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Strategies and Rubrics for Teaching Chaos and Complex Systems Theories as Elaborating, Self-Organizing, and Fractionating Evolutionary Systems

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Title: Strategies and Rubrics for Teaching Chaos and Complex Systems Theories as Elaborating, Self-Organizing, and Fractionating Evolutionary Systems


1
Strategies and Rubrics for Teaching Chaos and
Complex Systems Theories as Elaborating,
Self-Organizing, and Fractionating Evolutionary
Systems
Fichter, Lynn S., Pyle, E.J., and Whitmeyer,
S.J., 2010, Journal of Geoscience Education (in
press)
2
Self-Similarity Fractals
3
Universality
Properties of Complex Evolutionary Systems
Fractal Organization - Xnext
patterns, within patterns, within patterns
Red box in 1 Stretched and Enlarged in 2
4
Universality
Properties of Complex Evolutionary Systems
Fractal Organization - Xnext
patterns, within patterns, within patterns
Red box in 2 Stretched and Enlarged in 3
5
Universality
Properties of Complex Evolutionary Systems
Fractal Organization - Xnext
patterns, within patterns, within patterns
Red box in 3 Stretched and Enlarged in 4
6
Universality
Properties of Complex Evolutionary Systems
Fractal Organization - Xnext
patterns, within patterns, within patterns
Red box in 4 Stretched and Enlarged here.
7
Universality
Properties of Complex Evolutionary Systems
Fractal Organization - Xnext
The closer we zoom in the more the detail we
see, and we see similar patterns repeated again
and again.
patterns, within patterns, within patterns
Red box in 4 Stretched and Enlarged here.
8
This is Self Similarity
Similarities at all scales of observation
Patterns, within patterns, within patterns
FRACTAL
9
Learning Outcomes
7. Self Similarity
Self-similarity is patterns, within patterns,
within patterns, so that you see complex detail
at all scales of observation, all generated by an
iterative process.
10
Euclidean and Fractal Geometry
Things that are fractal are characterized by two
distinctive characteristics
1. Non-whole Dimensions
Log N (number of similar pieces)
Fractal Dimension

Log M (magnification factor)
N M D
N of new pieces
M magnification
D dimension
Fractal dimensions are never whole numbers.
11
Euclidean and Fractal Geometry
Things that are fractal are characterized by two
distinctive characteristics
2. Generated by iteration
Fractal objects are generated by iteration of an
algorithm, or formula. The Koch Curve is an
example, generated by 4 steps, which are then
repeated-iterated -over and over indefinitely, or
as long as you want.
Koch Curve
First Iteration
1. Begin with a line
2. Divide line into thirds
3. Remove middle portion
4. Add two lines to form a triangle in
middle third of original line
Repeat Steps 1 - 4
12
Universality
Fractal Geometry
Koch Curve
13
Koch Curve Fractal Dimensions
2
3
1
4
1
2
3
Log 4
.602
Log N (number of new pieces)
D

