Curves {fractal curve}| can have non-integral dimension.
dimension
Dimension d depends on unit-copy number m needed to make shape that number n of times bigger: m = n^d. For example, line segments can be two times longer using two line segments, so dimension is 1: 2 = 2^1.
If fractal unit has _/|_ shape, next larger self-similar shape is _/|_, and line segments look like original unit. The next-larger shape is three times bigger and needs four unit copies, making dimension 1.26186...: 4 = 3^1.26186...
self-similarity
For fractals, whole shape is similar to part shape {self-symmetry, polar} {self-similarity}. Scale changes do not change pattern. Fractals can model objects that have same shape at different scales. Fractals can model renormalization.
non-linear
Fractals are non-linear.
fractal limits
Fractal shape is the limit of iteratively applying mapping rules.
rule
Given fractal shapes, using same shape at smallest scale can induce rules for making the shape {collage theorem}.
examples: Mandelbrot curve
Fractal curves {Mandelbrot curve} can enclose finite or zero area but have infinite length. Infinite length fills two-dimensional space. Fractal curve has physical dimension 1. If fractal curve fills two-dimensional plane, it has fractal dimension 2.
examples: Peano curve
Fractals can be curves {Peano curve}.
examples: Koch curve
Starting with a triangle, repeatedly adding triangle one-third the size to line-segment middles makes curves {Koch curve}. Boundary has infinite length but finite area.
examples: Sierpinski carpet
Starting with square, making nine squares inside, removing central square, and then repeating makes surfaces {Sierpinski carpet}.
examples: Sierpinski gasket
Starting with equilateral triangle, making nine equilateral triangles inside, removing center equilateral triangle, and then repeating makes surfaces {Sierpinski gasket}.
examples: Menger sponge
Menger sponges are three-dimensional Sierpinski carpets.
examples: nature
Natural fractals are coastlines, rivers, islands, seas, lakes, mountains, arteries, music, Brownian motion paths, critical points, elasticity, turbulence, snowflakes, clouds, disconnected star-cluster points, temperature, spectra, and all intensive properties.
uses
Relief maps, Cantor infinite sets, computer designs, and Fourier analysis can use fractals.
Mathematical Sciences>Geometry>Fractal Geometry
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Date Modified: 2022.0224