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.
Mathematical Sciences>Geometry>Fractal Geometry
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Date Modified: 2022.0224