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 il ...

. 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 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 a ...

longer than a single point has

infinite length Infinite may refer to: Mathematics *Infinite set, a set that is not a finite set *Infinity, an abstract concept describing something without any limit Music * Infinite (group), a South Korean boy band *''Infinite'' (EP), debut EP of American m ...

. A famous example is the boundary of the

Mandelbrot set The Mandelbrot set () is the set of complex numbers c for which the function f_c(z)=z^2+c does not diverge to infinity when iterated from z=0, i.e., for which the sequence f_c(0), f_c(f_c(0)), etc., remains bounded in absolute value. This ...

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 ar ...

, 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 clas ...

snowflakes A snowflake is a single ice crystal that has achieved a sufficient size, and may have amalgamated with others, which falls through the Earth's atmosphere as snow.Knight, C.; Knight, N. (1973). Snow crystals. Scientific American, vol. 228, no. ...

, 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 throughout the world. They range from . Geckos ...

, frost crystals, 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 approxi ...

, dendrite crystal, trees, fractals,

Hofstadter's butterfly In condensed matter physics, Hofstadter's butterfly is a graph of the spectral properties of non-interacting two-dimensional electrons in a perpendicular magnetic field in a lattice. The fractal, self-similar nature of the spectrum was discovere ...

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 Self-organized criticality (SOC) is a property of dynamical systems that have a critical point as an attractor. Their macroscopic behavior thus displays the spatial or temporal scale-invariance characteristic of the critical point of a phase ...

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 coor ...

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 meas ...

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 illust ...

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 phy ...

and others have studied

self-similarity __NOTOC__ In mathematics, a self-similar object is exactly or approximately similar to a part of itself (i.e., the whole has the same shape as one or more of the parts). Many objects in the real world, such as coastlines, are statistically s ...

of fractal curves, and have applied theory of fractals to modelling

natural phenomena 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 p ...

. 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 analy ...

, #

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 The human body is the structure of a human being. It is composed of many different types of cells that together create tissues and subsequently organ systems. They ensure homeostasis and the viability of the human body. It comprises a head, ...

, ''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. Lingu ...

. As examples, "landscapes" revealed by

microscopic view A microscope () is a laboratory instrument used to examine objects that are too small to be seen by the naked eye. Microscopy is the science of investigating small objects and structures using a microscope. Microscopic means being invisible ...

s of

surface A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is t ...

s 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 network The blood circulatory system is a system of organs that includes the heart, blood vessels, and blood which is circulated throughout the entire body of a human or other vertebrate. It includes the cardiovascular system, or vascular system, tha ...

s, 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 *

Coastline paradox The coastline paradox is the counterintuitive observation that the coastline of a landmass does not have a well-defined length. This results from the fractal curve-like properties of coastlines; i.e., the fact that a coastline typically has a ...

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 s ...

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 p ...

Koch snowflake The Koch snowflake (also known as the Koch curve, Koch star, or Koch island) is a fractal curve and one of the earliest fractals to have been described. It is based on the Koch curve, which appeared in a 1904 paper titled "On a Continuous Cur ...

* 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 S ...

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 injective ...

Sierpiński triangle The Sierpiński triangle (sometimes spelled ''Sierpinski''), also called the Sierpiński gasket or Sierpiński sieve, is a fractal attractive fixed set with the overall shape of an equilateral triangle, subdivided recursively into smaller equi ...

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 ...

* 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 Weierst ...

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