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=> Apollonian Gasket
=> Barnsley's Fern
=> Barnsley's Tree
=> Batrachion
=> Blancmange Function
=> Box Fractal
=> Brown Function
=> Cactus Fractal
=> Cantor Dust
=> Cantor Function
=> Cantor Set
=> Cantor Square Fractal
=> Capacity Dimension
=> Carotid-Kundalini Fractal
=> Cesàro Fractal
=> Chaos Game
=> Circles-and-Squares Fractal
=> Coastline Paradox
=> Correlation Exponent
=> Count
=> Cross-Stitch Curve
=> Curlicue Fractal
=> Delannoy Number
=> Dendrite Fractal
=> Devil's Staircase
=> Douady's Rabbit Fractal
=> Dragon Curve
=> Elephant Valley
=> Exterior Snowflake
=> Gosper Island
=> H-Fractal
=> Haferman Carpet
=> Hénon Map
=> Hilbert Curve
=> Householder's Method
=> Ice Fractal
=> Julia Set
=> Koch Antisnowflake
=> Koch Snowflake
=> Lévy Fractal
=> Lévy Tapestry
=> Lindenmayer System
=> Mandelbrot Set
=> Mandelbrot Set Lemniscate
=> Mandelbrot Tree
=> Menger Sponge
=> Minkowski Sausage
=> Mira Fractal
=> Newton's Method
=> Peano Curve
=> Peano-Gosper Curve
=> Pentaflake
=> Plane-Filling Function
=> Pythagoras Tree
=> Randelbrot Set
=> Rep-Tile
=> Reverend Back's Abbey Floor
=> San Marco Fractal
=> Sea Horse Valley
=> Siegel Disk Fractal
=> Sierpiński Arrowhead Curve
=> Sierpiński Carpet
=> Sierpiński Curve
=> Sierpiński Sieve
=> Star Fractal
=> Strange Attractor
=> Tetrix
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Ziyaretçi defteri
 

Cantor Set

 
CantorSet

The Cantor set T_infty, sometimes also called the Cantor comb or no middle third set (Cullen 1968, pp. 78-81), is given by taking the interval [0,1] (set T_0), removing the open middle third (T_1), removing the middle third of each of the two remaining pieces (T_2), and continuing this procedure ad infinitum. It is therefore the set of points in the interval [0,1] whose ternary expansions do not contain 1, illustrated above.

Iterating the process 1 -> 101, 0 -> 000 starting with 1 gives the sequence 1, 101, 101000101, 101000101000000000101000101, .... The sequence of binary bits thus produced is therefore 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, ... (Sloane's A088917) whose nth term is amazingly given by D(n,n)=P_n(3) (mod 3), where D(n,n) is a (central) Delannoy number and P_n(x) is a Legendre polynomial (E. W. Weisstein, Apr. 9, 2006). The recurrence plot for this sequence is illustrated above.

This produces the set of real numbers {x} such that

 x=(c_1)/3+...+(c_n)/(3^n)+...,
(1)

where c_n may equal 0 or 2 for each n. This is an infinite, perfect set. The total length of the line segments in the nth iteration is

 l_n=(2/3)^n,
(2)

and the number of line segments is N_n=2^n, so the length of each element is

 epsilon_n=l/N=(1/3)^n
(3)

and the capacity dimension is

d_(cap) = -lim_(epsilon->0^+)(lnN)/(lnepsilon)
(4)
= log_32
(5)
= (ln2)/(ln3)
(6)
= 0.630929...
(7)

(Sloane's A102525). The Cantor set is nowhere dense, and has Lebesgue measure 0.

A general Cantor set is a closed set consisting entirely of boundary points. Such sets are uncountable and may have 0 or positive Lebesgue measure. The Cantor set is the only totally disconnected, perfect, compact metric space up to a homeomorphism (Willard 1970).


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