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Fractals
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Fractals
"Fractals are infinitely complex patterns that are selfsimilar across different scales. They are created by repeating a simple process over and over in an ongoing feedback loop. Driven by recursion, fractals are images of dynamic systems – the pictures of Chaos."
FractalFoundation.org
Fractals in Our Everyday Lives
Ferns
Mountain Ranges
Lungs
Peacocks
The Discovery of Fractals
FRACTALS
Snowflakes
Electricity
Romanescu
Clouds
And Many
More!
Sierpinski Triangle
Koch Snowflakes
Dragon Curve
Mandelbrot Set
Julia Set
Mandelbrot and Julia
What Are Fractals Useful For?
Rivers
“Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line." Benoit Mandelbrot in his book The Fractal Geometry of Nature.
n+1
Z
Weather Prediction
 By using fractals, meteorologists see patterns and are able to predict the weather more accurately.
The Universe
 Many Scientists believe the universe is a fractal.
Special Effects
 Helps special effect artists create reallife landscapes and characters.
Antennas
 Bending an antenna in small, complex shapes you will get more wire with less
area.
Digital Images
 Reduces pixelation.
Solar Power
 "Fractal trees" found in silver can help for a new kind of
solar cell.
Medicine
 Helping medics study the body and help with
cancer.
=Z
n
²
Galaxy
Hurricane
Animation
Antenna
Clear images
Silver
The Brain
Videos
Discovered by Gaston Maurice Julia.
Fractals were not really discovered, since they have been around in nature. In 1975, Benoît Mandelbrot, an IBM employee and math genius, truly found out what fractals were. Fractals were not called fractals before this discovery. Mandelbrot came up with the name by looking at a Latin dictionary. He found the word "fractus", which meant "broken". He thought this word fit it well, but changed it up a little and thus the word fractal was created and now the word is in every dictionary.
The Mandelbrot set was discovered by no other than Benoît Mandelbrot.
Discovered by Niels Fabian Helge von Koch.
Constructed by sectioning each side into thirds and drawing an equilateral tangle on each side.
Discovered by Wacław Sierpiński
Boundary of a Koch Snowflake
The Kotch Curve makes up the sides of a Koch Snowflake.
(Shown in
green
,
blue
, and
red
)
The Koch Snowflake can't extend beyond the circle, so it has a boundary.
The Koch Curve
Divide a line into three segments and draw a equilateral triangle
Repeat the same steps on all edges of the drawing.
You end up with a image like this. The Koch Curve is placed on all sides of an equilateral triangle to create the Koch Snowflake.
.
.
.
Types of Fractals
Mandelbrot Sets
Julia Sets
Dragon Curves
Sierpinski Triangles
Koch Curves
Binary Trees
Parts of the Julia Fractal Are in the Mandelbrot Fractal.
Peano Curves
Kleinian Group Fractals
Newton Method Fractals
Quaternion 3D Fractals
HyperComplex and
Three Dimensional Fractals
They are related!
Peano curve
Kleinian Group Fractal
Newton Method Fractal
Quaternion 3D Fractal
HyperComplex and Three Dimensional Fractal
Binary Trees
(2, 3i)
Others have worked with fractals before Mandelbrot
made his discovery and gave them a name
Z =Z ²+Z
[fraktl]
noun, Mathematics, Physics.
1.
a geometrical or physical structure having an irregular or fragmented shape at all scales of measurement between a greatest and smallest scale such that certain mathematical or physical properties of the structure, as the perimeter of a curve or the flow rate in a porous medium, behave as if the dimensions of the structure (fractal dimensions) are greater than the spatial dimensions.
fractal
n+1
n
0
Z
0
The Binary Tree is created making two branches off of each branch at the same angle as the first two branches. We make the length of the trunk 1 and the length of 0 < < 1. Also 0° < < 180°. This is for a symmetric binary tree.
1
r
r
2
r
r
r
r
2
2
2
Binary trees can keep going to infinity! Trees are fractals, as we can tell from this, but of course they can't go out to infinity. If they did we would have some pretty big trees!
