KOCH SNOWFLAKE

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.
One can imagine that it was created by starting with a line segment, then recursively altering each line segment as follows:
# divide the line segment into three segments of equal length.
# draw an equilateral triangle that has the middle segment from step 1 as its base and points outward.
# 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.
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).
The Koch curve is continuous but not differentiable anywhere.
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]

Contents

Representation as Lindenmayer System

Implementations

Logo

Web Turtle

Python

Popular Culture

Other

See also

References

Representation as Lindenmayer System

The Koch Curve can be expressed by a rewrite system (Lindenmayer System).
:'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).

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.
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.
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.
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.
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.

Other

Eric Haines has developed the 'sphereflake fractal', which is a three-dimensional version of the Koch snowflake.

See also

★ Fractal

★ List of fractals by Hausdorff dimension

References

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