And this describes the Koch Curve - it's wigglier than a straight line, but it doesn't fill up a whole 2-Dimensional plane either.Īs we'll see soon, the more of a plane that a fractal covers the closer its dimensions is to 2. If a line is 1-Dimensional, and a plane is 2-Dimensional, thenĪ fractional dimension of 1.26 falls somewhere in between a line and a plane. We can make some sense out of the dimension, by comparing it to the simple, whole number dimensions. In fact, all fractals have dimensions that are fractions, not whole numbers. What could a fractional dimension mean?įractional dimensions are very useful for describing fractal shapes. We're used to dimensions that are whole numbers, 1,2 or 3. Use a calculator (or Google) to find the value for log(4): So according to the formula D = log(N) / log(r), we can say that D = log(4) / log(3) = 1.26 This means if the length multiplier is 1 then the all the branches will have the same length. Length Multiplier After the first set of branches are drawn, at each iteration the current length is multiplied by this value. Order 4 has four times as many pieces as order 3, and each piece is 1/3 the scale. Start Length The length value of the first set of branches.
#Fractal time calculator generator
In this case, we can see that the number of pieces in the generator, N, is 4, and the magnification factor is 3, because each section of the generator is 1/3 of the unit length.This same relationship holdsīetween each of the orders of the curve. The FD of the images was determined using a simple Canny edge detection algorithm, a manual calculation method, and an indirect approach based on spectral decay. Remembering that D = log(N) / log(r), we can calculate the dimension D by seeing how the number of units, N, changes with the magnification factor, r. The third order curve follows the same pattern, and it has 64 tiny segments, each of which is 1/27 of the unit length, making a total length of 64/27.Īs the progression continues, the curve gets longer and longer, and eventually becomes infinitely long! Now, it is not very useful to know that a curve is infinitely long,Īnd this is where the concept of Fractal Dimension becomes very useful. Of the unit length, That means the total length of the second order curve is 16/9. The second order of the Koch Curve has had each of the 4 sections of the generator replaced with the same shape, so it has 16 small segments, and each segment is 1/9 The generator (order 1) is made of 4 sections, and each section is 1/3 of the length of the initiator (order 0), which has a unit length of 1. Let's look at the way the length of the curve changes as we iterate the fractal. As we learned in Chapter 2, geometric fractals can be made by starting with a simple generator pattern and replacing every section of the pattern with a smaller copy
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