fractal cards

Fractal CardsFractal Cards

Mathematics & Science Center


What is this?


Fractals:

A fractalfractal is a mathematical object that exhibits self similarityself similarity — it looks the same at any scale.

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


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

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


Fractal properties:


IterationFractals are often formed by what is called an

iterative processto repeat exactly the same step over and over

Fractal properties:


Step 0 Step 1

Step 2 Step 3

??


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 !


Paper Dragon

Keep the dot facing down

in your left hand.


For Each New Stage:

• Bend each segment in half

• Open 2 new segments to 90

degree angle


iteration

Paper Dragon: Stage 0: a strip of paperStage 1: right angle


Paper DragonStage 2


Paper Dragon

Stage 3


Paper Dragon

Stage 4


Paper Dragon

Stage 4Let’s look at more computer-generated iterations


Paper Dragon

Stage 5


Paper Dragon

Stage 6


Paper Dragon

Stage 7


Paper Dragon

Stage 8


Paper Dragon

Stage 9


Paper Dragon13 iterations


Self Similarity

Similar: same shapeSelf similarity: same shape as

itself

Fractal properties:


Fractal DimensionBased on 2 things:

Number of new pieces Ratio between lengths of

consecutive steps

Fractal properties:


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


By: Corey Kisner and Cari FaszewskiBy: Corey Kisner and Cari FaszewskiMay 18, 2000May 18, 2000

Sierpinski Carpet

Mathematics & Science Center http://astronomy.swin.edu.au/~pbourke/povray/povfrac/final/

Golden Sectionby Georg Mogk2004


Eight Tangent Reflecting Spheresby Dave Matthews

2004

http://astronomy.swin.edu.au/~pbourke/povray/povfrac/final/


Frost on my windshieldby Betsey Davis2001


Tie-dyeby Betsey Davis2005


Seven Zebra Polesby Betsey Davis2005


Seven Yellow Feathersby Betsey Davis2005


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 ?


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.


Fractal Card


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


Fold the center portion forward and down, creasing it parallel to the fold.

Fractal Card


Time for second iteration.

Fractal Card


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



Fractal Card


Continue for a total of 4 iterations

( 4 sets of cuts)


Fractal Card


Open carefully and re-fold steps

Fractal Card


Refold and put glue ONLY on the flat part

Fractal Card


Carefully slide into color paper

Fractal Card


Close color paper to glue into place

Fractal Card


Thank You !Thank You !

fractal cards

Documents