A fractal is a complex geometric shape with "fractional dimension." Coined by Benoit B. Mandelbrot, the term fractal comes from the Latin word fractus , meaning fragmented or broken. Unlike simple Euclidean geometry figures, fractals can describe irregularly shaped objects or nonuniform spatial phenomena in nature, like coastlines and mountain ranges. Many fractals have self-similarity, where parts resemble the whole at smaller scales. This scaling symmetry means the object remains similar under scale changes. A key characteristic of fractals is their fractal dimension, a noninteger that indicates a figure's complexity. Fractal geometry is used in statistical mechanics, fluid mechanics, computer graphics, and other fields.

What is an example of a simple fractal in nature or art?

A simple fractal example in nature is a snowflake. The microscopic crystals that form a snowflake resemble the overall flake shape, illustrating self-similarity. Similarly, tree bark and broccoli buds exhibit fractal patterns. The Koch snowflake is a mathematical example with six-fold symmetry, like a natural snowflake. It consists of three identical parts, each made of four parts that are scaled-down versions of the whole. Fractals' scaling symmetry and fractional dimension are used to model complex systems with seemingly random features.

What does it mean for a fractal shape to repeat itself at different scales?

A fractal shape repeating itself at different scales means it possesses self-similarity, where its components resemble the whole. This repetition occurs at smaller scales, and with purely abstract fractals, it can continue indefinitely. Each part, when magnified, looks like a part of the whole object. Self-similar objects remain the same despite scale changes, displaying scaling symmetry. Natural fractals like snowflakes and tree barks often show this phenomenon and scale statistically. This self-similarity and fractal dimension are key in statistical mechanics for analyzing physical systems with random features, such as galaxy cluster distribution and fluid turbulence.

How are fractals created using mathematical rules?

Fractals are complex geometric shapes with fractional dimensions, distinct from Euclidean geometry. Benoit Mandelbrot introduced the term fractal in 1975, using the Latin word fractus ("fragmented" or "broken"). Many fractals exhibit self-similarity, where components resemble the whole, with reiteration at smaller scales. This scaling symmetry means the object remains unchanged regardless of scale. Another key characteristic is fractal dimension, a noninteger. An example is the Mandelbrot set, a fractal that, when plotted, consists of connected points c in a complex plane. This is based on the recursive expression (z_{n+1} = z_n^2 + c) for (n = 0, 1, 2, 3), with (z_0 = 0), remaining bounded as (n) approaches infinity.

What is the difference between a regular geometric shape and a fractal?

Euclidean geometry uses simple figures such as squares, circles, and spheres, while fractals are complex geometric shapes with fractional dimensions. Many fractals exhibit self-similarity, where their parts resemble the whole, with repeating details at smaller scales. This scaling symmetry means the object remains the same regardless of scale. Regular geometric shapes do not have these properties. Benoit B. Mandelbrot coined the term fractal from the Latin word fractus , meaning fragmented or broken.

How are fractals used in technology and science today?

Fractals are complex geometric shapes with fractional dimensions, differing from Euclidean geometry. Benoit Mandelbrot introduced the term fractal in 1975, noting its utility in applied mathematics for modeling phenomena from physical objects to stock market behavior. Fractals exhibit self-similarity, where parts resemble the whole, recurring at smaller scales. This scaling symmetry is evident in objects like snowflakes and tree barks. Fractal dimension, unlike Euclidean dimension, is expressed as a noninteger and indicates a figure's complexity. Fractal geometry is applied in statistical mechanics for analyzing random systems, such as galaxy cluster distribution and fluid turbulence. Fractal algorithms also generate realistic images of complex natural objects like mountains and trees.

