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

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

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

•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

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

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