In mathematics, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales, as illustrated in successive magnifications of the Mandelbrot set. [1] ...

A fractal is a never-ending pattern. Fractals are infinitely complex patterns that are self-similar across different scales. They are created by repeating a simple process over and over in an ongoing feedback loop. Driven by recursion, fractals are images of dynamic systems – the pictures of Chaos. Geometrically, they exist in between our ...

Jan 29, 2024 · This is used to construct shapes like trees and plants etc. A classic example of a self-similar fractal is the Sierpinski triangle, where smaller triangles within the larger triangle are identical in shape. Self-Affine Fractals. Self-affine fractals use different scaling factors for all …

Mar 28, 2025 · fractal, in mathematics, any of a class of complex geometric shapes that commonly have “fractional dimension,” a concept first introduced by the mathematician Felix Hausdorff in 1918. Fractals are distinct from the simple figures of classical, or Euclidean, geometry—the square, the circle, the sphere, and so forth. They are capable of describing …

Aug 30, 2024 · A fractal is a complex geometric shape that can be split into parts, each of which is a reduced-scale copy of the whole. This property is known as self-similarity. Fractals are typically created by repeating a simple process over and over in an ongoing feedback loop. They often exhibit intricate patterns that are self-similar across different ...

Oct 31, 2023 · Fractal geometry throws this concept a curve by creating irregular shapes in fractal dimension; the fractal dimension of a shape is a way of measuring that shape's complexity. Now take all of that, and we can plainly see that a pure fractal is a geometric shape that is self-similar through infinite iterations in a recursive pattern and through ...

Have you ever seen an object which seems to repeat itself when you zoom in? No? Well, today's is a great day for you. Today, you will learn about fractals. So, you might be asking what exactly is a fractal? Well, a fractal, by definition, is a curve or geometric figure, each part of which has the same statistical character as the whole. Fractals are useful in modeling …

Dec 11, 2012 · The shape of a fractal can be completely captured by a small list of mathematical mappings that describe exactly how the smaller copies are arranged to form the whole fractal.

Sep 12, 2020 · Here is what is known as the Mandelbrot set, defined using the iteration of a certain complex-valued function. Note that this well-known set, discovered in 1980, also shows a fractal-like shape. The boundary of this shape exhibits quasi-self-similarity, in that portions look very similar to the whole.

Another famous fractal is the Sierpinski triangle. In this case, we start with a large, equilateral triangle, and then repeatedly cut smaller triangles out of the remaining parts. Notice how the final shape is made up of three identical copies of itself, and each of these is made up of even smaller copies of the entire triangle!