Rémy Sigrist wrote the paper “The Kochawave curve, a variant of the Koch curve,” so I thought I’d try it out.
KochwavePoints[n_Integer?NonNegative]:=​​Module[{ω=Exp[IPi/3],steps=4^n,z,uk,vk},​​z=ConstantArray[0,steps+1];​​Do[​​uk=DigitCount[k,4,1];​​vk=DigitCount[k,4,2];​​z[[k+2]]=z[[k+1]]+(1+ω)^uk/ω^(2vk),​​{k,0,steps-1}];​​Transpose[{Re[z/3^n],Im[z/3^n]}]];​​KochwaveCurve[n_Integer?NonNegative]:=Line[KochwavePoints[n]];
In[]:=
kp=KochwavePoints[6];Graphics[{EdgeForm[{Thin,Black}],Cyan,Polygon[Append[kp,{0,0}]]},PlotRange->All,AspectRatio->Automatic,ImageSize->800]
Out[]=
A simpler method, but something goes wrong with the last segment:
In[]:=
Graphics@KochCurve2,{1,0},
3
3
,30°,{1,-150°},{1,120°}
Out[]=
A method to get KochCurve working correctly:
In[]:=
Graphics@KochCurve5,{1,0},
3
3
,30°,{1,-150°},{1,120°},{1/100000,0}
Out[]=
Benoit Mandelbrot was born Nov 20, 1924. For his 101st birthday, I’ll be giving a free online talk about fractals at 11AM Chicago time. There are thousands of fractals written in the Wolfram Language, so I’ll be highlighting about a hundred of them.
For example, the Schmidt Arrangements function.
In[]:=
ResourceFunction["SchmidtArrangements"][-3,"MinCurvature"->0,"MaxCurvature"->10,"Color"->{"ColorSaul",{2,1,1}},"Transparency"->"Increasing","Filter"->Rectangle[{-2.1,-Sqrt[3.0]-0.1},{2.1,2*Sqrt[3.0]+0.1}],PlotRange->{{-2,2},{0,Sqrt[3.0]}},ImageSize->Large]
Out[]=
And the Pinwheel fractal:
In[]:=
ResourceFunction["AlgebraicSubstitutionTiling"]["Pinwheel",4,{"N","Start"->{{{0,0},{0,1},{2,1}},{{2,1},{2,0},{0,0}}}}]
Out[]=
Or the supergolden triangle built from the PsiQuad fractal. If you find a complex solution of
2
ψ
+1=
3
ψ
, then the powers of the value give the vertices of a substitution system. It’s a self-generating fractal.
In[]:=
{ψ,ψn,ψp}=x/.Solve[x^2+1==x^3];​​ψvals=ReIm/@{
0
ψp
,
1
ψp
,
2
ψp
,
3
ψp
,
4
ψp
,-
2
ψp
};​​ψtext={"
0
ψ
","
1
ψ
","
2
ψ
","
3
ψ
","
4
ψ
","-
2
ψ
"};​​pols={{1,2,3,4},{2,3,4,5},{5,4,1,6}};​​Graphics[{Opacity[.5],MapIndexed[{Hue[#2[[1]]/3],Polygon[ψvals[[#1]]]}&,pols],​​Opacity[1],Table[Style[Text[ψtext[[n]],ψvals[[n]]],40],{n,1,6}],​​Style[Text["+ 1 =",Mean[ψvals[[{3,4}]]]],40],Disk[{0,0},.02]}]
Out[]=
These lead to the supergolden fractal triangle:
In[]:=
ResourceFunction["AlgebraicSubstitutionTiling"]["PsiQuad",16,{"N","ImageSize"{600,410}}]
Out[]=
The supergolden ratio is an infinite sum of its own reciprocal powers:
Leading to the supergolden triangle nautilus spiral:
For these and more, come to my free online talk about fractals at 11AM Chicago time on Thursday, 20 November.

CITE THIS NOTEBOOK

The Kochawave curve and other fractals​
by Ed Pegg​
Wolfram Community, STAFF PICKS, November 19, 2025
​https://community.wolfram.com/groups/-/m/t/3577688