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.

Fractals have detail at arbitrarily small scales and display irregularity that cannot be described by traditional geometrical language. In other words, fractals are objects which, at any magnification, will never “smooth out” to look like Euclidean space.

2024年1月29日 · Fractal geometry deals with complexity and irregularity. While on the other hand, traditional Euclidean geometry, deals primarily with simple shapes such as circles, squares, and triangles. Fractals have three basic types which are below. Now we explain all of them briefly.

2010年11月24日 · In this paper we investigate how, using the language of Geometric Algebra [7, 4], the common escape-time Julia and Mandelbrot set fractals can be extended to arbitrary dimension and, uniquely, non-Eulidean geometries. We develop a geometric analog of complex numbers and show how existing ray-tracing techniques [2] can be extended.

Overview Measures and some examples Hutchinson’s theorem and its proof Applications of fractal geometry

分形 （英語： fractal，源自 拉丁語： frāctus，有「零碎」、「破裂」之意），又稱 碎形 、 殘形，通常被定義為「一個粗糙或零碎的 幾何形狀，可以分成數個部分，且每一部分都（至少近似地）是整體縮小後的形狀」 [2]，即具有 自相似 的性質。

In this paper we investigate how, using the language of Geo-metric Algebra [7, 4], the common escape-time Julia and Mandelbrot set fractals can be extended to arbitrary dimension and, uniquely, non-Eulidean geometries. We develop a geometric analog of complex numbers and show how existing ray-tracing techniques [2] can be extended.

Fractal geometry and its sibling, geometric measure theory, are branches of analysis which study the structure of “irregular” sets and measures in metric spaces, primarily

Fractals are usually too irregular to be described using usual Eu-clidean geometry. Let us now give an example of a well known fractal called the Sierpinski Triangle. We construct the triangle as follows: Start with an equilateral triangle of length 1 and denote it by S0.

Fractals can be found in nature, such as in the pattern of the leaves of a fern or a branches of a tree, and they can be generated geometrically and algebraically. Geometric fractals are created by repeating the same pattern of shapes over and over, like the Sierpinski triangle. The fractals shown above are well-known geometric fractals.