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 Irregular, irregulars or irregularity may refer to any of the following: Astronomy * Irregular galaxy * Irregular moon * Irregular variable, a kind of star Language * Irregular inflection, the formation of derived forms such as plurals in ...

, regardless of how high it is magnified, that is, its graph takes the form of a

fractal 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 ill ...

. In general, fractal curves are nowhere

rectifiable curve Rectification has the following technical meanings: Mathematics * Rectification (geometry), truncating a polytope by marking the midpoints of all its edges, and cutting off its vertices at those points * Rectifiable curve, in mathematics * Rec ...

s — 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 distance. In most systems of measurement a base unit for length is chosen, from which all other units are derived. In the Inte ...

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

Fractal curves in nature

Fractal curves and fractal patterns are widespread, in

nature Nature, in the broadest sense, is the physical world or universe. "Nature" can refer to the phenomena of the physical world, and also to life in general. The study of nature is a large, if not the only, part of science. Although humans are ...

, found in such places as

broccoli Broccoli (''Brassica oleracea'' var. ''italica'') is an edible green plant in the cabbage family (family Brassicaceae, genus ''Brassica'') whose large flowering head, stalk and small associated leaves are eaten as a vegetable. Broccoli is cla ...

, snowflakes, feet of geckos,

frost crystals Ice crystals are solid ice exhibiting atomic ordering on various length scales and include hexagonal columns, hexagonal plates, dendritic crystals, and diamond dust. Formation The hugely symmetric shapes are due to depositional growth, na ...

, and lightning bolts. See also

Romanesco broccoli Romanesco broccoli (also known as Roman cauliflower, Broccolo Romanesco, Romanesque cauliflower, Romanesco or broccoflower) is an edible flower bud of the species ''Brassica oleracea''. It is chartreuse in color, and has a form naturally approx ...

, dendrite crystal, trees, fractals, Hofstadter's butterfly,

Lichtenberg figure A Lichtenberg figure (German ''Lichtenberg-Figuren''), or Lichtenberg dust figure, is a branching electric discharge that sometimes appears on the surface or in the interior of insulating materials. Lichtenberg figures are often associated w ...

, 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 coord ...

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

fractal dimension In mathematics, more specifically in fractal geometry, a fractal dimension is a ratio providing a statistical index of complexity comparing how detail in a pattern (strictly speaking, a fractal pattern) changes with the scale at which it is me ...

and

list of fractals by Hausdorff dimension According to Benoit Mandelbrot, "A fractal is by definition a set for which the Hausdorff-Besicovitch dimension strictly exceeds the topological dimension." Presented here is a list of fractals, ordered by increasing Hausdorff dimension, to illus ...

Relationships of fractal curves to other fields

Starting in the 1950s

Benoit Mandelbrot Benoit B. Mandelbrot (20 November 1924 – 14 October 2010) was a Polish-born French-American mathematician and polymath with broad interests in the practical sciences, especially regarding what he labeled as "the art of roughness" of p ...

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 the social science that studies the production, distribution, and consumption of goods and services. Economics focuses on the behaviour and interactions of economic agents and how economies work. Microeconomics anal ...

, #

fluid mechanics Fluid mechanics is the branch of physics concerned with the mechanics of fluids ( liquids, gases, and plasmas) and the forces on them. It has applications in a wide range of disciplines, including mechanical, aerospace, civil, chemical and ...

, #

geomorphology Geomorphology (from Ancient Greek: , ', "earth"; , ', "form"; and , ', "study") is the scientific study of the origin and evolution of topographic and bathymetric features created by physical, chemical or biological processes operating at or ...

# human physiology, ''and'', #

linguistics Linguistics is the scientific study of human language. It is called a scientific study because it entails a comprehensive, systematic, objective, and precise analysis of all aspects of language, particularly its nature and structure. Ling ...

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

Brownian motion Brownian motion, or pedesis (from grc, πήδησις "leaping"), is the random motion of particles suspended in a medium (a liquid or a gas). This pattern of motion typically consists of random fluctuations in a particle's position insi ...

, vascular networks, and shapes of

polymer molecule A polymer (; Greek '' poly-'', "many" + ''-mer'', "part") is a substance or material consisting of very large molecules called macromolecules, composed of many repeating subunits. Due to their broad spectrum of properties, both synthetic and ...

s all relate to fractal curves.

Examples

Blancmange curve In mathematics, the blancmange curve is a self-affine curve constructible by midpoint subdivision. It is also known as the Takagi curve, after Teiji Takagi who described it in 1901, or as the Takagi–Landsberg curve, a generalization of the cur ...

* Coastline paradox *

De Rham curve In mathematics, a de Rham curve is a certain type of fractal curve named in honor of Georges de Rham. The Cantor function, Cesàro curve, Minkowski's question mark function, the Lévy C curve, the blancmange curve, and Koch curve are all spe ...

Dragon curve A dragon curve is any member of a family of self-similar fractal curves, which can be approximated by recursive methods such as Lindenmayer systems. The dragon curve is probably most commonly thought of as the shape that is generated from rep ...

Fibonacci word fractal The Fibonacci word fractal is a fractal curve defined on the plane from the Fibonacci word. Definition This curve is built iteratively by applying, to the Fibonacci word 0100101001001...etc., the Odd–Even Drawing rule: For each digit at po ...

* Koch snowflake * Boundary of the Mandelbrot set * Menger sponge *

Peano curve In geometry, the Peano curve is the first example of a space-filling curve to be discovered, by Giuseppe Peano in 1890. Peano's curve is a surjective, continuous function from the unit interval onto the unit square, however it is not in ...

* Sierpiński triangle *

Trees In botany, a tree is a perennial plant with an elongated stem, or trunk, usually supporting branches and leaves. In some usages, the definition of a tree may be narrower, including only woody plants with secondary growth, plants that are u ...

* Natural fractals *

Weierstrass function In mathematics, the Weierstrass function is an example of a real-valued function that is continuous everywhere but differentiable nowhere. It is an example of a fractal curve. It is named after its discoverer Karl Weierstrass. The Weierstr ...

References

External links and references

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