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Escape-time fractals – use a formula or recurrence relation at each point in a space (such as the complex plane); usually quasi-self-similar; also known as "orbit" fractals; e.g., the Mandelbrot set, Julia set, Burning Ship fractal, Nova fractal and Lyapunov fractal. The 2d vector fields that are generated by one or two iterations of escape ...
1.6667: 32-segment quadric fractal (1/8 scaling rule) see also: File:32 Segment One Eighth Scale Quadric Fractal.jpg: Generator for 32 segment 1/8 scale quadric fractal. Built by scaling the 32 segment generator (see inset) by 1/8 for each iteration, and replacing each segment of the previous structure with a scaled copy of the entire generator.
Edge, a 1-dimensional element; Face, a 2-dimensional element; Cell, a 3-dimensional element; Hypercell or Teron, a 4-dimensional element; Facet, an (n-1)-dimensional element; Ridge, an (n-2)-dimensional element; Peak, an (n-3)-dimensional element; For example, in a polyhedron (3-dimensional polytope), a face is a facet, an edge is a ridge, and ...
In fractal geometry, some fractals with Hausdorff dimension have zero or infinite -dimensional Hausdorff measure. For example, almost surely the image of planar Brownian motion has Hausdorff dimension 2 and its two-dimensional Hausdorff measure is zero. In order to "measure" the "size" of such sets, the following variation on the notion of the ...
Fractal fern in four states of construction. Highlighted triangles show how the half of one leaflet is transformed to half of one whole leaf or frond.. Though Barnsley's fern could in theory be plotted by hand with a pen and graph paper, the number of iterations necessary runs into the tens of thousands, which makes use of a computer practically mandatory.
On the other hand, for =, the sequence is 0, −1, 0, −1, 0, ..., which is bounded, so −1 does belong to the set. The Mandelbrot set can also be defined as the connectedness locus of the family of quadratic polynomials f ( z ) = z 2 + c {\displaystyle f(z)=z^{2}+c} , the subset of the space of parameters c {\displaystyle c} for which the ...
The theoretical fractal dimension for this fractal is 5/3 ≈ 1.67; its empirical fractal dimension from box counting analysis is ±1% [8] using fractal analysis software. A fractal dimension is an index for characterizing fractal patterns or sets by quantifying their complexity as a ratio of the change in detail to the change in scale.
Area#Area formulas – Size of a two-dimensional surface; Perimeter#Formulas – Path that surrounds an area; List of second moments of area; List of surface-area-to-volume ratios – Surface area per unit volume; List of surface area formulas – Measure of a two-dimensional surface; List of trigonometric identities
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