Gosper Island

Author

Eric W. Weisstein
January 6, 2011

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Fractals/GosperIsland.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/GosperIsland.html.

©2010 Wolfram Research, Inc. except for portions noted otherwise

Figure

V5

CurveArray[GosperIsland,{0,3},1000,ImageSize->500]

V6

GraphicsRow[GosperIsland/@Range[0,3]]

Area

Show[Table[Graphics[{Hue[i/4],GosperIsland[i][[1]]}],{i,0,3}],AspectRatio->Automatic]

area=Table[Area[Evaluate[Polygon[Most[GosperIsland[n,{0,0},8000][[1,1]]]]]],{n,0,5}]

N[%]

Clusters

FlowSnakeNoPlot[n_,pos_List]:= 
  TurtlePath[{"F"->"F+F-F"},"-F+F+F+F+F+F",n,60,3.^-n,{pos[[1]],pos[[2]]}]

FlowSnakeArray[2]=Module[{},
  dx=N[2/Sqrt[3]]; dy=1.351;
  pos={{0.,0.},{0.,dy},{dx,-.5dy},{dx,.5dy},{dx,1.5dy},{2dx,0.},{2dx,dy}};
  Show[Graphics[
    Table[Line[FlowSnakeNoPlot[2,pos[[i]]]],{i,1,Length[pos]}]],AspectRatio->Automatic]
]

FlowSnakeArray[3]=Module[{},
  dx=1.05; ddx=.045; dy=1.18; ddy=.045;
  pos={{0.,0.},{ddx,dy},{dx-ddx,-.5dy-ddy},{dx,.5dy-ddy},{dx+ddx,1.5dy-ddy},
    {2dx-ddx,-2ddy},{2dx,dy-2ddy}};
  Show[Graphics[
    Table[Line[FlowSnakeNoPlot[3,pos[[i]]]],{i,1,Length[pos]}]],AspectRatio->Automatic]
]