# Fractal

**In mathematics, a fractal is a self-similar subset of Euclidean space. The term ‘fractal’ was first used by mathematician Benoit Mandelbrot in 1975. Analytically, fractals are usually nowhere differentiable. An infinite fractal curve can be conceived of as winding through space differently from an ordinary line.**

### About Fractal in brief

In mathematics, a fractal is a self-similar subset of Euclidean space whose fractal dimension strictly exceeds its topological dimension. The term ‘fractal’ was first used by mathematician Benoit Mandelbrot in 1975. Analytically, fractals are usually nowhere differentiable. An infinite fractal curve can be conceived of as winding through space differently from an ordinary line – although it is still 1-dimensional, its fractaldimension indicates that it also resembles a surface. Fractals are of particular relevance in the field of chaos theory, since the graphs of most chaotic processes are fractals. Many real and model networks have been found to have fractal features such as self similarity. The concept is difficult to define formally, even for mathematicians, but key features can be understood with a little mathematical background. The feature of’self-similarity’, for instance, is by analogy to zooming in with a lens or other device that zooms on finer, previously invisible, detail. If this is done on fractals, however, no new pattern appears; nothing changes and the same pattern repeats over and over. The difference for fractals is that the pattern must be detailed. A straight line is usually perceived as perceived as straight, for instance. This idea of being detailed relates to another feature that must be understood without much mathematical background: having a fractological dimension greater than itsTopological dimension, for example, refers to how a geometric shape is perceived compared to how it scales to how geometric shapes are usually perceived.

The word ‘f Fractal’ often has different connotations for the lay public as opposed to mathematicians. The public is more likely to be familiar with fractal art than the mathematical concept. Fractal patterns with various degrees of self- Similarity have been rendered or studied in images, structures and sounds and found in nature, technology, art, architecture and law. The idea of ‘f fractals’ is to uncover invisible, new structure in digital images for digital images to uncover new, previously unseen, structure for digital photos and videos. For instance, in the Menger sponge, the pattern repeats and reappears over andover and over again, as in the same way as in a photograph of the sponge. The pattern is called affine self-Similarity, also known as expanding symmetry or unfolding symmetry, and is a type of symmetry. It is the same at different levels, as illustrated in successive magnifications of the MandelBrot set. It can be seen in the image below, which shows the same fractal pattern at various levels of magnification, as well as the size of a circle and a circle in the shape of a sphere. The shape of the sphere is shown in the right-hand side of the image, which is called the ‘Manger sponge’ The shape is called a’manger fractal’ and can be found in the gallery of images at the bottom of the page, where it can be viewed by clicking on the image to see it in full.

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This page is based on the article Fractal published in Wikipedia (as of Dec. 09, 2020) and was automatically summarized using artificial intelligence.