Fractal Project Mariellen Hemmerling. Fractals “A fractal is "a rough or fragmented geometric...

Fractal Project

Mariellen Hemmerling

Fractals• “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.” – Wikipedia

• “An irregular geometric object that is self-similar to its substructure at any level of refinement. The fractal dimension is a measure of the irregularity of the boundary of the object.” - amsglossary.allenpress.com/glossary/browse

– Meaning the fractal zoomed in is smaller size of the whole picture, no matter how many times you zoom in you will always get the same picture.

Sierpinski’s Triangle

• Discovered by Waclaw Sierpinski, in 1915.

• This fractal is a bunch of tiny triangles in one big triangle, which is how it makes it a fractal.

The Golden Ratio• This is divided so that the smaller part fits

in the slightly larger part etc.• This is found in many famous paintings,

and many buildings such as the Parthenon.

Pascal’s Triangle• “A geometric representation of the

binomial coefficients in a triangle”• The top numbers added together to make

the bottom numbers.

Pythagoras Tree• It’s a fractal constructed from squares.

• It was named after Pythagoras because each triple of touching squares makes a right triangle.

Lorenz Fractal• It’s named after Edward N. Lorenz.

• It’s a 3-D system that creates chaotic flow/ Unpredictable.

Van Koch• Founded by Helge Von Koch.• This is basically a snowflake, but the

repeat is the star points. This fractal is a curve.

IFS Fractal• A method of constructing fractals, that make a

whole fractal self similar. • In the picture below, the leaf has tiny leaves

making the big leaf.

Cesaro’s Sweep• This fractal has a “base” and a “motif”, when

together it makes an upright peano Curve. • So if you place the motif in different spots it will

create a sweep.

Levy Curve• This fractal is a self similar fractal, meaning the

smaller shapes make the whole fractal. • It makes a sort of boat like figure, with shapes

that are similar to a loop, that repeats itself to make a curve.

Dragon Fractal• This Fractal was discovered by William G.

Harter. • It’s a pretty cool fractal, and uses many cool

colors.• It resembles the Julia set, and in the Mandelbrot

set zoomed in it appeared.

Star fractal• This is made by a five cornered star, and

connecting smaller stars at each corner.• This fractal can be made of stars, triangles, or

squares.

Polya’s Sweep• This is very much alike with the Cesaro’s

Sweep, although it makes a different type of triangle.

• It has the base and motif.

Circle fractal• It closely resembles the Cantor set.• The Circle Fractal is basically one big

circle, with a bunch of smaller circles inside.

Mandelbrot Set• Named after Benoit Mandelbrot in 1978• It is similar to the Julia Set when modified.• It has a similar process of iteration

(reapeating itself when zoomed in)

Julia Set• When modified, it is found on the exterior

points of the Mandelbrot.

• The math controls the foundation of the map.

Nature

Software• http://www.ultrafractal.com/

• This software supports animation, and makes fractals.

Common Terms• Iteration: A repeating pattern.

• Trees: A fractal tree, make branches grow off of branches and repeat themselves.

• Base/Motif: Any shape that is composed as a line.

• Base: A line that is just straight

• Motif: Lines that switch directions that starts with a base.

Thoughts.• I think fractals are really cool, I never

thought art could be a mathematical thing.

• I love the colors, and the shapes and the fact that they repeat when zoomed in.

How this can help people• This can help people mathematically by

learning Trig, Algebra, and more mathematical terms.

• It can also help artists.

Research• Pythagoras Tree:

http://en.wikipedia.org/wiki/Pythagoras_treehttp://www.citizendia.org/Pythagoras_tree

• Sierpinski's Triangle: http://en.wikipedia.org/wiki/Sierpinski_triangle

 • Pascal’s Triangle:

http://en.wikipedia.org/wiki/Pascal's_triangle • The Golden Ratio:

http://en.wikipedia.org/wiki/Golden_ratio http://jwilson.coe.uga.edu/emt669/student.folders/

frietag.mark/homepage/goldenratio/goldenratio.html

• Star Fractal: http://library.thinkquest.org/26242/full/fm/fm34.html

 • Levy Curve: http://en.wikipedia.org/wiki/L

%C3%A9vy_C_curve http://www.nongnu.org/gsl-shell/examples.html• Cesaro’s Sweep:

http://library.thinkquest.org/26242/full/fm/fm6.html • Dragon Fractal:

http://en.wikipedia.org/wiki/Dragon_curvehttp://www.cs.unm.edu/~joel/PaperFoldingFractal/

construct.html

• Polya’s sweep: http://library.thinkquest.org/26242/full/fm/fm26.html

• Julia Set: http://en.wikipedia.org/wiki/Julia_set

• Lambda Fractal:http://bugman123.com/Fractals/Fractals2.html

 • Circle Fractal:

http://www.mymathforum.com/viewtopic.php?f=13&t=517&start=0

 http://classes.yale.edu/Fractals/CircInvFrac/welcome.html

• Lorenz Fractal: http://library.thinkquest.org/26242/full/fm/fm20.html

http://www.greatdreams.com/cosmic/101706.htm

• IFS fractal: http://en.wikipedia.org/wiki/Iterated_function_system

http://charels.vassallo.pagesperso-orange.fr/en/art/ifs.html

• Van Koch Fractal: http://en.wikipedia.org/wiki/Koch_snowflake

http://www.myspace.com/vankoch