Fractals of rational dimension and fractals of integral powers

In summary, the conversation discussed generalizations of fractals with nonzero rational dimensions, as well as the comparison of fractals with non-integral and integral dimensions. The concept of raising a fractal space to the Nth power to obtain a whole number dimensional space was also mentioned. The example of the Serpinski gasket was brought up, with a disagreement on its exact fractal dimension being either two or 1.58.
  • #1
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What generalizations can be made concerning fractals of nonzero rational dimensions M/N (where M and N are nonzero integers)?

How does a fractal of non-integral dimension F compare geometrically to a fractal of dimension GF, where G is a nonzero integer?
 
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  • #2
In the first paragraph, I was concerned with a fractal space that could be raised to the integer Nth power to obtain a whole number dimensional space.

Similarly, in the second paragraph, I wondered about the geometry of an "axis" of fractal dimension F extended to G axes to produce a GF fractal dimensional space, or moreso, comparing the geometry of int[GF] dimensional spaces of int[GF] axes for G=1, 2, 3... .
 
  • #3
Believe the Serpinski gasket has a fractal dimension of exactly two.
 
  • #4
Originally posted by Ben-CS
Believe the Serpinski gasket has a fractal dimension of exactly two.

no - triangle has a dimension of 1.58. Carpet has a dimension of 1.89.

If its dimension was 2 it wouln't be a fractal.

Cheers,

ron.

Can't help with the earlier Q.
 

1. What is the difference between fractals of rational dimension and fractals of integral powers?

Fractals of rational dimension have a fractal dimension that is expressed as a fraction, while fractals of integral powers have a fractal dimension that is expressed as a whole number.

2. How are fractal dimensions calculated for these types of fractals?

The fractal dimension for both fractals of rational dimension and fractals of integral powers can be calculated using a variety of methods, such as the box counting method or the Hausdorff dimension formula.

3. Can fractals of rational dimension and fractals of integral powers be found in nature?

Yes, both types of fractals can be found in nature, as many natural phenomena exhibit self-similar and recursive patterns that can be described using fractal geometry.

4. How are fractals of rational dimension and fractals of integral powers used in real-world applications?

Fractals of rational dimension and fractals of integral powers have various practical applications, including in image compression, computer graphics, and modeling complex systems in fields such as physics, biology, and economics.

5. Are there any famous examples of fractals of rational dimension and fractals of integral powers?

Yes, there are many famous examples, such as the Mandelbrot set, the Koch snowflake, and the Sierpinski triangle, which all exhibit self-similar and recursive patterns that can be described using fractal geometry.

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