Is It Live, or Is It Fractal?

Bergren ForumSeptember 3, 2009

Addison Frey, Presenter

Chaos Under Control: The Art and Science of Complexity

David Peak, Physics Department, Utah State University, Logan, UT 84322-4415(PeakD@cc.usu.edu)andMichael Frame, Mathematics Department, Union College, Schenectady, NY 12308(FrameM@union.edu)

W.H. Freeman, Publishers1994ISBN 0-7167-2429-4

What Is a Fractal?

• Many fractals are “self-similar” (or nearly so).– But what is “self-similar”?

What Is a Fractal?• Many fractals are “self-similar” (or nearly so).

– But what is “self-similar”?

A self-similar object is exactly or approximately similar to a part of itself (i.e. the whole has the same shape as one or more of the parts).

What Is a Fractal?

• Many fractals are “self-similar” (or nearly so).– But what is “self-similar”?– Example:

What Is a Fractal?

• Many fractals are “self-similar” (or nearly so).– But what is “self-similar”?– Example:

What Is a Fractal?

• Many fractals are “self-similar” (or nearly so).– But what is “self-similar”?– Example:

But a square is not a fractal!

What Is a Fractal?

• Many fractals are “self-similar” (or nearly so)• A fractal has a fine structure at arbitrarily

small scales.

What Is a Fractal?• Many fractals are “self-similar” (or nearly so)• A fractal has a fine structure at arbitrarily small

scales.– Example:

What Is a Fractal?• Many fractals are “self-similar” (or nearly so)• A fractal has a fine structure at arbitrarily small

scales.– Example:

What Is a Fractal?

• Many fractals are “self-similar” (or nearly so)• A fractal has a fine structure at arbitrarily small scales.• A fractal has a Hausdorff dimension greater than its

Euclidean dimension

What Is Hausdorff Dimension?

Example:

What Is Hausdorff Dimension?

Example:

What Is Hausdorff Dimension?

Example:

What Is Hausdorff Dimension?

Example:

What Is Hausdorff Dimension?

Example:

What Is Hausdorff Dimension?

Example:

What Is Hausdorff Dimension?

Example:

What Is Hausdorff Dimension?

Example:

What Is Hausdorff Dimension?

Example:

WARNING!

WARNING!

LOGARITHMS AHEAD!

What is the Hausdorff dimension of the Sierpinski carpet?

What is the Hausdorff dimension of the Sierpinski carpet?

What is the Hausdorff dimension of the Sierpinski carpet?

Fractals in Nature

Fractals in Nature

Fractals in Nature

Canacadea Creek, Alfred

James Cahill

Process

• Box Counting Method– Place object in a single large box that leaves little

to no extra space on the ends.– Apply a grid with smaller boxes over object, and

count the number of boxes the object is in.– Repeat the second step with increasingly

increasingly smaller boxes, and least twice more.

James Cahill

The Math

• The number of boxes = N• The scale factor (s) = Big box / Little box• Plot graph with log(N) on the y-axis and log(s)

on the x-axis.• Create a best-fit line of the points. The slope

of that line is the dimension of the object.

James Cahill

N=1

S=1

N=17

S=11.02

N=38

S=20.84

James Cahill

N=116S=60.8

James Cahill

The Creek is 1.166 Dimensional

James Cahill

• The box counting method is a slow, painstaking, but all together fairly accurate way to find the dimensions of natural objects. The idea behind it is that we take and average of the smaller and larger N values, and hope that it smoothes out any wrinkles in the results. Unlike mathematical fractals like the Sierpinski Gasket, our rivers and cracks lose definition at high magnification, so there comes a point when smaller S values are completely pointless, as the boxes are smaller than the thickness of the line we are examining.

Concludatory

James Cahill

What Is a Fractal?• Many fractals are “self-similar” (or nearly so)• It has a fine structure at arbitrarily small scales.• It has a Hausdorff dimension greater than its Euclidean dimension• It has a simple and recursive definition.

What Is a Fractal?• Many fractals are “self-similar” (or nearly so)• It has a fine structure at arbitrarily small scales.• It has a Hausdorff dimension greater than its Euclidean dimension• It has a simple and recursive definition.

Example (A Deterministic Approach):

Generating the Sierpinski Gasket (Deterministic Approach)

Generating the Sierpinski Gasket (Deterministic Approach)

Generating the Sierpinski Gasket (Deterministic Approach)

Generating the Sierpinski Gasket (Deterministic Approach)

Generating the Sierpinski Gasket (Deterministic Approach)

Generating the Sierpinski Gasket (Deterministic Approach)

Generating the Sierpinski Gasket (Deterministic Approach)

Generating the Sierpinski Gasket (Deterministic Approach)

Generating the Sierpinski Gasket (Deterministic Approach)

Generating the Sierpinski Gasket(Random Approach)

0.Start with the point 1.Randomly choose one of the following:

a.b.c.

2.Plot3.Let 4.Go back to Step 1

Generating the Sierpinski Gasket(Random Approach)

Generating the Sierpinski Gasket(Random Approach)

Generating the Sierpinski Gasket(Random Approach)

Generating the Sierpinski Gasket(Random Approach)

Generating the Sierpinski Gasket(Random Approach)

A Maple Leaf

A Tree

Jarrett Lingenfelter

A Tree

Jarrett Lingenfelter

A Tree

Jarrett Lingenfelter

A Tree

Jarrett Lingenfelter

A Tree

Jarrett Lingenfelter

A Tree

Jarrett Lingenfelter

A Tree

Jarrett Lingenfelter

Another Tree

Another Tree

Another Tree

WARNING!

WARNING!

TRIGONOMETRY AHEAD!

Another Tree(Random Approach)

0. Start with the point 1. Randomly choose one of the following:

a.b.c.d.e.f.

2. Plot3. Let 4. Go back to Step 1

Another Tree(Random Approach)

Another Tree(Random Approach)

Another Tree(Random Approach)

Another Tree(Random Approach)

Another Tree(Random Approach)

Another Tree(Second Attempt)

Another Tree(Second Attempt)

Another Tree(Second Attempt)

Another Tree(Second Attempt)

Another Tree(Second Attempt)

Another Tree(With Some Fruit)

                                                

A fractal landscape created by Professor Ken Musgrave (Copyright: Ken Musgrave).

A fractal planet.

Is It Live, or Is It Fractal?