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| - | The '''Koch snowflake''' (or '''Koch star''') is a [[mathematics|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 [[Sweden|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]].
| + | You grab one of the woman's ankles and stuff her hosed foot into the ex-virgin's mouth. The deflowered virgin moans and the woman coos passionately. You grab the woman's other foot and inhale deeply of her aroma. The scent makes your dick swell inside the ex-virgin under you. The woman slides her hand inside her skirt and frigs herself furiously as the virgin hungrily sucks on her foot. You thrust your dick in and out of the ex-virgin and the woman curls her hosed toes around your nose. The ex-virgin bobs her head up and down on the woman's foot and if she were giving the woman's toes a blowjob. You pump the ex-virgin, inhale the woman's odor and the combination makes your dick fire off blast after blast of sperm inside the deflowered virgin. The virgin shivers and her pussy grips you tightly as she cums. The woman stiffens and her toes clench around your nose as she gets herself off. |
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| - | One can imagine that it was created by starting with a line segment, then recursively altering each line segment as follows:
| + | Do you: |
| - | # divide the line segment into three segments of equal length.
| + | *[[Take them both back to your apartment]] |
| - | # draw an [[equilateral]] triangle that has the middle segment from step 1 as its base and points outward.
| + | *[[Keep smelling the woman's hose and hump the ex-virgin harder]] |
| - | # remove the line segment that is the base of the triangle from step 2.
| + | {{SexRompStatus|Location=''[[The Food Mart]]''|Health=Horny|MP=0|Level=1}} |
| - | | + | [[Category: Smutty Sex Romp]] |
| - | After doing this once the result is a shape similar to the [[Star of David]].
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| - | The Koch curve is the limit approached as the above steps are followed over and over again.
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| - | 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)<sup>n</sup>: 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).
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| - | The Koch curve is [[continuous function|continuous]] but not [[differentiable]] anywhere.
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| - | The area of the Koch snowflake is <math>\\frac{2\\sqrt{3}s^2}{5}</math>, where ''s'' is the measure of one side of the original triangle, and so an infinite perimeter encloses a finite area.<ref>http://ecademy.agnesscott.edu/~lriddle/ifs/ksnow/ksnow.htm</ref>
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| - | As noted in the article on [[geometric series]], the area of the Koch snowflake is 8/5 times the area of the base triangle.
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| - | | + | |
| - | ==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
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| - | | + | |
| - | Here, ''F'' means "draw forward", ''+'' means "turn right 60°", and ''-'' means "turn left 60°" (see [[turtle graphics]]).
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| - | | + | |
| - | ==Implementations==
| + | |
| - | Below are a variety of implementations of the Koch snowflake.
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| - | | + | |
| - | ===Logo===
| + | |
| - | Below is a recursive implementation in [[Logo (programming language)|Logo]]. It can be tried out with most implementations of Logo, or online with the [[Java (programming language)|Java]] implementation [http://xlogo.tuxfamily.org/en/index-en.html XLogo].
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| - | Try start, call <code>rt 30 koch 100</code>.
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| - | to koch :x
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| - | 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
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| - | | + | |
| - | ===Web Turtle=== | + | |
| - | Here follows a sample implementation of the Koch curve for a '''Turtle robot''' written in a [[Logo (programming language)|Logo]]-like language. It can be tried out online with [http://www.sonic.net/~nbs/webturtle/webturtle.cgi 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.
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| - | | + | |
| - | LET A 5
| + | |
| - | ; calculate adjusted side-length
| + | |
| - | LET B 243
| + | |
| - | REPEAT A
| + | |
| - | LET B B/3
| + | |
| - | NEXT
| + | |
| - | ; place pointer
| + | |
| - | POINT 150
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| - | MOVE 140
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| - | POINT 0
| + | |
| - | ; start
| + | |
| - | GO SIDE
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| - | RIGHT 120
| + | |
| - | GO SIDE
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| - | RIGHT 120
| + | |
| - | GO SIDE
| + | |
| - | ; finished.
| + | |
| - | END
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| - | | + | |
| - | ; main loop
| + | |
| - | # SIDE
| + | |
| - | GO F
| + | |
| - | LEFT 60
| + | |
| - | GO F
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| - | RIGHT 120
| + | |
| - | GO F
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| - | LEFT 60
| + | |
| - | GO F
| + | |
| - | RETURN
| + | |
| - | | + | |
| - | ; forward
| + | |
| - | # F
| + | |
| - | IF A > 1
| + | |
| - | ; go deeper depending on level
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| - | LET A A-1
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| - | GO SIDE
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| - | LET A A+1
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| - | ELSE
| + | |
| - | ; or just do a single line
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| - | DRAW B
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| - | ENDIF
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| - | RETURN
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| - | ===Python=== | + | |
| - | Here is the Koch curve in Python.
