Pentaflake

Author

Eric W. Weisstein
February 2, 2011

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

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

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

Additional Authors

Recursive growth code by
Michael Trott
The Mathematica GuideBook for Graphics
New York: Springer-Verlag, 2004

Graphic

V5

Show[GraphicsArray[Pentaflake/@Range[0,3]],AspectRatio->Automatic,ImageSize->500]

V6

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

Dimension

-Limit[((Log[(6^n)])/(Log[(((1+GoldenRatio))^(-n))])),n->∞]

N[%,20]

Recursive Growth

This code by Michael Trott

newPolygons[poly_Polygon,fac_]:=Module[{l,mp,r,dir,p0},l=Length[poly[[1]]];mp=Plus@@poly[[1]]/l;
r=Sqrt[#.#]&[mp-poly[[1,1]]]; (* next layer *)Function[p,dir=(p-mp);p0=ArcTan@@-dir;
Polygon[Table[p+fac (dir+r{Cos[p1],Sin[p1]}),{p1,p0,p0+(l-1)/l N[2Pi],N[2Pi]/l}]]]/@If[r==1,poly[[1]],Rest[poly[[1]]]]]
(* display all pentagons *)Show[Graphics[(* iterate pentagon construction *)FoldList[Function[{polys,fac},Flatten[newPolygons[#,fac]&/@polys]],{Polygon[Table[{Cos[φ],Sin[φ]},{φ,N[Pi/2],N[Pi/2]+4/5 N[2Pi],N[2Pi]/5}]]},Table[1/2.6,{5}]]],PlotRange->All,AspectRatio->Automatic];