I fail to see how the planting patterns are fractal. A fractal pattern is one which repeats itself at different scales. I realize that the repetition does not need to be exact but I don't see how there is any at all in this situation.
Actually, that is only one type of fractals. Perfectly self similar patterns like the Triforce or Sierpensky triangle are used as toy models for learning . Typically, fractals in nature are not self similar(Branching of your blood vessels for example) . Mandelbrot developed fractal geometry as a way to model nature that captures roughness. It was kind of a rebellion against calculus, where if you zoom in eventually you get smoothness.
It's funny that many people think that only self similar patterns are fractals when his desire was to get away from idealized models towards more pragmatic ones
EDIT: I'm actually just learning about fractals, so I'm certainly no expert, but I'm excited to share what I've learned so far.
Look at a line, square, and cube in 1, 2, and 3 dimensions respectively. If you scale down each by 1/2 in all its dimensions, you need to look at how much of the "mass" (for lack of a better term is scaled down
If you cut a line in 1/2, it's mass is scaled by 1/2 as it takes 2 one-half length lines to make the original line.
If you scale a square down 1/2 along all its dimensions, you break into/scale it down into 4 smaller squares, it's mass scaling factor is 1/4....scaling a cube down 1/2 along all its dimensions (1/2)^2 breaks it into 8 smaller cubes...it's scaling factor is 1/8 and you can see the progression here. (1/2)^3. (I imagine a tesseract/hypercube has a scaling of 1/16 as it is the 4D analogue of a cube) The exponent represents the concept of dimension and this is how you can have non-integer dimensions.
So, back to the Sierpenski triangle or triforce example. Let's scale it down 1/2. When we do that, we know we get 1/3 the "mass" of the original since there are 3 triangles contained in the larger triangle at each level. The dimensionality is then (1/2)^x = 1/3 where x is the dimensionality. We rewrite this as log 3 base 2 which gives it a dimensionality of ~1.585.
And that is the definition of a fractal, a shape with a non-integer dimension which gives the shape roughness at every scale. I don't quite understand the more technical definition, but hopefully this helps!
Note, you cannot use length or area as a measurment for a scaling factor, as the length would be infinite and area would be 0. Also, I say mass because the more correct concept is difficult for me to explain without a whiteboard, but basically, it has to do with putting the fractal on a 2d grid, scaling it, and seeing what the ratio of boxes touched is. See, told you I'm bad at explaining it :D
I forgot to add that fractal dimensionality is a clear differentiator between natural and man made things. So, it does make some sense to separate the natural from the synthetic.
Off topic, but does anyone know how this would affect the Fermi Paradox if aliens produce technology inherently fractal in nature?
I did enjoy it, if you can't tell from the content of my original post. I never quite knew what fractals were until yesterday afternoon. After 30 minutes of youtube, I'm explaining it to someone else, although at a 10 yr old's level.
There's no One True Definition of what it means to be fractal, but a lot of working mathematicians use the criterion that a space's Hausdorf dimension is not equal to its topological dimension. Oftentimes (but not always), the Hausdorf dimension will be fractional, which is where the word comes from. Self-similarity is an easy way to satisfy that requirement in a way that's easy to explain, but it's far from the only way.
I was under the impression that the fractal dimension must be greater than the topological dimension for the space to be a fractal.
And that the fractal dimension is usually equivalent to the mincowski dimension (the limit of the area measured by finite boxes as the size of the boxes grows arbitrarily small) and hausdorf dimension (the limit of the perimeter as measured by an aproximting polygon with equal length sides as the number of sides grows arbitrarily large) along with others.
Something as simple as a contour map could be thought of as intrinsically fractal. It isn't as mathematically rigorous as the fractals we are used to seeing described as such, but it fulfils the criteria.
