Fractal Cards
description
Transcript of Fractal Cards
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Fractal CardsFractal Cards
Mathematics & Science Center
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What is this?
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What is this?
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Fractals:
A fractalfractal is a mathematical object that exhibits self similarityself similarity — it looks the same at any scale.
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Benoit Mandelbrot
Professor of Mathematical SciencesYale University
Coined the Coined the word word “fractal” in “fractal” in 19751975
http://www-gap.dcs.st-and.ac.uk/~history/PictDisplay/Mandelbrot.html
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Fractal Sequence:
A fractal sequencefractal sequence
contains an infinite number of copies of itself, embedded within itself.
It is a pattern within a pattern within a pattern
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Clark Kimberling
Professor of Mathematics
University of Evansville
http://faculty.evansville.edu/ck6/
First wrote about First wrote about
Fractal Sequences Fractal Sequences
in 1995in 1995
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Fractal properties:
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IterationFractals are often formed by what is called an
iterative processto repeat exactly the same step over and over
Fractal properties:
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Step 0 Step 1
Step 2 Step 3
??
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Paper Dragon
Sometimes called: Jurassic Fractal
Stage 0: start with a segment
Let’s add a dot to our strip to insure that we all fold the same way !
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Paper Dragon
Keep the dot facing down
in your left hand.
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For Each New Stage:
• Bend each segment in half
• Open 2 new segments to 90
degree angle
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iteration
Paper Dragon: Stage 0: a strip of paperStage 1: right angle
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Paper DragonStage 2
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Paper Dragon
Stage 3
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Paper Dragon
Stage 4
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Paper Dragon
Stage 4Let’s look at more computer-generated iterations
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Paper Dragon
Stage 5
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Paper Dragon
Stage 6
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Paper Dragon
Stage 7
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Paper Dragon
Stage 8
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Paper Dragon
Stage 9
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Paper Dragon13 iterations
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Self Similarity
Similar: same shapeSelf similarity: same shape as
itself
Fractal properties:
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Fractal DimensionBased on 2 things:
Number of new pieces Ratio between lengths of
consecutive steps
Fractal properties:
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Recap: Fractal Characteristics Iteration: to repeat the same process over and over
Self Similarity: Within the whole you find smaller versions of
the same shape
Fractal Dimension: Non-integer values
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By: Corey Kisner and Cari FaszewskiBy: Corey Kisner and Cari FaszewskiMay 18, 2000May 18, 2000
Sierpinski Carpet
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Mathematics & Science Center http://astronomy.swin.edu.au/~pbourke/povray/povfrac/final/
Golden Sectionby Georg Mogk2004
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Eight Tangent Reflecting Spheresby Dave Matthews
2004
http://astronomy.swin.edu.au/~pbourke/povray/povfrac/final/
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Frost on my windshieldby Betsey Davis2001
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Tie-dyeby Betsey Davis2005
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Seven Zebra Polesby Betsey Davis2005
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Seven Yellow Feathersby Betsey Davis2005
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1 1 2 1 3 2 4 1 5 3 6 2 7 4 8 1 9 5 10 3
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
1 2 1 3 2 4 1 5 3 6 2 7 4 8 1 9 5 10
3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
2 1 3 2 4 1 5 3 6 2 7 4 8 1 9 5
4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
1 3 2 4 1 5 3 6 2 7 4 8 1 9
5 6 7 8 9 10 11 12 13 14 15 16 17
3 2 4 1 5 3 6 2 7 4 8 1
6 7 8 9 10 11 12 13 14 15 16
2 4 1 5 3 6 2 7 4 8
7 8 9 10 11 12 13 14 15
4 1 5 3 6 2 7 4
8 9 10 11 12 13 14
1 5 3 6 2 7
9 10 11 12 13
5 3 6 2
10 11 12
3 6
11
What patterns do you see ?
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1 1 2 1 3 2 4 1 5 3 6 2 7 4 8 1 9 5 10 3
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
1 2 1 3 2 4 1 5 3 6 2 7 4 8 1 9 5 10
3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
2 1 3 2 4 1 5 3 6 2 7 4 8 1 9 5
4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
1 3 2 4 1 5 3 6 2 7 4 8 1 9
5 6 7 8 9 10 11 12 13 14 15 16 17
3 2 4 1 5 3 6 2 7 4 8 1
6 7 8 9 10 11 12 13 14 15 16
2 4 1 5 3 6 2 7 4 8
7 8 9 10 11 12 13 14 15
4 1 5 3 6 2 7 4
8 9 10 11 12 13 14
1 5 3 6 2 7
9 10 11 12 13
5 3 6 2
10 11 12
3 6
11
•Cross out each number in the first line only the first time it appears
•On the line above the sequence, write the resulting sequence.
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Fractal Card
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Fractal Card
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Measure ¼ way¼ way along the fold from each end.
Cut perpendicular to the fold.
Each cut should be exactly ¼ of the length of the fold.
Fractal Card
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Fold the center portion forward and down, creasing it parallel to the fold.
Fractal Card
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Fold the center portion forward and down, creasing it parallel to the fold.
Fractal Card
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Fold the center portion forward and down, creasing it parallel to the fold.
Fractal Card
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Time for second iteration.
Fractal Card
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Measure ¼ way¼ way along the fold from each end.
Cut perpendicular to the fold
Each cut should be exactly ¼ of the length of the fold.
Fractal Card
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Fold the center portion forward and down, creasing it parallel to the fold.
Fractal Card
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Fractal Card
Fold the center portion forward and down, creasing it parallel to the fold.
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Continue for a total of 4 iterations
( 4 sets of cuts)
Fold the center portion forward and down, creasing it parallel to the fold.
Fractal Card
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Open carefully and re-fold steps
Fractal Card
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Refold and put glue ONLY on the flat part
Fractal Card
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Carefully slide into color paper
Fractal Card
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Close color paper to glue into place
Fractal Card
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Thank You !Thank You !