Pick a value for Z . Let's choose (2+3i).
Using the equation Z = Z +Z
we plug in the value for Z .
n+1
n
²
0
0
Z = (2+3i)² + (2+3i)
1
* Because Z is our first point, we also use it for Z
0
n
Z = (4+12i9)+2+3i
Z = 3+15i
1
1
Your new point is (3, 15i)
θ
r

To find your next point you would plug in your value for Z for Z and keep using your previous answer until infinity.
1
n
For a dragon curve you draw a line, as shown, then rotate that 90 to get the red line in the second picture. You keep rotating what you got 90 to get your next picture and so on. The blue lines show what you rotated and the red lines show what you made by rotating the blue part 90 .
°
How to calculate it
.
.
°
°
.
.
.
Fun fact:
The dragon curve is featured in "Jurasstic Park" books. It shows different iterations after a few chapters.
The dragon curve was discovered by John E. Heighway, a physicist, by folding a piece of paper in the same direction multiple times and laying it out on it's edge making every fold 90 .
°
Equation:
Draw an equilateral triangle
.
Take the midpoint of each side, and draw a triangle within the original triangle
midpoints
midpoints
Use the equation Z = Z ²+ C
Ex) Z = 1+2i and c = 2+i
So, Z = (1+2i)²+(2+i)
Z = 1+4i4+2+i
Z = 1+5i
Then you repeat the process choosing different values for Z, keeping C constant.
+ C
Draw three more triangles ,inside each of the three you just created, using the midpoints of each side.
Continue adding triangles inside of triangles.
Constant
You end up with this.
.
.
.
.
*Z must be a complex number (contain i)
0
0
n+1
n
Z has to be a complex number (a+bi)
C is a constant that you chose
*there is an infinite amount of different Julia Fractals because there is are infinite values for C that you can choose from.
.
Z (1,2i)
0
Equation:
By Morgan & Rachel Flynn
Analytic Geometry 1st hourFull transcript
by
TweetMorgan Flynn
on 27 May 2015Transcript of Fractals
+C
Fractals
"Fractals are infinitely complex patterns that are selfsimilar across different scales. They are created by repeating a simple process over and over in an ongoing feedback loop. Driven by recursion, fractals are images of dynamic systems – the pictures of Chaos."
FractalFoundation.org
Fractals in Our Everyday Lives
Ferns
Mountain Ranges
Lungs
Peacocks
The Discovery of Fractals
FRACTALS
Snowflakes
Electricity
Romanescu
Clouds
And Many
More!
Sierpinski Triangle
Koch Snowflakes
Dragon Curve
Mandelbrot Set
Julia Set
Mandelbrot and Julia
What Are Fractals Useful For?
Rivers
“Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line." Benoit Mandelbrot in his book The Fractal Geometry of Nature.
n+1
Z
Weather Prediction
 By using fractals, meteorologists see patterns and are able to predict the weather more accurately.
The Universe
 Many Scientists believe the universe is a fractal.
Special Effects
 Helps special effect artists create reallife landscapes and characters.
Antennas
 Bending an antenna in small, complex shapes you will get more wire with less
area.
Digital Images
 Reduces pixelation.
Solar Power
 "Fractal trees" found in silver can help for a new kind of
solar cell.
Medicine
 Helping medics study the body and help with
cancer.
=Z
n
²
Galaxy
Hurricane
Animation
Antenna
Clear images
Silver
The Brain
Videos
Discovered by Gaston Maurice Julia.
Fractals were not really discovered, since they have been around in nature. In 1975, Benoît Mandelbrot, an IBM employee and math genius, truly found out what fractals were. Fractals were not called fractals before this discovery. Mandelbrot came up with the name by looking at a Latin dictionary. He found the word "fractus", which meant "broken". He thought this word fit it well, but changed it up a little and thus the word fractal was created and now the word is in every dictionary.
The Mandelbrot set was discovered by no other than Benoît Mandelbrot.
Discovered by Niels Fabian Helge von Koch.
