Slide 1

Download Report

Transcript Slide 1 

Fractal Cards
Mathematics & Science Center
What is this?
Mathematics & Science Center
What is this?
Mathematics & Science Center
Fractals:
 A fractal is a mathematical object
that exhibits self similarity — it
looks the same at any scale.
Mathematics & Science Center
Coined the
word
“fractal” in
1975
http://www-gap.dcs.st-and.ac.uk/~history/PictDisplay/Mandelbrot.html
Benoit Mandelbrot
Professor of Mathematical Sciences
Yale University
Fractal Sequence:
 A fractal sequence
contains an infinite number of copies of
itself, embedded within itself.
 It is a pattern within a pattern within a
pattern
Mathematics & Science Center
First wrote about
Fractal Sequences
in 1995
http://faculty.evansville.edu/ck6/
Clark Kimberling
Professor of Mathematics
University of Evansville
Fractal properties:
Mathematics & Science Center
Fractal properties:
Iteration
 Fractals are often formed by what
is called an iterative process
 to repeat exactly the same step
over and over
Mathematics & Science Center
Step 0
Step 1
Step 3
Step 2
?
Mathematics & Science Center
Paper Dragon
 Sometimes called Jurassic Fractal
Stage 0: Start with a segment
Mathematics & Science Center
Paper Dragon
 Sometimes called: Jurassic Fractal
Stage 0: start with a segment
Mathematics & Science Center
Paper Dragon
 Sometimes called: Jurassic Fractal
Stage 0: start with a segment
Mathematics & Science Center
Paper Dragon
Keep the dot facing down
in your left hand.
Mathematics & Science Center
For Each New Stage:
• Bend each segment in half
• Open 2 new segments to 90
degree angle
Mathematics & Science Center
Iteration
 Paper Dragon:
 Stage 0: a strip of paper
 Stage 1: right angle
Mathematics & Science Center
Paper Dragon
 Stage 2
Mathematics & Science Center
Paper Dragon
 Stage 3
Mathematics & Science Center
Paper Dragon
 Stage 4
Mathematics & Science Center
Paper Dragon
 Stage 4
Let’s look at more computergenerated iterations
Mathematics & Science Center
Paper Dragon
 Stage 5
Mathematics & Science Center
Paper Dragon
 Stage 6
Mathematics & Science Center
Paper Dragon
 Stage 7
Mathematics & Science Center
Paper Dragon
 Stage 8
Mathematics & Science Center
Paper Dragon
 Stage 9
Mathematics & Science Center
Paper Dragon
 13 iterations
Mathematics & Science Center
Fractal properties:
Self Similarity
 Similar: same shape
 Self similarity: same shape as
itself
Mathematics & Science Center
Fractal properties:
Fractal Dimension
 Based on 2 things:
 Number
of new pieces
 Ratio between lengths of
consecutive steps
Mathematics & Science Center
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
Mathematics & Science Center
By: Corey Kisner and Cari Faszewski
May 18, 2000
Sierpinski Carpet
Mathematics & Science Center
Golden Section
by Georg Mogk
2004
Mathematics & Science Center
http://astronomy.swin.edu.au/~pbourke/povray/povfrac/final/
Mathematics & Science Center
Eight Tangent
Reflecting
Spheres
by Dave
Matthews
2004
Mathematics & Science Center
http://astronomy.swin.edu.au/~pbourke/povray/povfrac/final/
Frost on my windshield
by Betsey Davis
2001
Mathematics & Science Center
Tie-dye
by Betsey Davis
2005
Mathematics & Science Center
Seven
Zebra
Poles
by
Betsey
Davis
2005
Mathematics & Science Center
Seven Yellow
Feathers
by Betsey Davis
2005
Mathematics & Science Center
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
What patterns do
you see ?
Mathematics & Science Center
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
•Cross out each 2
number in the
first line only the
first time it
appears
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
•On the line above the
sequence, write the
resulting sequence.
Mathematics & Science Center
Fractal Card
Mathematics & Science Center
Fractal Card
Mathematics & Science Center
Fractal Card
Measure ¼ way along the fold from each end.
Cut perpendicular to the fold.
Each cut should be exactly ¼ of the length of
the fold.
Mathematics & Science Center
Fractal Card
Fold the center portion forward and
down, creasing it parallel to the fold.
Mathematics & Science Center
Fractal Card
Fold the center portion forward and
down, creasing it parallel to the fold.
Mathematics & Science Center
Fractal Card
Fold the center portion forward and
down, creasing it parallel to the fold.
Mathematics & Science Center
Fractal Card
Time for second iteration.
Mathematics & Science Center
Fractal Card
Measure ¼ way along the fold from each end.
Cut perpendicular to the fold
Each cut should be exactly ¼ of the length of
the fold.
Mathematics & Science Center
Fractal Card
Fold the center portion forward and
down, creasing it parallel to the fold.
Mathematics & Science Center
Fractal Card
Fold the center portion forward and
down, creasing it parallel to the fold.
Mathematics & Science Center
Fractal Card
Fold the center portion forward and
down, creasing it parallel to the fold.
Continue for a total of 4 iterations
( 4 sets of cuts)
Mathematics & Science Center
Fractal Card
Open carefully and re-fold steps
Mathematics & Science Center
Fractal Card
Refold and put glue ONLY on the flat part
Mathematics & Science Center
Fractal Card
Carefully slide into color paper
Mathematics & Science Center
Fractal Card
Close color paper to glue into place
Mathematics & Science Center
Thank You !
Mathematics & Science Center