Log M (Magnification factor of finer resolution)
Log 3
.477
Koch's Curve has a dimension of 1.2618595071429
14
Universality
Fractal Geometry in the The Mandelbrot Set
Geometrical Self Similarity
Mandelbrot Equation Z  Z 2  C C is a constant,
one point on the complex plain. Z starts out as
zero, but with each iteration a new Z forms
that is equal to the old Z squared plus the
constant C .
Take a point on the complex number plain, place
its value into the Mandelbrot equation and
iterate it 1000 times. If the number resulting
from the equation settles down to one value,
color the pixel black. If the number enlarges
towards infinity then color it something else,
say fast expanding numbers red, slightly slower
ones magenta, very slow ones blue, and so on.
Thus, if you have a sequence of pixels side by
side, of different colors, that means that each
of those values expanded toward infinity at a
different rate in the iterated equation. The
discs, swirls, bramble-like bushes, sprouts and
tendrils spiraling away from a central disc you
see are the results of calculating the Mandelbrot
set.
15
The Mandelbrot Set Cascade
16
The Mandelbrot Set Cascade
17
The Mandelbrot Set Cascade
18
The Mandelbrot Set Cascade
19
Universality
Properties of Complex Evolutionary Systems
Fractal Organization Dow Jones Average
patterns, within patterns, within patterns
20
What you can see and understand . . .
Depends on Your Scale of Observation
21
Fractal Temperature Patterns in Time
20,000 Year Record
1,000 Year Record
22
Fractal Temperature Patterns in Time
450,000 Year Record
20,000 Year Record
23
Universality
Properties of Complex Evolutionary Systems
Fractal Organization Drainage Patterns
patterns, within patterns, within patterns
Careful geologists always include a scale or
scale reference (a coin, a hammer, a camera lens
cap or a human) when taking a picture of geologic
interest. The reason is that if they didnt, the
actual size of the object pictured could not be
determined. This is because many natural forms,
such as coastlines, fault and joint systems,
folds, layering, topographic features, turbulent
water flows, drainage patterns, clouds, trees,
etc. look alike on many scales.
http//www.earthscape.org/t1/ems01/link03Txt-03.ht
ml
24
Universality
Properties of Complex Evolutionary Systems
Fractal Organization Sea Level Changes
25
Universality
Properties of Complex Evolutionary Systems
Fractal Organization Landscapes
patterns, within patterns, within patterns
26
Scale and Observation
What you can measure depends on the scale of your
ruler.
The time you can resolve depends on the accuracy
of your clock.
The size of what you can see depends on the power
of your measuring instrument microscopes for
small things, eyes, for intermediate things,
telescopes for very distant things.
The Earth events you can witness, or even the
human species can witness, depends on how long
you live.
There is no typical or average size for events.
27
How Long is the Coast of Great Britain?
It depends on the length of your ruler
The red ruler measures a longer coastline.
http//en.wikipedia.org/wiki/List_of_fractals_by_H
ausdorff_dimension
28
(No Transcript)
29
How Long is the Coast of Great Britain?
It depends on the length of your ruler
The coast line is actually infinitely long
Fractal Dimension 1.24
http//en.wikipedia.org/wiki/List_of_fractals_by_H
ausdorff_dimension
30
Euclidean and Fractal Geometry
Things that are fractal are characterized by two
distinctive characteristics
1. Non-whole Dimensions
N M D
Self similarity dimension
Number of smaller self similar objects generated
by the iterative process
Magnification factor number each new division
must be multiplied by to yield size of original
segment
Log N (number of new pieces)
D

Log M (Magnification factor of finer resolution)
How much we zoom in on or magnify each new piece
to view it the same size as the original.
31
Euclidean and Fractal Geometry
D
Log N (number of new pieces)

Log M (Magnification factor of finer resolution)
How much we zoom in on or magnify each new piece
to view it the same size as the original.
Original object
Divided into 3 new pieces N
a line
Magnification Factor 3
How much we have to magnify each piece to get
object of original size
N M D
3 3 1
1
Dimension
32
Euclidean and Fractal Geometry
D
Log N (number of new pieces)

Log M (Magnification factor of finer resolution)
How much we zoom in on or magnify each new piece
to view it the same size as the original.
Original object
Divided into 9 new pieces N
a square
Magnification Factor 9
How much we have to magnify each piece to get
object of original size
N M D
9 3 2
2
Dimension
33
Euclidean and Fractal Geometry
D
Log N (number of new pieces)

Log M (Magnification factor of finer resolution)
How much we zoom in on or magnify each new piece
to view it the same size as the original.
Original object
Divided into 27 new pieces N
Magnification Factor 27
a cube
How much we have to magnify each piece to get
object of original size
N M D
27 3 3
3
Dimension
34
Learning Outcomes
8. Fractal Geometry
There is no typical or average size of events, or
objects they come nested inside each other,
patterns within patterns within patterns, all
generated by an iterative process.
9. Non-whole Number Dimensions
Unlike Euclidian geometry (plane or solid
geometry) most natural objects have non-whole
number dimensions, something between, for
example, 2 and 3.
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