Enter the number of miniature pieces in the final figure and the scaling factor into the Calculator. The calculator will evaluate the Fractal Dimension. 

Fractal Dimension Calculator

Self-Similar
Box-Counting

Enter any 2 values to calculate the missing variable

In the self-similar formula, S is the magnification factor: each miniature piece has linear size 1/S of the original.

Fractal Dimension Formula

D = log (N) / log (S)

Variables:

  • D is the fractal dimension (dimensionless)
  • N is the number of self-similar miniature pieces in the final figure
  • S is the scaling (magnification) factor, where each piece is scaled to a linear size of 1/S of the original (typically S > 1)

To calculate the self-similar fractal dimension, divide log(N) by log(S). Any consistent logarithm base may be used.

How to Calculate Fractal Dimension?

The following steps outline how to calculate the Fractal Dimension.

  1. First, determine the number of miniature pieces in the final figure. 
  2. Next, determine the scaling factor. 
  3. Next, gather the formula from above = D = log (N) / log (S).
  4. Finally, calculate the Fractal Dimension.
  5. After inserting the variables and calculating the result, check your answer with the calculator above.

Example Problem : 

Use the following variables as an example problem to test your knowledge. (Result: D = log(3)/log(2) ≈ 1.5850.)

number of miniature pieces in the final figure = 3

scaling factor = 2

FAQ

What is a Fractal Dimension?

Fractal Dimension is a (generally non-integer) scalar value that gives an indication of how completely a fractal appears to fill space as one zooms down into finer scales. It provides a quantitative measure of the complexity of a fractal shape.

Why is the Fractal Dimension formula important in calculating antenna size?

Fractal dimension is sometimes used as a descriptive metric in fractal antenna research and design because it relates to how “space-filling” a geometry is. However, antenna size is typically determined by wavelength and the specific geometry; fractal dimension alone does not directly determine an antenna’s physical size.

How does scaling factor influence the Fractal Dimension?

In the self-similar formula D = log(N) / log(S), S is the magnification factor (original length divided by the copy length). For a fixed N, increasing S increases the denominator and results in a smaller D. (If you instead define a reduction ratio r with 0 < r < 1, then S = 1/r and the equivalent formula is D = log(N) / log(1/r).)

Can the Fractal Dimension be applied to fields other than antenna design?

Yes, the concept of Fractal Dimension is applied across various fields, including geology, biology, and physics, to analyze patterns and structures that exhibit self-similarity at different scales. It is used to study the complexity of coastlines, clouds, tree branches, and even the distribution of galaxies.