ClassicalFractals.pdf

HistoricalFractals

Lecture notes for Access 2008

GeorgFerdinandLudwigPhilippCantor,

1845(Russia)-1918(Germany)

Georg Cantor is best known for the Cantor set. To construct the Cantor set, start with

the unit interval, delete the (open) middle 1 3 of this interval, and then successively delete the

open middle thirds from the remaining set, and so on. The points which are never deleted

in this process comprise the Cantor Set. A related function is the \Devil's Staircase" which

is continuous and has derivative equal to zero on 100% (but not all!) of it's domain - yet

it is not constant. Cantor loved set theory and the logical underpinnings of mathematics.

Hilbert described Cantor's work as \...the nest product of mathematical genius and one

of the supreme achievements of purely intellectual human activity."

http://www-history.mcs.st-and.ac.uk/Mathematicians/Cantor.html

Figure 1: Cantor's Set and the Devil's Staircase

How long is the Cantor set?

NielsFabianHelgevonKoch,

1870-1924(Sweden)

Koch is best known for the Koch snowake. It is constructed as follows: Begin with

the unit interval. Delete the middle open third and replace with two intervals of length 1 3 ,

forming two sides of an equilateral triangle. Iterate this process, and at each stage replace

each edge with 4 sub-edges with 1 3 the original edge length.

Take the limit of this process, and glue three of the limit pieces together in an equilateral

fashion to create the Koch snowake.

http://www-history.mcs.st-and.ac.uk/Mathematicians/Koch.html

How long isK?

WaclawSierpinski

1882-1969(Poland)

You construct the Sierpinski Triangle as follows: Begin with a closed equilateral tri-

angular region with side length 1. At stage 1, subdivide this triangular region into four

equilateral subregions, with new edges having half the previous side lengths. Remove the

open central sub-triangular region (keeping the edges). Continue this process inductively.

The Sierpinski Triangle is the collection of points from the original closed region which

never got deleted.

You can also make fractals in 3-dimensional space ...

The Menger Sponge

KarlMenger

1902(Austria)-1985(ChicagoUSA)

How much volume does the Menger Sponge have?

Scaling(Self-Similarity)Dimension

The concept of scaling dimension builds on the ideas we used to understand how area,

length, and volume of objects changed under uniform scaling of the background space.

There is a much more general notion of dimension, called \Hausdor dimension", which is

also much harder to explain.

Examples:

1. If you scale a line segment by an integer factor ofM, you getM 1 pieces which are

congruent to the original line segment.

2. If you rescale a square by an integer factor ofM, you getM 2 congruent pieces.

3. If you rescale a cube by a factor ofMyou getM 3 congruent pieces.

Scaling Dimension: If there is a uniform scaling factorMwhich transforms an object

intoN(non-overlapping) pieces congruent to the original object, then the solutiondto the

equationN=M d is the scaling dimension. Computationally,d= ln(N)

ln(M) . Note:MandN

could be fractions rather than integers.

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