Modified internal rate of return: Difference between revisions

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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]].
 
{|
|[[Image:Gosper curve 3.svg]]||[[Image:Gosper curve 1.svg]]
|-
|A fourth-stage Gosper curve ||The line from the red to the green point shows a single step of the Gosper curve construction.
|}
 
== Algorithm ==
 
===Lindenmayer System===
 
The Gosper curve can be represented using an [[L-System]] with rules as follows:
* Angle: 60°
* Axiom: <math>A</math>
* Replacement rules:
** <math>A \mapsto A-B--B+A++AA+B-</math>
** <math>B \mapsto +A-BB--B-A++A+B</math>
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]].
 
=== Logo ===
A [[Logo programming language|Logo]] program to draw the Gosper curve using [[turtle graphics]] ([http://logo.twentygototen.org/mJjiNzK0 online version]):
<code>
to rg :st :ln
make "st :st - 1
make "ln :ln / 2.6457
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]
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]
end
to gl :st :ln
make "st :st - 1
make "ln :ln / 2.6457
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]
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]
end
</code>
 
The program can be invoked, for example, with <code>rg 4 300</code>, or alternatively <code>gl 4 300</code>.
 
The constant 2.6457 in the program code is an approximation of √7.
 
==Properties==
The space filled by the curve is called the '''Gosper island'''. The first few iterations of it are shown below:
{| align=center
|[[Image:Gosper Island 0.svg|180px]]
|[[Image:Gosper Island 1.svg|180px]]
|[[Image:Gosper Island 2.svg|180px]]
|[[Image:Gosper Island 3.svg|180px]]
|[[Image:Gosper Island 4.svg|180px]]
|}
 
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.
 
{| align=center
|[[Image:Gosper Island Tesselation 2.svg|240px]]
|[[image:Gosper Island Tesselation.svg|240px]]
|}
 
==See also==
*[[List of fractals by Hausdorff dimension]]
 
==References==
{{reflist}}
 
==External links==
*http://kilin.u-shizuoka-ken.ac.jp/museum/gosperex/343-024.pdf
*http://kilin.clas.kitasato-u.ac.jp/museum/gosperex/343-024.pdf
*http://www.mathcurve.com/fractals/gosper/gosper.shtml (in French)
*http://mathworld.wolfram.com/GosperIsland.html
*http://logo.twentygototen.org/mJjiNzK0
*http://80386.nl/projects/flowsnake/
 
[[Category:Fractal curves]]

Revision as of 12:39, 13 December 2013

The Gosper curve, also known as Peano-Gosper Curve,[1] named after Bill Gosper, also known as the flowsnake (a spoonerism of snowflake), is a space-filling curve. It is a fractal object similar in its construction to the dragon curve and the Hilbert curve.

A fourth-stage Gosper curve The line from the red to the green point shows a single step of the Gosper curve construction.

Algorithm

Lindenmayer System

The Gosper curve can be represented using an L-System with rules as follows:

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.

A Logo program to draw the Gosper curve using turtle graphics (online version): 

to rg :st :ln
make "st :st - 1
make "ln :ln / 2.6457
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]
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]
end

to gl :st :ln
make "st :st - 1
make "ln :ln / 2.6457
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]
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]
end

The program can be invoked, for example, with rg 4 300, or alternatively gl 4 300.

The constant 2.6457 in the program code is an approximation of √7.

Properties

The space filled by the curve is called the Gosper island. The first few iterations of it are shown below:

The Gosper Island can tile the plane. In fact, seven copies of the Gosper island can be joined together to form a shape that is 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.

See also

References

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External links

  1. ↑ Template:Cite web
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