In mathematics, the Lévy C curve is a self similar fractal that was first described and investigated by French mathematician Paul Lévy in his 1938 paper Plane or Space Curves and Surfaces Consisting of Parts Similar to the Whole. Mathematics, often abbreviated maths in Commonwealth English and math in American English, is the study of abstraction. ... A self-similar object is exactly or approximately similar to a part of itself. ... The Mandelbrot set, named after its discoverer, is a famous example of a fractal. ... A mathematician is a person whose area of study and research is mathematics. ... Paul Pierre Lévy (September 15, 1886 - December 15, 1971) was a French mathematician who was active especially in probability theory, introduced martingales and Lévy flights. ...

L-system construction:

Lévy C curve (from a L-system, after the first 12 stages)

If using a Lindenmayer system then the construction of the C curve starts with a straight line. An isosceles triangle with angles of 45°, 90° and 45° is built using this line as its hypotenuse. The original line is then replaced by the other two sides of this triangle. Levy C Curve Created by Gandalf61 using Fractint File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... Levy C Curve Created by Gandalf61 using Fractint File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... An L-system or Lindenmayer system is a formal grammar (a set of rules and symbols) most famously used to model the growth processes of plant development, though able to model the morphology of a variety of organisms. ... For alternate meanings, such as the musical instrument, see triangle (disambiguation). ... For alternate meanings, such as the musical instrument, see triangle (disambiguation). ...

At the second stage, the two new lines each form the base for another right-angled isosceles triangle, and are replaced by the other two sides of their respective triangle. So, after two stages, the curve takes the appearance of three sides of a rectangle with the same length as the original line, but only half as wide.

At each subsequent stage, each straight line segment in the curve is replaced by the other two sides of a right-angled isosceles triangle built on it. After n stages the curve consists of 2ⁿ line segments, each of which is smaller than the original line by a factor of 2^n/2.

The fractal curve that is the limit of this "infinite" process is the Lévy C curve. It takes its name from its resemblance to a highly ornamented version of the letter "C".

Variations:

The standard C curve is built using 45° isosceles triangles. Variations of the C curve can be constructed by using isosceles triangles with angles other than 45°. As long as the angle is less than 60°, the new lines introduced at each stage are each shorter than the lines that they replace, so the construction process tends towards a limit curve. Angles less than 45° produce a fractal that is less tightly "curled".

IFS construction:

Lévy C curve (from IFS, infinite levels)

If using a iterated function system (IFS, or the random game IFS-method actually), then the construction of the C curve is a bit easier. It will need two sets of "rules" and that is: Two points in a plane (the translators), each associated with a scale factor of 1 / sqrt(2). The first rule will have a rotation of 45° and the second -45°. This set will iterate a point [x, y] from randomly choosing any of the two rules and use the parameters associated with the rule to scale/rotate and translate the point using a 2D-transform function. Menger sponge, created by using IFS. Iterated function systems or IFS, are a kind of fractal that was conceived in its present form by John Hutchinson in 1981 and popularized by Michael Barnsleys book Fractals Everywhere. ... The word point can refer to: a location in physical space a unit of angular measurement; see navigation point is a typographic unit of measure in typography equal inch or sometimes approximated as inch; on computer displays it should be equal to point in typography if the correct display resolution... Plane may refer to: An Aeroplane or airplane, a type of fixed-wing aircraft. ... Translation is an activity comprising the interpretation of the meaning of a text in one language—the called the source text—and the production of a new, equivalent text in another language—called the target text, or the translation. ... Scale (botany) Scale (zoology) Scale (medical) Scale (music) Scale (measurement) Scale (chemical) Scale (social sciences) Scale (spatial) Scale (computing) Order of magnitude Logarithmic scale Scale model Architects scale Engineers scale This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same... A factor can be: a person acting as a mercantile agent a number that is a divisor of another number in mathematics; an enzyme in biochemistry. ... Iteration is the repetition of a process, typically within a computer program. ... Transformation is the genetic alteration of a cell resulting from the introduction, uptake and expression of foreign genetic material (DNA or RNA). ...