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| The '''Gosper curve''', also known as '''Peano-Gosper Curve''',<ref>{{cite web|last=Weisstein, Eric W.|title=Peano-Gosper Curve|url=http://mathworld.wolfram.com/Peano-GosperCurve.html|publisher=[[MathWorld]]|accessdate=31 October 2013}}</ref> named after [[Bill Gosper]], also known as the '''flowsnake''' (a [[spoonerism]] of [[Koch snowflake|snowflake]]), is a [[space-filling curve]]. It is a [[fractal]] object similar in its construction to the [[dragon curve]] and the [[Hilbert curve]].
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| {|
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| |[[Image:Gosper curve 3.svg]]||[[Image:Gosper curve 1.svg]]
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| |-
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| |A fourth-stage Gosper curve ||The line from the red to the green point shows a single step of the Gosper curve construction.
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| |}
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| == Algorithm ==
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| ===Lindenmayer System===
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| The Gosper curve can be represented using an [[L-System]] with rules as follows:
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| * Angle: 60°
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| * Axiom: <math>A</math>
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| * Replacement rules:
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| ** <math>A \mapsto A-B--B+A++AA+B-</math>
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| ** <math>B \mapsto +A-BB--B-A++A+B</math>
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| In this case both A and B mean to move forward, + means to turn left 60 degrees and - means to turn right 60 degrees - using a "turtle"-style program such as [[Logo programming language|Logo]].
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| === Logo ===
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| A [[Logo programming language|Logo]] program to draw the Gosper curve using [[turtle graphics]] ([http://logo.twentygototen.org/mJjiNzK0 online version]):
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| <code> | |
| to rg :st :ln
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| make "st :st - 1
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| make "ln :ln / 2.6457
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| if :st > 0 [rg :st :ln rt 60 gl :st :ln rt 120 gl :st :ln lt 60 rg :st :ln lt 120 rg :st :ln rg :st :ln lt 60 gl :st :ln rt 60]
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| if :st = 0 [fd :ln rt 60 fd :ln rt 120 fd :ln lt 60 fd :ln lt 120 fd :ln fd :ln lt 60 fd :ln rt 60]
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| end
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|
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| to gl :st :ln
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| make "st :st - 1
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| make "ln :ln / 2.6457
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| if :st > 0 [lt 60 rg :st :ln rt 60 gl :st :ln gl :st :ln rt 120 gl :st :ln rt 60 rg :st :ln lt 120 rg :st :ln lt 60 gl :st :ln]
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| if :st = 0 [lt 60 fd :ln rt 60 fd :ln fd :ln rt 120 fd :ln rt 60 fd :ln lt 120 fd :ln lt 60 fd :ln]
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| end
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| </code>
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| The program can be invoked, for example, with <code>rg 4 300</code>, or alternatively <code>gl 4 300</code>.
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| The constant 2.6457 in the program code is an approximation of √7.
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| ==Properties==
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| The space filled by the curve is called the '''Gosper island'''. The first few iterations of it are shown below:
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| {| align=center
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| |[[Image:Gosper Island 0.svg|180px]]
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| |[[Image:Gosper Island 1.svg|180px]]
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| |[[Image:Gosper Island 2.svg|180px]]
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| |[[Image:Gosper Island 3.svg|180px]]
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| |[[Image:Gosper Island 4.svg|180px]]
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| |}
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| The Gosper Island can [[tessellation|tile]] the [[Plane (mathematics)|plane]]. In fact, seven copies of the Gosper island can be joined together to form a shape that is [[Similarity (geometry)|similar]], but scaled up by a factor of √7 in all dimensions. As can be seen from the diagram below, performing this operation with an intermediate iteration of the island leads to a scaled-up version of the next iteration. Repeating this process indefinitely produces a [[tessellation]] of the plane. The curve itself can likewise be extended to an infinite curve filling the whole plane.
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| {| align=center
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| |[[Image:Gosper Island Tesselation 2.svg|240px]]
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| |[[image:Gosper Island Tesselation.svg|240px]]
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| |}
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| ==See also==
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| *[[List of fractals by Hausdorff dimension]]
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| ==References==
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| {{reflist}}
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| ==External links==
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| *http://kilin.u-shizuoka-ken.ac.jp/museum/gosperex/343-024.pdf
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| *http://kilin.clas.kitasato-u.ac.jp/museum/gosperex/343-024.pdf
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| *http://www.mathcurve.com/fractals/gosper/gosper.shtml (in French)
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| *http://mathworld.wolfram.com/GosperIsland.html
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| *http://logo.twentygototen.org/mJjiNzK0
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| *http://80386.nl/projects/flowsnake/
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| [[Category:Fractal curves]]
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