoundation.org/OFC/OFC-index.htm
• GIMP
• http//www.ehow.com/how_2196594_original-images
  -using-fractals.html

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Homework

1. Create a Fractal on your own and print it out.
   (You can use the links that shown in Extra(fun)
   slide.)
2. The following fractal is formed by reducing the
   unit square by the factor one-third and placing
   the images in form of a cross. Calculate the area
   for this Fractal for step nth.

2nd
1st
Originally
3rd
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Homework

• 3. Extra In Fractal geometry, it introduces
  that dimensions could be in fractions. Could
  you find out the fractal dimension for the Koch
  Snowflake? Explain your answer briefly if
  possible.

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Online discussion

• Search a Fractal Art in the internet and share
  the link with us.
• Search another Fractals application in the
  internet and explain a little bit with it.
• Can you think of other applications or topics
  involving scaling or self-similarities?

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Summary

• What fractals are
• Properties of fractals
• How to create fractals
• Math concepts
• Appreciations and applications

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References

• http//fractalfoundation.org/OFC/OFC-index.htm
• http//fractalfoundation.org/images/
• http//library.thinkquest.org/26242/full/
• http//www.tursiops.cc/fm/
• http//webecoist.com/2008/09/07/17-amazing-example
  s-of-fractals-in-nature/
• http//www.miqel.com/fractals_math_patterns/visual
  -math-natural-fractals.html
• http//library.thinkquest.org/26242/full/

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References

• http//hk.myblog.yahoo.com/jw!vjC_crWaHwNGi_.lEaAt
  KIIfZPg-/article?mid2180
• http//kitoba.com/pedia/Fractal20Shell.html
• http//gut.bmj.com/content/57/7.cover-expansion
• http//www.homepages.ucl.ac.uk/sjjgnle/
• http//www.nutralegacy.com/blog/general-healthcare
  /the-basics-of-the-circulatory-system/
• http//nerdnirvana.org/wp-content/uploads/2010/08/
  4391-Amazing-Recursive-Painting.jpg

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Thankyou