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 5corner 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
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.
Polya’s sweep
• Polya’s Sweep is a base-motif fractal
formed with a very simple base and motif
To 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/silwenkaabstract-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%20Spac
e.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/30
926867/fxdpre5.gif
• http://milan.milanovic.org/math/english/fibo/fibo4_files/ca
talan_58786_pascal-.jpg
More resources
• http://milan.milanovic.org/math/english/fibo/fibo4_files/ca
talan_58786_pascal-.jpg
• http://library.wolfram.com/images/infocenter/PeanoCurve
794.gif
• http://images.clipartof.com/small/49694-Royalty-FreeRF-Clipart-Illustration-Of-A-Blue-And-Green-LiquidPlasma-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/967
799/2/istockphoto_967799-snowflakes-vector.jpg
• http://www.paxcam.com/imgs/library/18/crystals.jpg
• http://www.referencesystemk.com/images/supportingart/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.gi
f
• http://library.thinkquest.org/26242/full/images/leaf.gif