Generalized Cantor Sets and Their Hausdorff Dimensions

number of iterations

10

color:

green

type:

middle sevenths

show dimension

Hausdorff dimension ≈ 0.818068

To construct a generalized Cantor set iteratively, remove from the interval[0,1] a specified middle portion of every subinterval at each stage of the construction. This Demonstration runs up to 10 iterations of the Cantor set for five different middle interval variations. For each set, select "show dimension" to give the Hausdorff (or fractal) dimension

d

. This is defined by

d=-log2/logf

, where

f=

1

4

,

1

3

,

3

8

,

2

5

,

3

7

for middle halves, middle thirds, middle fourths, middle fifths and middle sevenths, respectively.

The original Cantor set was defined for

f=

1

3

and therefore had

d=log2/log3≈0.630930

.

External Links

Cantor Set (Wolfram MathWorld)

Cantor Set

The Sum of Two Cantor Sets

Sums of Generalized Cantor Sets

Permanent Citation

Erin K. Kline, Matthew A. Morena

"Generalized Cantor Sets and Their Hausdorff Dimensions"
http://demonstrations.wolfram.com/GeneralizedCantorSetsAndTheirHausdorffDimensions/
Wolfram Demonstrations Project
Published: November 30, 2021