Kronecker product based fractals

Kronecker product based fractals
You are encouraged to solve this task according to the task description, using any language you may know.

This task is based on Kronecker product of two matrices. If your language has no a built-in function for such product then you need to implement it first.
The essence of fractals is self-replication (at least, self-similar replications). So, using n times self-product of the matrix (filled with 0/1) we will have a fractal of the n-th order.
Even just looking at the resultant matrix you can see it.
There are virtually infinitely many fractals of this type. You are limited only by your creativity and the power of your computer.

Task

Using Kronecker product implement and show two popular and well-known fractals, i.e.:

• Vicsek fractal;
• Sierpinski carpet fractal.

The last one ( Sierpinski carpet) is already here on RC, but built using different approaches.

Test cases

These 2 fractals (order 4 at least) should be built using the following 2 simple matrices:

|0 1 0|	and |1 1 1|
|1 1 1|	    |1 0 1|
|0 1 0|	    |1 1 1|

Note

• Output could be a graphical or ASCII-art representation, but if an order is set > 4 then printing is not suitable.
• The orientation and distortion of the fractal could be your language/tool specific.
• It would be nice to see one additional fractal of your choice, e.g., based on using a single (double) letter(s) of an alphabet, any sign(s) or alredy made a resultant matrix of the Kronecker product.

See implementations and results below in JavaScript, PARI/GP and R languages. They have additional samples of "H", "+" and checkerboard fractals.

gnuplot

File for the load command is the only possible imitation of the fine function in the gnuplot.

Note

• Find plotff.gp here on RC
• dat-files are PARI/GP generated output files. They are too big to post them here on RC.
  

Works with: gnuplot version 5.0 (patchlevel 3) and above

File:Pkf1.png

Output pkf1.png

File:Pkf2.png

Output pkf2.png

File:Pkf3.png

Output pkf3.png

<lang gnuplot>

1. 1. KPF.gp 4/8/17 aev
   2. Plotting 3 KPF pictures.
   3. dat-files are PARI/GP generated output files:
2. cd 'C:\gnupData'

1. 1. PKF1 from PARI/GP created file pkf1.dat

ttl = "Vicsec fractal"; clr = '"blue"'; filename = "pkf1"; load "plotff.gp"

1. 1. PKF2 from PARI/GP created file pkf2.dat

ttl = "Sierpinski carpet fractal"; clr = '"navy"';filename = "pkf2"; load "plotff.gp"

1. 1. PKF3 from PARI/GP created file pkf3.dat

ttl = "Sierpinski triangle fractal"; clr = '"dark-green"'; filename = "pkf3"; load "plotff.gp" </lang>

Output:

3 plotted files: pkf1.png, pkf2.png and pkf3.png.

JavaScript

Using Version #1 of Kronecker product in JavaScript.

Works with: Chrome

File:VicsekFractaljs.png

Output VicsekFractaljs.png

File:SierpCarpetFractaljs.png

Output SierpCarpetFractaljs.png

File:CheckbrdFractaljs.png

Output CheckbrdFractaljs.png

<lang javascript> // KPF.js 6/23/16 aev // HFJS: Plot any matrix mat (filled with 0,1) function pmat01(mat, color) {

 // DCLs
 var cvs = document.getElementById('canvId');
 var ctx = cvs.getContext("2d"); 
 var w = cvs.width; var h = cvs.height;
 var m = mat[0].length; var n = mat.length;
 // Cleaning canvas and setting plotting color 
 ctx.fillStyle="white"; ctx.fillRect(0,0,w,h);
 ctx.fillStyle=color;
 // MAIN LOOP
 for(var i=0; i<m; i++) {
   for(var j=0; j<n; j++) {
     if(mat[i][j]==1) { ctx.fillRect(i,j,1,1)};
   }//fend j
 }//fend i

}//func end // Prime functions: // Create Kronecker product based fractal matrix rm from matrix m (order=ord) function ckpbfmat(m,ord) {

 var rm=m;
 for(var i=1; i<ord; i++) {rm=mkp(rm,m)};
 //matpp2doc('R 4 ordd',rm,'*'); // ASCII "plotting" - if you wish to try.
 return(rm);

} // Create and plot Kronecker product based fractal from matrix m (filled with 0/1) function cpmat(m,ord,color) {

 var kpr;
 kpr=ckpbfmat(m,ord);
 pmat01(kpr,color);

} // Fractal matrix "pretty" printing to document. // mat should be filled with 0 and 1; chr is a char substituting 1. function matpp2doc(title,mat,chr) {

 var i,j,re=,e; var m=mat.length; var n=mat[0].length;

document.write('  '+title+':

');
  for(var i=0; i<m; i++) {
    for(var j=0; j<n; j++) {
      e=' '; if(mat[i][j]==1) {e=chr}; re+=e; 
    }//fend j
    document.write('  '+re+'<br />'); re='';
  }//fend i
  document.write('

');

