Definition of Fractal Dimension
"A fractal dimension is an index for characterizing fractal patterns or sets by quantifying their complexity as a ratio of the change in detail to the change in scale."
Benoit B. Mandelbrot (1983). The fractal geometry of nature. Macmillan. ISBN 978-0-7167-1186-5. Retrieved 1 February 2012.
In other words "Create a geometric structure from a given D-dimensional object (e.g., line, square, cube, etc) by repeatedly dividing the lenght of its sides by a number M."
Then each level is made up of MD copies of the previous level.
Call the number of copies N.
Then N=MD.
Now we can define D taking the logarithm on both sides, so there we have log N = D log M.
Now we can say that D = log N / log M
Therefore, dimension is defined by the logarithm of the number of copies we get divided by the amount by which we reduced the length of the sides.
Mitchell, Melanie (2009). Complexity: A Guided Tour. Oxford, UK: Oxford University Press. ISBN 0-19-512441-3.
Introducing Chaos to a Fractal Dimension
Having interesting detail at all levels applies to perfect mathematical fractals, but in nature, these are extremely complex and also irregular. how can we mimic the way nature is designing fractals? can we use fractals as a design tool?.
To answer these questions we have constructed a system that measures the perimeter of a given geometric structure and divides its length into any number of copies (N) creating regular fractal dimensions.
The hard part was to introduce Chaos to this system in order to generate unpredictable results inside the matrix we determined. To solve this problem, we establish a disruptive constant inverting a critical part of the FDL construction logic after reaching a limit of calculations.
We mutate the division of the area, iterating (N) looking for a regular configuration in an already distorted Voronoi tessellation, therefore the first disruptive constant is smoothed, giving us a more harmonic result.
Finally, we repeated the operation with different regular and irregular geometric structures. Iterating, we found the resultant number of copies (N) that generates this regularity inside the irregularity and we extrapolated to Sub Fractal Dimensions (FDL1,FDL2,FDL3).
The graph represents the heterogeneity between the polygons generated as FDL1 parcels. 1. It takes the lowest and the highest numbers in the area analysis of every polygon, 2. creates a range of colors from green to red, and 3.- assigns a RGB value according to the area of the FDL1 polygons.
With this new "Chapter" we add a new way of creating the Fractal Dimension Levels, which are: 1.- Nested K-means of BFP Population. 2.- Nested K-means of BFP Population + human manipulation. 3.- Natural Fractals with "Chaotic Nature Algorithm". Depending on the state of the project and master plans development, we can engage a strategy according to the needs.
Isaías Baruch.