The first four iterations of the Koch snowflake.

The first seven iterations in animation.

The Koch curve.

The Koch snowflake (or Koch star) is a mathematical curve and one of the earliest fractal curves to have been described. It appeared in a 1904 paper entitled "On a continuous curve without tangents, constructible from elementary geometry" (original French title: "Sur une courbe continue sans tangente, obtenue par une construction géométrique élémentaire") by the Swedish mathematician Helge von Koch. The lesser known Koch curve is the same as the snowflake, except it starts with a line segment instead of an equilateral triangle. The Koch curve is a special case of the de Rham curve. Image File history File links This is a lossless scalable vector image. ... Image File history File links This is a lossless scalable vector image. ... The word iteration is sometimes used in everyday English with a meaning virtually identical to repetition. ... Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... Image File history File links This is a lossless scalable vector image. ... Image File history File links This is a lossless scalable vector image. ... Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... In mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object. ... The boundary of the Mandelbrot set is a famous example of a fractal. ... 1904 (MCMIV) was a leap year starting on a Friday (see link for calendar). ... Leonhard Euler, considered one of the greatest mathematicians of all time A mathematician is a person whose primary area of study and research is the field of mathematics. ... Niels Fabian Helge von Koch (January 25, 1870 - March 11, 1924) was a Swedish mathematician, who gave his name to the famous fractal known as the Koch curve, which was one of the earliest fractal curves to have been described. ... The geometric definition of a line segment In geometry, a line segment is a part of a line that is bounded by two end points, and contains every point on the line between its end points. ... For alternate meanings, such as the musical instrument, see triangle (disambiguation). ... In mathematics, a de Rham curve is a certain type of fractal curve. ...

One can imagine that it was created by starting with a line segment, then recursively altering each line segment as follows:

1. divide the line segment into three segments of equal length.
2. draw an equilateral triangle that has the middle segment from step 1 as its base and points outward.
3. remove the line segment that is the base of the triangle from step 2.

After doing this once the result should be a shape similar to the Star of David. In geometry, an equilateral polygon has all sides of the same length. ... The Star of David The Star of David in the oldest surviving complete copy of the Masoretic text, the Leningrad Codex, dated 1008. ...

The Koch curve is the limit approached as the above steps are followed over and over again.

The Koch curve has an infinite length because each time the steps above are performed on each line segment of the figure there are four times as many line segments, the length of each being one-third the length of the segments in the previous stage. Hence the total length increases by one third and thus the length at step n will be (4/3)ⁿ: the fractal dimension is log 4/log 3 ≈ 1.26, greater than the dimension of a line (1) but less than Peano's space-filling curve (2). In fractal geometry, the fractal dimension, D, is a statistical quantity that gives an indication of how completely a fractal appears to fill space, as one zooms down to finer and finer scales. ... Giuseppe Peano (August 27, 1858 – April 20, 1932) was an Italian mathematician and philosopher. ... Space-filling curves or Peano curves are curves, first described by Giuseppe Peano, whose ranges contain the entire 2-dimensional unit square (or the 3-dimensional unit cube). ...

The Koch curve is continuous but not differentiable anywhere. In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ... In mathematics, the derivative of a function is one of the two central concepts of calculus. ...

The area of the Koch snowflake is , when s is the measure of one side of the original triangle, and so an infinite perimeter encloses a finite area.^[1]

Representation as Lindenmayer System

The Koch Curve can be expressed by a rewrite system (Lindenmayer System). Rewriting in mathematics, computer science and logic covers a wide range of methods of transforming strings, written in some fixed alphabet, that are not deterministic but are governed by explicit rules. ... 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. ...

Alphabet : F

Constants : +, −

Axiom : F++F++F

Production rules:

F → F−F++F−F

Here, F means "draw forward", + means "turn right 60°", and - means "turn left 60°" (see turtle graphics). The Logo programming language is an adaptation by Wally Feurzeig and Seymour Papert of the Lisp programming language that is easier to read. ...

Implementations

Below are a variety of implementations of the Koch snowflake.

Logo

Below is a recursive implementation in Logo. It can be tried out with most implementations of Logo, or online with the Java implementation XLogo. The Logo programming language is a functional programming language. ... “Java language” redirects here. ...

Try start, call rt 30 koch 100.

 to koch :x repeat 3 [triline :x rt 120] end to triline :x if :x < 1 [fd :x] [triline :x/3 lt 60 triline :x/3 rt 120 triline :x/3 lt 60 triline :x/3] end 

Web Turtle

Here follows a sample implementation of the Koch curve for a Turtle robot written in a Logo-like language. It can be tried out online with Web Turtle. Change the value of A in the first line to any number from 1 to 5 to see the different levels of complexity. The Logo programming language is a functional programming language. ...

 LET A 5 ; calculate adjusted side-length LET B 243 REPEAT A LET B B/3 NEXT ; place pointer POINT 150 MOVE 140 POINT 0 ; start GO SIDE RIGHT 120 GO SIDE RIGHT 120 GO SIDE ; finished. END ; main loop # SIDE GO F LEFT 60 GO F RIGHT 120 GO F LEFT 60 GO F RETURN ; forward # F IF A > 1 ; go deeper depending on level LET A A-1 GO SIDE LET A A+1 ELSE ; or just do a single line DRAW B ENDIF RETURN 

Python

Here is the Koch curve in Python.

 import turtle set="F" for i in range(5): set=set.replace("F","FLFRFLF") turtle.down() for move in set: if move is "F": turtle.forward(100.0/3**i) if move is "L": turtle.left(60) if move is "R": turtle.right(120) input () 

Popular Culture

In popular culture, the investigation of Koch's Snowflake can form a piece of Coursework at GCSE level in Great Britain. Coursework refers to work carried out by students while they are studying a course at university or school that contributes towards their overall grade, but which is assessed separately from their final exams. ... GCSE is an acronym that can refer to: General Certificate of Secondary Education global common subexpression elimination - an optimisation technique used by some compilers This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ...

It is also an optional Portfolio Task for the IB's (International Baccalaureate) Standard Level Math coursework, or an IB Higher Level Type 1 Portfolio task. It has been suggested that this article or section be merged into International Baccalaureate Organization. ...

Alan Moore compares the multitude of increasingly outlandish Ripper theories to a Koch snowflake in his graphic novel From Hell, where a finite, fixed location, event and era (London, in late 1888) can have an infinite number of nooks and crannies. Alan Moore (born November 18, 1953, in Northampton) is an English writer most famous for his influential work in comics, including the acclaimed graphic novels Watchmen, V for Vendetta and From Hell. ... From Hell is a graphic novel by writer Alan Moore and artist Eddie Campbell speculating upon the identity and motives of Jack the Ripper. ...

Other

Eric Haines has developed the sphereflake fractal, which is a three-dimensional version of the Koch snowflake. 2-dimensional renderings (ie. ...

See also

Wikimedia Commons has media related to:

Koch curve

Wikimedia Commons has media related to:

Koch snowflake

• Fractal
• List of fractals by Hausdorff dimension

Image File history File links Commons-logo. ... Image File history File links Commons-logo. ... The boundary of the Mandelbrot set is a famous example of a fractal. ... A fractal is a geometric object whose Hausdorff dimension (δ) strictly exceeds its topological dimension. ...

References

1. ^ http://ecademy.agnesscott.edu/~lriddle/ifs/ksnow/ksnow.htm