} // mkp function (exotic arrow function): Return the Kronecker product // of the a and b matrices mkp=(a,b)=>a.map(a=>b.map(b=>a.map(y=>b.map(x=>r.push(y*x)),t.push(r=[]))),t=[])&&t; </lang>

Required tests

<lang html> <html> <head>

 <title>Vicsek fractal</title>
 <script src="KPF.js"></script>

</head> <body onload="cpmat([[0,1,0],[1,1,1],[0,1,0]],6,'navy')">

Vicsek fractal

  <a href="SierpCarpetFractal.html"> Next: Sierpinski carpet fractal</a>

  <canvas id="canvId" width="750" height="750" style="border: 1px outset;"></canvas>

</body></html> </lang>

<lang html> <html> <head>

 <title>Sierpinski carpet fractal</title>
 <script src="KPF.js"></script>

</head> <body onload="cpmat([[1,1,1],[1,0,1],[1,1,1]],6,'brown')">

Sierpinski carpet fractal

  <a href="Checkerboard.html"/> Next: Checkerboard </a>

  <canvas id="canvId" width="750" height="750" style="border: 1px outset;"></canvas>

</body></html>

<lang html> <html> <head>

 <title>Checkerboard</title>
 <script src="KPF.js"></script>

</head> <body onload="cpmat([[0,1,0,1],[1,0,1,0],[0,1,0,1],[1,0,1,0]],5,'black')">

Checkerboard

  <a href="VicsekFractal.html"/> Next: Vicsek fractal </a>

  <canvas id="canvId" width="750" height="750" style="border: 1px outset;"></canvas>

</body></html> </lang>

Output:

Page VicsekFractal.html with VicsekFractaljs.png
Page SierpCarpetFractal.html with SierpCarpetFractaljs.png
Page Checkerboard.html with CheckbrdFractaljs.png

PARI/GP

Note

• Find iPlotmat() here on Helper Functions page.
• Find matkronprod() here on Kronecker product page.

Works with: PARI/GP version 2.9.1 and above

File:VicsekFractalgp.png

Output VicsekFractalgp.png

File:SierpCarpetFractalgp.png

Output SierpCarpetFractalgp.png

File:SierpTriFractalgp.png

Output SierpTriFractalgp.png

<lang parigp> \\ Build block matrix applying Kronecker product to the special matrix m \\ (n times to itself). Then plot Kronecker fractal. 4/25/2016 aev pkronfractal(m,n=2,clr)={

 my(r=m);
 for(i=1,n, r=matkronprod(r,m));
 iPlotmat(r,clr);

} \\Requireq tests: {\\ Vicsek fractal: VicsekFractalgp.png

 my(M=[0,1,0;1,1,1;0,1,0]); print(" *** Vicsek fractal, order 4:");
 pkronfractal(M,4,6);

} {\\ Sierpinski carpet fractal: SierpCarpetFractalgp.png

 my(M=[1,1,1;1,0,1;1,1,1]); print(" *** Sierpinski carpet fractal, order 4:");
 pkronfractal(M,4,5);

} {\\ Sierpinski triangle fractal: SierpTriFractalgp.png

 my(M=[1,1;0,1]); print(" *** Sierpinski triangle fractal, order 7:");
 pkronfractal(M,7,6);

} </lang>

Output:

 *** Vicsek fractal, order 4:
 *** matrix(243x243) 3125 DOTS

 *** Sierpinski carpet fractal, order 4:
 *** matrix(243x243) 32768 DOTS 

 *** Sierpinski triangle fractal, order 7:
 *** matrix: 256x256, 6561 DOTS

R

Generate and plot 3 Kronecker product based fractals.
Note: Find plotmat() and plotv2() here on Helper Functions page.

Works with: R version 3.3.2 and above

File:VicsekFractalR.png

Output VicsekFractalR.png

File:SierpCarpetFR.png

Output SierpCarpetFR.png

File:PlusSignFR.png

Output PlusSignFR.png

<lang r>

1. 1. Generate and plot Kronecker product based fractals. aev 8/12/16
   2. gpKronFractal(m, n, pf, clr, ttl, dflg=0, psz=600):
   3. Where: m - initial matrix (filled with 0/1); n - order of the fractal;
   4. pf - plot file name (without extension); clr - color; ttl - plot title;
   5. dflg - writing dump file flag (0/1); psz - picture size.

gpKronFractal <- function(m, n, pf, clr, ttl, dflg=0, psz=600) {

 cat(" *** START:", date(), "n=", n, "clr=", clr, "psz=", psz, "\n");
 cat(" *** Plot file -", pf, "\n");
 r <- m;
 for(i in 1:n) {r = r%x%m};
 plotmat(r, pf, clr, ttl, dflg, psz);
 cat(" *** END:", date(), "\n");

}

1. 1. Required tests:
2. 1. Vicsek Fractal

M <- matrix(c(0,1,0,1,1,1,0,1,0), ncol=3, nrow=3, byrow=TRUE); gpKronFractal(M, 4, "VicsekFractalR","red", "Vicsek Fractal n=4")