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Introduction References & Edit History Related Topics

Images

$Mandelbrot setDuring the late 20th century, Polish mathematician Benoit Mandelbrot helped popularize the fractal that bears his name. The fundamental set contains all complex numbers C such that the iterative equation Zn + 1 = Zn2 + C stays finite for all n starting with Z0 = 0. As shown here, the set of points that remain finite through all iterations is white, with darker colours showing how quickly other values diverge to infinity. The fractal edge between points that remain finite and those that diverge to infinity is extremely complicated, with self-repeating features that can be seen at all scales.$ $Sierpiński gasketPolish mathematician Wacław Sierpiński described the fractal that bears his name in 1915, although the design as an art motif dates at least to 13th-century Italy. Begin with a solid equilateral triangle, and remove the triangle formed by connecting the midpoints of each side. The midpoints of the sides of the resulting three internal triangles are connected to form three new triangles that are then removed to form nine smaller internal triangles. The process of cutting away triangular pieces continues indefinitely, producing a region with a Hausdorf dimension of a bit more than 1.5 (indicating that it is more than a one-dimensional figure but less than a two-dimensional figure).$

Quick Facts

Key People:: Benoit Mandelbrot; Wacław Sierpiński; Gaston Maurice Julia

Related Topics:: Mandelbrot set; fractal curve; fractal dimension; Sierpiński gasket; Julia set

On the Web:: IBM - Fractal geometry (Oct. 30, 2026)

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Top Questions

What is a fractal?
A fractal is a complex geometric shape with "fractional dimension." Coined by Benoit B. Mandelbrot, the term fractal comes from the Latin word fractus, meaning fragmented or broken. Unlike simple Euclidean geometry figures, fractals can describe irregularly shaped objects or nonuniform spatial phenomena in nature, like coastlines and mountain ranges.

Many fractals have self-similarity, where parts resemble the whole at smaller scales. This scaling symmetry means the object remains similar under scale changes. A key characteristic of fractals is their fractal dimension, a noninteger that indicates a figure's complexity. Fractal geometry is used in statistical mechanics, fluid mechanics, computer graphics, and other fields.
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What is an example of a simple fractal in nature or art?
A simple fractal example in nature is a snowflake. The microscopic crystals that form a snowflake resemble the overall flake shape, illustrating self-similarity. Similarly, tree bark and broccoli buds exhibit fractal patterns. The Koch snowflake is a mathematical example with six-fold symmetry, like a natural snowflake. It consists of three identical parts, each made of four parts that are scaled-down versions of the whole.

Fractals' scaling symmetry and fractional dimension are used to model complex systems with seemingly random features.
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1. fractal
2. Did you know?
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What does it mean for a fractal shape to repeat itself at different scales?
A fractal shape repeating itself at different scales means it possesses self-similarity, where its components resemble the whole. This repetition occurs at smaller scales, and with purely abstract fractals, it can continue indefinitely. Each part, when magnified, looks like a part of the whole object.

Self-similar objects remain the same despite scale changes, displaying scaling symmetry. Natural fractals like snowflakes and tree barks often show this phenomenon and scale statistically. This self-similarity and fractal dimension are key in statistical mechanics for analyzing physical systems with random features, such as galaxy cluster distribution and fluid turbulence.
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1. fractal
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How are fractals created using mathematical rules?
Fractals are complex geometric shapes with fractional dimensions, distinct from Euclidean geometry. Benoit Mandelbrot introduced the term fractal in 1975, using the Latin word fractus ("fragmented" or "broken").

Many fractals exhibit self-similarity, where components resemble the whole, with reiteration at smaller scales. This scaling symmetry means the object remains unchanged regardless of scale. Another key characteristic is fractal dimension, a noninteger. An example is the Mandelbrot set, a fractal that, when plotted, consists of connected points c in a complex plane. This is based on the recursive expression (z_{n+1} = z_n^2 + c) for (n = 0, 1, 2, 3), with (z_0 = 0), remaining bounded as (n) approaches infinity.
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What is the difference between a regular geometric shape and a fractal?
Euclidean geometry uses simple figures such as squares, circles, and spheres, while fractals are complex geometric shapes with fractional dimensions.

