What is the dimension of the Sierpinski triangle?
What is the dimension of the Sierpinski triangle?
The gasket is perfectly self similar, an attribute of many fractal images. Any triangular portion is an exact replica of the whole gasket. The dimension of the gasket is log 3 / log 2 = 1.5849, ie: it lies dimensionally between a line and a plane.
What is the dimension of the Koch curve?
The fractal dimension of the Koch curve is ln 4ln 3 ≈ 1.26186. This is greater than that of a line (=1) but less than that of Peano’s space-filling curve (=2). The Koch curve is continuous everywhere, but differentiable nowhere.
What is the dimensions of a fractal?
Fractal dimension is a measure of how “complicated” a self-similar figure is. In a rough sense, it measures “how many points” lie in a given set. A plane is “larger” than a line, while S sits somewhere in between these two sets.
What is the maximum dimension a fractal can have?
The theoretical fractal dimension for this fractal is 5/3 ≈ 1.67; its empirical fractal dimension from box counting analysis is ±1% using fractal analysis software.
How do you code a Sierpinski triangle?
The procedure for drawing a Sierpinski triangle by hand is simple. Start with a single large triangle. Divide this large triangle into three new triangles by connecting the midpoint of each side. Ignoring the middle triangle that you just created, apply the same procedure to each of the three corner triangles.
How are fractal dimensions derived?
The relation between log(L(s)) and log(s) for the Koch curve we find its fractal dimension to be 1.26. The same result obtained from D = log(N)/log(r) D = log(4)/log(3) = 1.26.
Why do fractals have fractional dimensions?
Fractional dimensions are very useful for describing fractal shapes. In fact, all fractals have dimensions that are fractions, not whole numbers. If a line is 1-Dimensional, and a plane is 2-Dimensional, then a fractional dimension of 1.26 falls somewhere in between a line and a plane.
What is the dimension of the Sierpinski carpet?
The Hausdorff dimension of the carpet is log 8log 3 ≈ 1.8928. Sierpiński demonstrated that his carpet is a universal plane curve.
https://www.youtube.com/watch?v=EeWnb56Tne8
