Top Banner
This presentation provides a broad and basic introduction to the subject of fractal geometry. My thanks to Michael Frame at Yale University for the use of many of the photos and graphics that appear here. His fascinating and comprehensive treatment of the subject can be found at: http://classes.yale.edu/Fractals/ . Fractals: A Brief Overview Fractals: A Brief Overview Harlan J. Brothers, Director of Technology The Country School Madison, CT
20
Welcome message from author
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Page 1: Fractals

This presentation provides a broad and basic introduction to thesubject of fractal geometry.

My thanks to Michael Frame at Yale University for the use of many of the photos and graphics that appear here. His fascinating and comprehensive treatment of the subject can be found at:

http://classes.yale.edu/Fractals/ .

Fractals: A Brief OverviewFractals: A Brief OverviewHarlan J. Brothers, Director of TechnologyThe Country School Madison, CT

Page 2: Fractals

We commonly recognize when shapes demonstrate symmetry under the three familiar transformations of reflection, rotation, and translation.

Familiar SymmetriesFamiliar Symmetries

ReflectionReflection RotationRotation TranslationTranslation

Page 3: Fractals

Fractals demonstrate a fourth type of symmetry; they possess “self-similarity.”

Self-similar objects appear the same under magnification. They are, in some fashion, composed of smaller copies of themselves. This characteristic is often referred to as “scaling symmetry” or “scale invariance.”

Scaling SymmetryScaling Symmetry

Sierpinski Gasket

Page 4: Fractals

Not all self-similarity, however, is of a fractal nature. Objects like spirals and nested dolls that are self-similar around a single point are NOT fractal.

Scaling SymmetryScaling Symmetry

Not fractalNot fractal Not fractalNot fractal

Page 5: Fractals

In the broadest sense, fractals can be divided into two categories:

objects that occur in Nature, andmathematical constructions.

FractalsFractals

Page 6: Fractals

Natural objects exhibit scaling symmetry, but only over a limited range of scales. They also tend to be “roughly” self-similar, appearing more or less the same at different scales of measurement. Sometimes this means that they are statistically self-similar; that is to say, they have a distribution of elements that is similar under magnification.

Fractals in NatureFractals in Nature

Page 7: Fractals

Fractals in NatureFractals in Nature

Trees Ferns

Page 8: Fractals

Fractals in NatureFractals in Nature

Mountains

Page 9: Fractals

Fractals in NatureFractals in Nature

Coastline and snow fields of Norway Waterfall

Page 10: Fractals

Fractals in NatureFractals in Nature

Clouds

Page 11: Fractals

Fractals in NatureFractals in Nature

LighteningBacterial colony(courtesy E. Ben-Jacob)

Page 12: Fractals

In contrast to naturally occurring fractals, mathematical fractals can possess an infinite range of scaling symmetry. The more common constructions also tend to be exactly self-similar.

Mathematical ConstructionsMathematical Constructions

Koch Curve

The Koch curve above is composed of exactly four copies of itself. Can you construct it from just two?

Page 13: Fractals

Mathematical ExamplesMathematical Examples

Mandelbrot SetMandelbrot Set

Cantor CombCantor Comb

Menger SpongeMenger Sponge

Koch SnowflakeKoch Snowflake

Sierpinski GasketSierpinski Gasket

Page 14: Fractals

Scale InvarianceScale Invariance

The fact that a fractal object is, in some sense, composed of smaller copies of itself, has interesting implications. One of these is that when we examine a fractal shape without a suitableframe of reference, it is often impossible to tell the scale of magnification at which it is being viewed.

For natural phenomena, this translates to uncertainty with respect to the distance, extent, or size of the object. We end with two examples of scale invariance.

What do the following two images look like to you?

Page 15: Fractals

Image 1Image 1

Page 16: Fractals

Image 2Image 2

Page 17: Fractals

Image 1 Image 1 (another view)(another view)

Page 18: Fractals

Image 2 Image 2 (another view)(another view)

Page 19: Fractals

DiscussionDiscussion

Depending on who you ask, the preceding images may look like satellite or aerial photos, rock formations, or photomicrographs.

This simply illustrates the fact that certain natural processes, like erosion or the formation ice crystals, follow patterns that can be repeated at many scales of measurement. Without a frame of reference, a photograph of a rock sitting one meter away can effectively look the same as a boulder several meters away or a cliff hundreds of meters distant.

With a knowledgeable eye, one sees a natural world that abounds in fractal shapes.

Page 20: Fractals

Author InformationAuthor Information

Harlan J. BrothersDirector of TechnologyThe Country SchoolMadison, CT 06443Tel. (203) 421-3113E-mail: [email protected]