fractals

Douglas reeves

Pythagoras Tree

• The Pythagoras tree is a plane fractal constructed from squares. It is named after Pythagoras because each triple of touching squares encloses a right triangle, in a configuration traditionally used to depict the Pythagorean theorem .

• it uses right triangles to decribe pythagreons therom witch is A squared plus B squared equals C squared

Mandelbrot set

• is a set of points in the complex plane, the boundary of which forms a fractal.

• When computed and graphed on the complex plane the Mandelbrot set is seen to have an elaborate boundary which does not simplify at any given magnification. This qualifies the boundary as a fractal.

Sierpinkis triangle

• it is a mathematically generated pattern that can be reproducible at any magnification or reduction.

• Comparing the Sierpinski triangle or the Sierpinski carpet to equivalent repetitive tiling arrangements, it is evident that similar structures can be built into any reptile arrangements.

• Also called Sierpinski gasket or the Sierpinski Sieve

Pascal's triangle

• Is a triangle that is formed with bionomical coefficients and this is done by starting and ending every line with 1.

• The triangle will have a set of color sequences so you can add to bigger numbers like if u add all the groups of green then u will get a number and if u add all the groups of red you will get another number.

Piano curves

• This is a curve that range that contains the entire 2 dimensional square unit or more represented by x dimension discovered by Giuseppe Piano that’s why its called Pianos curve

• this curve can transform simple fractals into complex fractals.

Plasma fractals

• Also known as Random Midpoint Displacement Fractals

• This is cause the fractal is be3cause the mid point is considered to be where the center of the coefficient corners meet.

The golden ratio

• This is a number often encountered when taking the ratios of distances in simple geometric figures such as the pentagon, pentagram, decagon and dodecahedron.

• This is considered to be the golden ratio 1.61803 39887 49894 84820 it is also the ratio for length to width of rectangles.

Star fractals

• The Star Fractal is formed by taking a 5-corner star and connecting stars that are 3 times smaller to every corner

• If you do the same to every one of the smaller stars and continue it infinitely, you will get a Star Fractal

• You can use different shapes to make star fractals also.

Snowflakes

• Made through design repeated and attached to each side of the original design or design before it.

• This design mostly uses base motif method.

Crystals

• Crystals in nature form a fractal naturally by growing in shapes and positions that tend to repeat in the crystal.

• This is very common in crystals though almost every crystal has a repeated pattern made out of shapes.

Quarternions • The quaternions are members of a

noncommutative division algebra first invented by William Rowan Hamilton. He spent years trying to find a three dimensional number systems, but with no success, but when he looked in 4 dimensions instead of 3 it worked.

• Quaternions form an interesting algebra where each object contains 4 scalar variables (sometimes known as Euler Parameters not to be confused with Euler angles), these objects can be added and multiplied as a single unit in a similar way to the usual algebra of numbers

Levy curves

• is a self-similar fractal that was first described and whose differentiability properties were analysed by Ernesto Cesàro in 1906 and in 1910, but now bears the name of French mathematician Paul Pierre Lévy, who was the first to describe its self-similarity properties

Cesaro’s sweep

• Cesaro’s Sweep is a base-motif fractal with the following simple base and motif:

• To form it using generator iteration, we take the base and substitute it with the motif. However, since it’s a sweep, we flip the motif over at the next iteration

Dragon fractal

• the fractal is Made with many smaller triangles that just keep getting infinitely smaller. the dragon fractal can go on for however long, getting smaller depending how big you want to make it.

• It has the basic base motif method in the design but it also keeps is.

Dragon

Polya’s sweep

• Polya’s Sweep is a base-motif fractal formed with a very simple base and motifTo form this fractal, you substitute the original line segment with the motif. In all of the next iterations, however, you alternate between the original and the flipped version of the motif, since the fractal is a sweep.

Resources

• http://www.abstractdigitalartgallery.com/silwenka-abstract-digital-art-fractal-Deep_Space.jpg

• http://media.photobucket.com/image/mandelbrot%20set%25255D/12_21_12/Nebulabrot-large.jpg

• http://www.sgeier.net/fractals/fractals/07/Kappa%20Space.jpg

• http://upload.wikimedia.org/wikipedia/commons/thumb/1/19/Pythagoras_tree.png/800px-Pythagoras_tree.png

• http://allencentre.wikispaces.com/file/view/fxdpre5.gif/30926867/fxdpre5.gif

• http://milan.milanovic.org/math/english/fibo/fibo4_files/catalan_58786_pascal-.jpg

More resources

• http://milan.milanovic.org/math/english/fibo/fibo4_files/catalan_58786_pascal-.jpg

• http://library.wolfram.com/images/infocenter/PeanoCurve794.gif

• http://images.clipartof.com/small/49694-Royalty-Free-RF-Clipart-Illustration-Of-A-Blue-And-Green-Liquid-Plasma-Fractal-Background.jpg

• http://cquestgarden.com/upload/3851/7286.jpg• http://fc05.deviantart.net/fs47/f/2009/201/1/c/Infinity_Star

_Fractal_Stockies_by_zananeichan.jpg

Even more Resources

• http://www.istockphoto.com/file_thumbview_approve/967799/2/istockphoto_967799-snowflakes-vector.jpg

• http://www.paxcam.com/imgs/library/18/crystals.jpg• http://www.referencesystemk.com/images/supporting-

art/Science%20Gallery/Fig147QS3-d1ib.jpg• http://upload.wikimedia.org/wikipedia/commons/thumb/

1/1e/Levy_C_Curve.svg/520px-Levy_C_Curve.svg.png• http://library.thinkquest.org/26242/full/types/images/

48.gif• http://library.thinkquest.org/26242/full/images/leaf.gif