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Fractals contain detailed structure at arbitrarily small scales and often display self-similarity.
A fractal is a geometric shape or mathematical structure with intricate detail at arbitrarily small scales. Fractals commonly exhibit self-similarity, meaning that parts resemble the whole when viewed at different magnifications. This similarity may be exact, approximate, statistical, or qualitative. Examples include the Mandelbrot set, Koch snowflake, Sierpiński carpet, and Menger sponge. A central feature of fractals is their fractal dimension, often measured using the Hausdorff dimension. Unlike ordinary geometric objects, fractals may have dimensions that are non-integer and greater than their topological dimensions, reflecting how efficiently they fill space as they scale. Many mathematical fractals are also irregular and nowhere differentiable, with patterns that continue to reappear at increasingly fine scales. Because the term is difficult to define universally, fractals are generally characterized by a combination of self-similarity, fine-scale detail, unusual scaling, and geometric irregularity.
Fractals contain detailed structure at arbitrarily small scales and often display self-similarity.
Fractal dimension describes how a fractal scales and may be non-integer or greater than its topological dimension.
Many fractals are irregular and nowhere differentiable, making their lengths or boundaries difficult to measure using conventional geometry.
Self-similarity can be exact, approximate, statistical, qualitative, or involve multiple scaling dimensions.
Common examples include the Mandelbrot set, Koch snowflake, Sierpiński carpet, Sierpiński gasket, and Menger sponge.
A geometric shape or mathematical structure characterized by fine detail, irregularity, and often self-similarity across scales.
The property whereby a whole fractal or parts of it resemble one another when viewed at different scales.
A measure of how a fractal's structure scales with size, often producing a non-integer value.
A mathematical dimension used to quantify the scaling complexity of sets, including fractals.
The conventional dimension of a space, such as one for a line, two for a surface, or three for a solid.
A property of a function or curve that has no well-defined derivative at any point.
Self-similarity in which the same pattern is reproduced identically at every scale.
Self-similarity in which statistical or numerical properties remain consistent across different scales rather than exact shapes being repeated.
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