| Line 11: | Line 11: | ||
Now we will prove some interesting properties of C. | Now we will prove some interesting properties of C. | ||
| + | The term fractal is used to describe objects that cannot be effectively described using normal Euclidean geometry. | ||
| + | The word “fractal” is somewhat synonymous with the word “broken”. Below is an example of a well-known | ||
| + | fractal object called the Cantor Set. | ||
| + | The Cantor Set is created by the following algorithm: | ||
| + | Start with the closed interval [0,1]. Call this set ��, or the 0th (initial) set. | ||
| − | + | [[Image:eg2.jpg]] | |
[Category:MA375Spring2012Walther] | [Category:MA375Spring2012Walther] | ||
Revision as of 14:32, 25 April 2012
The Cantor set is a famous set first constructed by Georg Cantor in 1883. It is simply a subset of the interval [0, 1], but the set has some very interesting properties. We will first describe how to construct this set, and then prove some interesting properties of the set.
Continue in this way always removing the middle third of each segment to get A3,A4, . . ..
Note that A1 ⊇ A2 ⊇ A3 ⊇ · · · . And for each k ∈ N, Ak is the union of 2k closed intervals, each of length
3−k.
Let C = ∩1
i=1Ai. Then C is the Cantor set.
Now we will prove some interesting properties of C.
The term fractal is used to describe objects that cannot be effectively described using normal Euclidean geometry. The word “fractal” is somewhat synonymous with the word “broken”. Below is an example of a well-known fractal object called the Cantor Set. The Cantor Set is created by the following algorithm: Start with the closed interval [0,1]. Call this set ��, or the 0th (initial) set.
[Category:MA375Spring2012Walther]