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| - | import turtle
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| - | set="F"
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| - | for i in range(5): set=set.replace("F","FLFRFLF")
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| - | turtle.down()
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| - | for move in set:
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| - | if move is "F": turtle.forward(100.0/3**i)
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| - | if move is "L": turtle.left(60)
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| - | if move is "R": turtle.right(120)
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| - | input ()
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| - | | + | |
| - | The program can be easily modified to show the entire snowflake:
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| - | | + | |
| - | import turtle
| + | |
| - | set="F"
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| - | for i in range(5): set=set.replace("F","FLFRFLF")
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| - | set=set+"R"+set+"R"+set
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| - | turtle.down()
| + | |
| - | for move in set:
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| - | if move is "F": turtle.forward(100.0/3**i)
| + | |
| - | if move is "L": turtle.left(60)
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| - | if move is "R": turtle.right(120)
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| - | input ()
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| - | == Variants of the von Koch curve ==
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| - | Following von Koch's concept, several variants of the von Koch curve were designed, considering right angles (quadratic), other angles (Cesaro) or circles and their extensions to higher dimensions (Sphereflake),
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| - | {| border="0" cellpadding="4" rules="all" style="border: 1px solid #999; background-color:#FFFFFF"
| + | |
| - | |- align="center" bgcolor="#cccccc"
| + | |
| - | ! Variant || Illustration|| Construction||
| + | |
| - | |-
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| - | | 1D & angle=85°|| [[Image:Koch Curve 85degrees.png|thumb|150px|Cesaro fractal. ]]|| The Cesaro fractal is a variant of the von Koch curve with an angle between 60° and 90° (here 85°).
| + | |
| - | |-
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| - | | 1D & 90° angle || [[Image:Quadratic Koch 2.png|thumb|150px|Quadratic type 1 curve]]|| align="left"| [[Image:Quadratic Koch curve type1 iterations.png|thumb|450px| The first 2 iterations.]]
| + | |
| - | |-
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| - | | 1D & 90° angle || [[Image:Quadratic Koch.png|thumb|150px|Quadratic type 2 curve]]|| align="left"| [[Image:Quadratic Koch curve type2 iterations.png|thumb|450px| The first 2 iterations. Its fractal dimension equals 1.5 and is exactly half-way between dimension 1 and 2. It is therefore often chosen when studying the physical properties of non-integer fractal objects.]]
| + | |
| - | |-
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| - | | 2D & triangles|| [[Image:Koch surface.png|thumb|150px|von Koch surface]]|| [[Image:Koch surface iterations.png|thumb|450px| The first 2 iterations. Natural extension of the von Koch curve in 2 dimensions.]]
| + | |
| - | |-
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| - | | 2D & 90° angle|| [[Image:Quadratic Koch 3D (type1).png|thumb|150px|Quadratic type 1 surface]]|| Extension of the quadratic type 1 curve. The Illustration on the left shows the fractal after the second iteration
| + | |
| - | |-
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| - | | 2D & 90° angle|| [[Image:Quadratic Koch 3D.png|thumb|150px|Quadratic type 2 surface]]|| Extension of the quadratic type 2 curve. The Illustration on the left shows the fractal after the first iteration.
| + | |
| - | |-
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| - | | 2D & spheres || || Eric Haines has developed the '''sphereflake fractal''', which is a three-dimensional version of the Koch snowflake, using spheres.
| + | |
| - | |}
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| - | == See also ==
| + | |
| - | {{Commons|Koch curve}}
| + | |
| - | {{Commons|Koch snowflake}}
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| - | | + | |
| - | * [[List of fractals by Hausdorff dimension]]
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| - | == References ==
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| - | <references/>
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| - | [[Category:Fractal curves]] | + | |
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| - | [[bn:কচ স্নোফ্লেক]]
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| - | [[ca:Floc de neu de Koch]]
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| - | [[cs:Kochova křivka]]
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| - | [[de:Koch-Kurve]]
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| - | [[es:Copo de nieve de Koch]]
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| - | [[eo:Neĝero de Koch]]
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| - | [[fr:Flocon de Koch]]
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| - | [[gl:Curva de Koch]]
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| - | [[ko:코흐 곡선]]
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| - | [[hr:Kochova krivulja]]
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| - | [[it:Curva di Koch]]
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| - | [[he:פתית השלג של קוך]]
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| - | [[hu:Koch-görbe]]
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| - | [[ja:コッホ曲線]]
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| - | [[pl:Krzywa Kocha]]
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| - | [[pt:Curva de Koch]]
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| - | [[ru:Кривая Коха]]
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| - | [[sk:Kochova krivka]]
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| - | [[sl:Kochova snežinka]]
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| - | [[sr:Кохова пахуља]]
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| - | [[fi:Kochin käyrä]]
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| - | [[sv:Von Kochs kurva]]
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| - | [[th:เกล็ดหิมะค็อค]]
| + | |
You grab one of the woman's ankles and stuff her hosed foot into the ex-virgin's mouth. The deflowered virgin moans and the woman coos passionately. You grab the woman's other foot and inhale deeply of her aroma. The scent makes your dick swell inside the ex-virgin under you. The woman slides her hand inside her skirt and frigs herself furiously as the virgin hungrily sucks on her foot. You thrust your dick in and out of the ex-virgin and the woman curls her hosed toes around your nose. The ex-virgin bobs her head up and down on the woman's foot and if she were giving the woman's toes a blowjob. You pump the ex-virgin, inhale the woman's odor and the combination makes your dick fire off blast after blast of sperm inside the deflowered virgin. The virgin shivers and her pussy grips you tightly as she cums. The woman stiffens and her toes clench around your nose as she gets herself off.