Constructed by sectioning each side into thirds and drawing an equilateral tangle on each side.
Discovered by Wacław Sierpiński
Boundary of a Koch Snowflake
The Kotch Curve makes up the sides of a Koch Snowflake.
(Shown in
green
,
blue
, and
red
)
The Koch Snowflake can't extend beyond the circle, so it has a boundary.
The Koch Curve
Divide a line into three segments and draw a equilateral triangle
Repeat the same steps on all edges of the drawing.
You end up with a image like this. The Koch Curve is placed on all sides of an equilateral triangle to create the Koch Snowflake.
.
.
.
Types of Fractals
Mandelbrot Sets
Julia Sets
Dragon Curves
Sierpinski Triangles
Koch Curves
Binary Trees
Parts of the Julia Fractal Are in the Mandelbrot Fractal.
Peano Curves
Kleinian Group Fractals
Newton Method Fractals
Quaternion 3D Fractals
HyperComplex and
Three Dimensional Fractals
They are related!
Peano curve
Kleinian Group Fractal
Newton Method Fractal
Quaternion 3D Fractal
HyperComplex and Three Dimensional Fractal
Binary Trees
(2, 3i)
Others have worked with fractals before Mandelbrot
made his discovery and gave them a name
Z =Z ²+Z
[fraktl]
noun, Mathematics, Physics.
1.
a geometrical or physical structure having an irregular or fragmented shape at all scales of measurement between a greatest and smallest scale such that certain mathematical or physical properties of the structure, as the perimeter of a curve or the flow rate in a porous medium, behave as if the dimensions of the structure (fractal dimensions) are greater than the spatial dimensions.
fractal
n+1
n
0
Z
0
The Binary Tree is created making two branches off of each branch at the same angle as the first two branches. We make the length of the trunk 1 and the length of 0 < < 1. Also 0° < < 180°. This is for a symmetric binary tree.
1
r
r
2
r
r
r
r
2
2
2
Binary trees can keep going to infinity! Trees are fractals, as we can tell from this, but of course they can't go out to infinity. If they did we would have some pretty big trees!
Pick a value for Z . Let's choose (2+3i).
Using the equation Z = Z +Z
we plug in the value for Z .
n+1
n
²
0
0
Z = (2+3i)² + (2+3i)
1
* Because Z is our first point, we also use it for Z
0
n
Z = (4+12i9)+2+3i
Z = 3+15i
1
1
Your new point is (3, 15i)
θ
r

To find your next point you would plug in your value for Z for Z and keep using your previous answer until infinity.
1
n
For a dragon curve you draw a line, as shown, then rotate that 90 to get the red line in the second picture. You keep rotating what you got 90 to get your next picture and so on. The blue lines show what you rotated and the red lines show what you made by rotating the blue part 90 .
°
How to calculate it
.
.
°
°
.
.
.
Fun fact:
The dragon curve is featured in "Jurasstic Park" books. It shows different iterations after a few chapters.
The dragon curve was discovered by John E. Heighway, a physicist, by folding a piece of paper in the same direction multiple times and laying it out on it's edge making every fold 90 .
°
Equation:
Draw an equilateral triangle
.
Take the midpoint of each side, and draw a triangle within the original triangle
midpoints
midpoints
Use the equation Z = Z ²+ C
Ex) Z = 1+2i and c = 2+i
So, Z = (1+2i)²+(2+i)
Z = 1+4i4+2+i
Z = 1+5i
Then you repeat the process choosing different values for Z, keeping C constant.
+ C
Draw three more triangles ,inside each of the three you just created, using the midpoints of each side.
Continue adding triangles inside of triangles.
Constant
You end up with this.
.
.
.
.
*Z must be a complex number (contain i)
0
0
n+1
n
Z has to be a complex number (a+bi)
C is a constant that you chose
*there is an infinite amount of different Julia Fractals because there is are infinite values for C that you can choose from.
.
Z (1,2i)
0
Equation:
By Morgan & Rachel Flynn
Analytic Geometry 1st hour