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.
Intervals {fractal interval} can be harmonic, as in logarithmic relations, instead of linear or vectorial. For example, remove interval middle third, then remove middle thirds of both remaining line segments, and so on, indefinitely, to make triadic set. Intervals can use other numbers, proportions, sizes, and positions of such cutouts. Cutouts can be random or fixed.
purposes
Triadic sets, and similar point distributions over intervals, model hierarchical errors, time-measurement errors, high signal-to-noise-ratio noise, negligible thermal noise {excess noise}, computing errors, and other errors in which, as time goes up, error chance goes down.
dimensions and fractals
Points have dimension zero. Sets of countable separated points have dimension zero. Sets of uncountable separated points have no density and have topological and fractal dimension zero. Line segments are point sets with linear density and have topological and fractal dimension one. Bounded surfaces are point sets with surface density and have topological and fractal dimension two. Bounded volumes are point sets with volume density and have topological and fractal dimension three.
Fractals are geometric figures with non-integer dimensions. Some geometric fractals start with a line segment and repeatedly remove intervals. Repeatedly removing intervals (to make Cantor sets, for example) keeps topological dimension one but reduces fractal dimension to less than one.
Some geometric fractals start with a line segment and repeatedly replace intervals with added values. Repeatedly replacing intervals with added values (to make Koch curves, for example) makes topological dimension two and fractal dimension greater than one.
To make more than one dimension, fractals use complex numbers, which have two components and so can graph to surfaces. Some geometric fractals start with a bounded surface and repeatedly remove inner regions, to make topological dimension two and fractal dimension less than two. Some geometric fractals start with a bounded surface and repeatedly replace inner regions with added values, to make topological dimension three and fractal dimension greater than two.
To make more than two dimensions, fractals use hypercomplex numbers, which have three or more components and so can graph to volumes and hypervolumes. Some geometric fractals start with a bounded volume and repeatedly remove inner regions, to make topological dimension three and fractal dimension less than three. Some geometric fractals start with a bounded volume and repeatedly replace inner regions with added values, to make topological dimension four and fractal dimension greater than three.
Generalized Brownian motion {Levy flight} has different-length jumps in all directions. Levy flights are isotropic. Jump-length distribution has fractal dimension.
If fractal-curve or figure scale changes, curve has same shape {self-symmetry, fractal}| {fractal hierarchy}. Whole shape is similar to part shape.
Sets {fractal set}| can have members related by fractal steps. All initial values used in Newton's method lead to square root. Following fractal paths, initial value lying between two roots can lead to any value. Feigenbaum functions on complex plane reach all frequencies {Julia set}. Julia sets can generalize. Take complex number, square it, add square to original complex number, square result, add square to original complex number, and so on. If result does not diverge from complex-plane origin, result has bound, and complex number is in set {Mandelbrot set}. If real or imaginary part becomes greater than 2 or smaller than -2, number diverges. Mandelbrot set is boundary between the regions {fractal basin boundary}. Mandelbrot set is not precisely self-similar.
Sets {Levy set}, such as rectifiable or connected curves, can have dimension between zero and one and density between zero and one. Points have dimension zero. Lines have dimension one. Lower dimension means more clustering.
Remove line-segment middle third, then remove middle thirds of both remaining line segments, and repeat indefinitely {triadic set}.
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Date Modified: 2022.0225