FractalPacks-EducatorsGuide.pdf

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Fractal Pack 1
Educators’ Guide
Fractals Are SMART: Science, Math & Art!
www.FractalFoundation.org
All contents copyright 2009 FractalFoundation
Fractals Are SMART: Science, Math & Art!
www.FractalFoundation.org
All contents copyright 2009 FractalFoundation
1034264140.011.png
 
Contents
Introduction
3
Natural Fractals
4
Geometrical Fractals
6
Algebraic Fractals
7
Patterns and Symmetry
8
Ideas of Scale
10
Fractal Applications
11
Fulldome Animations:
12
Crystaloon
Galanga
Pleoria
Morphalingus
Featherino
Peacock
Geometric Fractals
Fractivities:
Sierpinski Triangle Construction 16
Explore fractals with XaoS
18
Appendix:
Math & Science Education
Standards met with fractals
19
Fractals Are SMART: Science, Math & Art!
www.FractalFoundation.org
All contents copyright 2009 FractalFoundation
1034264140.012.png
INTRODUCTION:
WHAT IS A FRACTAL
A fractal is a never ending pattern that repeats itself at different scales.
This property is called “Self-Similarity.”
Fractals are extremely complex, sometimes ininitely complex - meaning you
can zoom in and ind the same shapes forever.
Amazingly, fractals are extremely simple to make.
A fractal is made by repeating a simple process again and again.
WHERE DO WE FIND FRACTALS
In Geometry
In Nature
In Algebra
Fractals Are SMART: Science, Math & Art!
www.FractalFoundation.org
All contents copyright 2009 FractalFoundation
1034264140.013.png 1034264140.001.png 1034264140.002.png 1034264140.003.png 1034264140.004.png
NATURAL FRACTALS
BRANCHING
Fractals are found all over nature, spanning a huge range of scales. We ind
the same patterns again and again, from the tiny branching of our blood vessels and
neurons to the branching of trees, lightning bolts, and river networks. Regardless of
scale, these patterns are all formed by repeating a simple branching process.
A fractal is a picture that tells the story of the process that created it.
Neurons from the human cortex.
The branching of our brain cells
creates the incredibly complex
network that is responsible for all
we perceive, imagine, remember.
Scale = 100 microns = 10 - 4 m.
Our lungs are branching fractals with a sur-
face area ~100 m 2 . The similarity to a tree is
signiicant, as lungs and trees both use their
large surface areas to exchange oxygen and
CO 2 . Scale = 30 cm = 3*10 -1 m.
Lichtenberg “lightning”, formed by
rapidly discharging electrons in
lucite. Scale = 10 cm = 10 -1 m.
Oak tree, formed by a sprout
branching, and then each of the
branches branching again, etc.
Scale = 30 m = 3*10 1 m.
River network in China, formed by erosion
from repeated rainfall lowing downhill for
millions of years.
Scale = 300 km = 3*10 5 m.
Fractals Are SMART: Science, Math & Art!
www.FractalFoundation.org
All contents copyright 2009 FractalFoundation
1034264140.005.png 1034264140.006.png 1034264140.007.png
NATURAL FRACTALS
SPIRALS
The spiral is another extremely common fractal in nature, found over a
huge range of scales. Biological spirals are found in the plant and animal king-
doms, and non-living spirals are found in the turbulent swirling of luids and in the
pattern of star formation in galaxies.
All fractals are formed by simple repetition, and combining expansion and
rotation is enough to generate the ubiquitous spiral.
A spiral galaxy is the largest natural
spiral comprising hundreds of billions
of stars. Scale = 100,000 ly = ~10 20 m.
A fossilized ammonite from
300 million years ago. A
simple, primitive organism, it
built its spiral shell by adding
pieces that grow and twist at a
constant rate. Scale = 1 m.
A hurricane is a self-organizing spiral in
the atmosphere, driven by the evaporation
and condensation of sea water.
Scale = 500 km = 5*10 5 m.
The plant kingdom is full of spirals. An
agave cactus forms its spiral by growing
new pieces rotated by a ixed angle. Many
other plants form spirals in this way, includ-
ing sunlowers, pinecones, etc.
Scale = 50 cm = 5*10 -1 m.
The turbulent motion of luids cre-
ates spirals in systems ranging
from a soap ilm to the oceans,
atmosphere and the surface of jupi-
ter. Scale = 5 mm = 5*10 -3 m.
A iddlehead fern is a self-similar
plant that forms as a spiral of
spirals of spirals.
Scale = 5 cm = 5*10 -2 m.
Fractals Are SMART: Science, Math & Art!
www.FractalFoundation.org
All contents copyright 2009 FractalFoundation
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