A fractal curve is, loosely, a mathematical

curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...

whose shape retains the same general pattern of irregularity, regardless of how high it is magnified, that is, its graph takes the form of a

fractal In mathematics, a fractal is a Shape, 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 scale ...

. In general, fractal curves are nowhere rectifiable curves — that is, they do not have finite

length Length is a measure of distance. In the International System of Quantities, length is a quantity with Dimension (physical quantity), dimension distance. In most systems of measurement a Base unit (measurement), base unit for length is chosen, ...

— and every subarc longer than a single point has infinite length. A famous example is the boundary of the

Mandelbrot set The Mandelbrot set () is a two-dimensional set (mathematics), set that is defined in the complex plane as the complex numbers c for which the function f_c(z)=z^2+c does not Stability theory, diverge to infinity when Iteration, iterated starting ...

Fractal curves in nature

Fractal curves and fractal patterns are widespread, in

nature Nature is an inherent character or constitution, particularly of the Ecosphere (planetary), ecosphere or the universe as a whole. In this general sense nature refers to the Scientific law, laws, elements and phenomenon, phenomena of the physic ...

, found in such places as

broccoli Broccoli (''Brassica oleracea'' var. ''italica'') is an edible green plant in the Brassicaceae, cabbage family (family Brassicaceae, genus ''Brassica'') whose large Pseudanthium, flowering head, plant stem, stalk and small associated leafy gre ...

snowflakes A snowflake is a single ice crystal that is large enough to fall through the Earth's atmosphere as snow.Knight, C.; Knight, N. (1973). Snow crystals. Scientific American, vol. 228, no. 1, pp. 100–107.Hobbs, P.V. 1974. Ice Physics. Oxford: C ...

, feet of

geckos Geckos are small, mostly carnivorous lizards that have a wide distribution, found on every continent except Antarctica. Belonging to the infraorder Gekkota, geckos are found in warm climates. They range from . Geckos are unique among lizards f ...

, frost crystals, and lightning bolts. See also Romanesco broccoli, dendrite crystal, trees, fractals, Hofstadter's butterfly, Lichtenberg figure, and self-organized criticality.

Dimensions of a fractal curve

Most of us are used to mathematical curves having

dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coo ...

one, but as a general rule, fractal curves have different dimensions, also see

fractal dimension In mathematics, a fractal dimension is a term invoked in the science of geometry to provide a rational statistical index of complexity detail in a pattern. A fractal pattern changes with the Scaling (geometry), scale at which it is measured. It ...

and list of fractals by Hausdorff dimension.

Relationships of fractal curves to other fields

Starting in the 1950s Benoit Mandelbrot and others have studied self-similarity of fractal curves, and have applied theory of fractals to modelling natural phenomena. Self-similarity occurs, and analysis of these patterns has found fractal curves in such diverse fields as

economics Economics () is a behavioral science that studies the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods and services. Economics focuses on the behaviour and interac ...

fluid mechanics Fluid mechanics is the branch of physics concerned with the mechanics of fluids (liquids, gases, and plasma (physics), plasmas) and the forces on them. Originally applied to water (hydromechanics), it found applications in a wide range of discipl ...

geomorphology Geomorphology () is the scientific study of the origin and evolution of topographic and bathymetric features generated by physical, chemical or biological processes operating at or near Earth's surface. Geomorphologists seek to understand wh ...

, human physiology and

linguistics Linguistics is the scientific study of language. The areas of linguistic analysis are syntax (rules governing the structure of sentences), semantics (meaning), Morphology (linguistics), morphology (structure of words), phonetics (speech sounds ...

. As examples, "landscapes" revealed by microscopic views of surfaces in connection with

Brownian motion Brownian motion is the random motion of particles suspended in a medium (a liquid or a gas). The traditional mathematical formulation of Brownian motion is that of the Wiener process, which is often called Brownian motion, even in mathematical ...

, vascular networks, and shapes of polymer molecules all relate to fractal curves.

Examples

* Blancmange curve * Coastline paradox *

De Rham curve In mathematics, a de Rham curve is a continuous fractal curve obtained as the image of the Cantor space, or, equivalently, from the base-two expansion of the real numbers in the unit interval. Many well-known fractal curves, including the Cantor ...

* Dragon curve * Fibonacci word fractal * Koch snowflake * Boundary of the

Menger sponge In mathematics, the Menger sponge (also known as the Menger cube, Menger universal curve, Sierpinski cube, or Sierpinski sponge) is a fractal curve. It is a three-dimensional generalization of the one-dimensional Cantor set and two-dimensional Sie ...

* Peano curve * Sierpiński triangle *

Weierstrass function In mathematics, the Weierstrass function, named after its discoverer, Karl Weierstrass, is an example of a real-valued function (mathematics), function that is continuous function, continuous everywhere but Differentiable function, differentiab ...

References

External links

Wolfram math on fractal curves

The Fractal Foundation's homepage

fractalcurves.com

Making a Kock Snowflake, from Khan Academy

Area of a Koch Snowflake, from Khan Academy

Youtube on space-filling curves

Youtube on the Dragon Curve
{{Fractals Types of functions