Fractals Alan CHEE Ka Ho Justina LAI Siu

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

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

Observation What are the similarities among these photos? ?

Characteristics: �Scaling ( <1) �Self-similar

Fractal is… �Latin: fractus – “broken” or “fractured” �Mathematically, based on an equation that undergoes iteration(重複) �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. "

Properties �Reduced scale �Self similar �Area? �Perimeter? �http: //fractalfoundation. org/resources/what-arefractals/

How to create fractals?

Fractals – Line Segment Generator �Koch curve Ø The middle one-third of the line segment is replaced by another two line segment formed as an equilateral triangle. Ø The triangles have length as the original one-third of the line segment.

Fractals – Line Segment Generator �Minkowski Sausage

Fractals – Line Segment Generator �Create your own Fractals ^^ �http: //www. dangries. com/Flash/Fractal. Maker. html

Math concepts

Calculation Time~ Sierpinski Gasket (Triangle) http: //www. csua. berkeley. edu/~raytrace/java/sierpinski/gasket. html

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

Rules: Sierpinski Triangle Keep repeating this process, we will eventually have something like this: http: //math. bu. edu/DYSYS/applets/fractalina. ht ml

Change in Area Originally: = S OR = =

Change in Area After 1 st iteration, After 2 nd iteration,

Change in Area After nth iteration, As the number of iteration tends to infinity, http: //www. csua. berkeley. edu/~raytrace/java/sierpinski/gasket. ht ml

Change in Perimeter Originally: S After 1 st iteration,

Change in Perimeter After 2 nd iteration, After 3 rd iteration,

Change in Perimeter After nth iteration, As the number of iteration tends to infinity, http: //www. csua. berkeley. edu/~raytrace/java/sierpinski/gasket. ht ml

More Examples �KOCH Snowflakes

More Examples Sierpinski Carpet http: //www. csua. berkeley. edu/~raytrace/java/sierpinski/carpet. html

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

Appreciation �Human body

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

Appreciation—Human Body

Appreciation—Human Body

Appreciation �Nature

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 God’s invisible qualities— his eternal power and divine nature—have been clearly seen, being understood from what has been made, so that people are without excuse. (Romans 1: 20)

Application

Application—Arts �http: //hk. myblog. yahoo. com/jw!vj. C_cr. Wa. Hw. NGi_. l. E a. At. KIIf. ZPg-/article? mid=2180 �http: //fractalfoundation. org/images/

Application--Music �Method of creating the self-similarity of fractals (LSystems): �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. )

Application--Music 1. C D Example: Start: C D Rules: C -> F E D D -> C E F F -> D E E -> F D 2. F E D C E F 3. D E F D C E F F E D F D D E …

Other Applications �Find out yourself and share in the online discussion~~

Extra and Homework

Extra(fun) �Fractal Maker: http: //www. dangries. com/Flash/Fractal. Maker. html �http: //fractalfoundation. org/OFC-index. htm �GIMP: http: //www. ehow. com/how_2196594_original-imagesusing-fractals. html

Homework 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 nthst. nd rd 1. Originally 1 2 3

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.

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

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

References �http: //fractalfoundation. org/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 -amazingexamples-of-fractals-in-nature/ �http: //www. miqel. com/fractals_math_patterns/visual -math-natural-fractals. html �http: //library. thinkquest. org/26242/full/

References �http: //hk. myblog. yahoo. com/jw!vj. C_cr. Wa. Hw. NGi_. l. E a. At. KIIf. ZPg-/article? mid=2180 �http: //kitoba. com/pedia/Fractal%20 Shell. html �http: //gut. bmj. com/content/57/7. cover-expansion �http: //www. homepages. ucl. ac. uk/~sjjgnle/ �http: //www. nutralegacy. com/blog/generalhealthcare/the-basics-of-the-circulatory-system/ �http: //nerdnirvana. org/wpcontent/uploads/2010/08/4391 -Amazing-Recursive. Painting. jpg

Thankyou