Many fractals exhibit self-similarity, where their parts resemble the whole, with repeating details at smaller scales. This scaling symmetry means the object remains the same regardless of scale. Regular geometric shapes do not have these properties. Benoit B. Mandelbrot coined the term fractal from the Latin word fractus, meaning fragmented or broken.
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How are fractals used in technology and science today?
Fractals are complex geometric shapes with fractional dimensions, differing from Euclidean geometry. Benoit Mandelbrot introduced the term fractal in 1975, noting its utility in applied mathematics for modeling phenomena from physical objects to stock market behavior. Fractals exhibit self-similarity, where parts resemble the whole, recurring at smaller scales. This scaling symmetry is evident in objects like snowflakes and tree barks. Fractal dimension, unlike Euclidean dimension, is expressed as a noninteger and indicates a figure's complexity. Fractal geometry is applied in statistical mechanics for analyzing random systems, such as galaxy cluster distribution and fluid turbulence. Fractal algorithms also generate realistic images of complex natural objects like mountains and trees.
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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 many irregularly shaped objects or spatially nonuniform phenomena in nature such as coastlines and mountain ranges. The term fractal, derived from the Latin word fractus (“fragmented,” or “broken”), was coined by the Polish-born mathematician Benoit B. Mandelbrot. See the animation of the Mandelbrot fractal set.

Although the key concepts associated with fractals had been studied for years by mathematicians, and many examples, such as the Koch or “snowflake” curve were long known, Mandelbrot was the first to point out that fractals could be an ideal tool in applied mathematics for modeling a variety of phenomena from physical objects to the behavior of the stock market. Since its introduction in 1975, the concept of the fractal has given rise to a new system of geometry that has had a significant impact on such diverse fields as physical chemistry, physiology, and fluid mechanics.

Many fractals possess the property of self-similarity, at least approximately, if not exactly. A self-similar object is one whose component parts resemble the whole. This reiteration of details or patterns occurs at progressively smaller scales and can, in the case of purely abstract entities, continue indefinitely, so that each part of each part, when magnified, will look basically like a fixed part of the whole object. In effect, a self-similar object remains invariant under changes of scale—i.e., it has scaling symmetry. This fractal phenomenon can often be detected in such objects as snowflakes and tree barks. All natural fractals of this kind, as well as some mathematical self-similar ones, are stochastic, or random; they thus scale in a statistical sense.

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Numbers and Mathematics

Another key characteristic of a fractal is a mathematical parameter called its fractal dimension. Unlike Euclidean dimension, fractal dimension is generally expressed by a noninteger—that is to say, by a fraction rather than by a whole number. Fractal dimension can be illustrated by considering a specific example: the snowflake curve defined by Helge von Koch in 1904. It is a purely mathematical figure with a six-fold symmetry, like a natural snowflake. It is self-similar in that it consists of three identical parts, each of which in turn is made of four parts that are exact scaled-down versions of the whole. It follows that each of the four parts itself consists of four parts that are-scaled down versions of the whole. There would be nothing surprising if the scaling factor were also four, since that would be true of a line segment or a circular arc. However, for the snowflake curve, the scaling factor at each stage is three. The fractal dimension, D, denotes the power to which 3 must be raised to produce 4—i.e., 3^D= 4. The dimension of the snowflake curve is thus D = log 4/log 3, or roughly 1.26. Fractal dimension is a key property and an indicator of the complexity of a given figure.

Fractal geometry with its concepts of self-similarity and noninteger dimensionality has been applied increasingly in statistical mechanics, notably when dealing with physical systems consisting of seemingly random features. For example, fractal simulations have been used to plot the distribution of galaxy clusters throughout the universe and to study problems related to fluid turbulence. Fractal geometry also has contributed to computer graphics. Fractal algorithms have made it possible to generate lifelike images of complicated, highly irregular natural objects, such as the rugged terrains of mountains and the intricate branch systems of trees.

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