1. 2. Sierpinski carpet fractal

M <- matrix(c(1,1,1,1,0,1,1,1,1), ncol=3, nrow=3, byrow=TRUE); gpKronFractal(M, 4, "SierpinskiCarpetFR", "maroon", "Sierpinski carpet fractal n=4")

1. 3. Plus sign fractal

M <- matrix(c(1,1,1,1,1,1,1,1,1,1,0,1,1,1,1,1,1,0,1,1,1,1,0,0,0,0,0,1,1,1,1, +0,1,1,1,1,1,1,0,1,1,1,1,1,1,1,1,1,1), ncol=7, nrow=7, byrow=TRUE); gpKronFractal(M, 3, "PlusSignFR", "maroon", "Plus sign fractal, n=3")

1. Also, try these 2. I bet you've never seen them before.
2. 4. Wider Sierpinski carpet fractal (a.k.a. Sierpinski carpet mutant)
3. Note: If your computer is not super fast it could take a lot of time.
4. Use dump flag = 1, to save generated fractal.
5. M <- matrix(c(1,1,1,1,1,1,0,0,0,1,1,0,0,0,1,1,0,0,0,1,1,1,1,1,1), ncol=5,
6. +nrow=5, byrow=TRUE);
7. gpKronFractal(M, 4, "SierpinskiCarpetFw", "brown", "Wider Sierpinski carpet fractal n=4", 1)
8. 5. "H" fractal (Try all other letters in the alphabet...)
9. M <- matrix(c(1,1,1,1,1,1,1,1,0,0,0,0,0,1,1,1,1,0,1,1,1,1,1,1,0,1,1,1,1,1,1,
10. +0,1,1,1,1,0,0,0,0,0,1,1,1,1,1,1,1,1), ncol=7, nrow=7, byrow=TRUE);
11. gpKronFractal(M, 3, "HFR", "maroon", "'H' fractal n=3", 1)

</lang>

Output:

> M <- matrix(c(0,1,0,1,1,1,0,1,0), ncol=3, nrow=3, byrow=TRUE);
> gpKronFractal(M, 4, "VicsekFractalR", "red", "Vicsek Fractal n=4")
 *** START: Mon Aug 29 16:14:14 2016 n= 4 clr= red 
 *** Plot file - VicsekFractalR 
 *** Matrix( 243 x 243 ) 3125  DOTS
 *** END: Mon Aug 29 16:14:14 2016 

> M <- matrix(c(1,1,1,1,0,1,1,1,1), ncol=3, nrow=3, byrow=TRUE);
> gpKronFractal(M, 4, "SierpinskiCarpetFR", "maroon", "Sierpinski carpet fractal n=4")
 *** START: Mon Aug 29 16:16:14 2016 n= 4 clr= maroon 
 *** Plot file - SierpinskiCarpetFR
 *** Matrix( 243 x 243 ) 32768  DOTS
 *** END: Mon Aug 29 16:16:32 2016 

> M <- matrix(c(1,1,1,1,1,1,1,1,1,1,0,1,1,1,1,1,1,0,1,1,1,1,0,0,0,0,0,1,1,1,1,
+0,1,1,1,1,1,1,0,1,1,1,1,1,1,1,1,1,1), ncol=7, nrow=7, byrow=TRUE);
> gpKronFractal(M, 3, "PlusSignFR", "maroon", "Plus sign fractal, n=3")
 *** START: Thu Apr 06 21:45:33 2017 n= 3 clr= maroon psz= 600 
 *** Plot file - PlusSignFR 
 *** Matrix( 2401 x 2401 ) 2560000 DOTS
 *** END: Fri Apr 07 09:31:07 2017 

zkl

Uses Image Magick and the PPM class from http://rosettacode.org/wiki/Bitmap/Bresenham%27s_line_algorithm#zkl <lang zkl>var [const] GSL=Import.lib("zklGSL"); // libGSL (GNU Scientific Library) fcn kronecker(A,B){ //--> new Matrix

  m,n, p,q := A.rows,A.cols, B.rows,B.cols;
  r:=GSL.Matrix(m*p, n*q);
  foreach i,j,k,l in (m,n,p,q){ r[p*i + k, q*j + l]=A[i,j]*B[k,l] }
  r

}

fcn kfractal(M,n,fname){

  R:=M;
  do(n){ R=kronecker(R,M) }
  r,c,img := R.rows, R.cols, PPM(r,c);
  foreach i,j in (r,c){ img[i,j]=(R[i,j] and 0x00FF00 or 0) }  // green dots
  println("%s: %dx%d with %,d points".fmt(fname,R.rows,R.cols,
       R.pump(0,Ref(0).inc,Void.Filter).value));   // count 1s in fractal matrix
  img.writeJPGFile(fname);

}</lang> <lang zkl>var [const] A=GSL.Matrix(3,3).set(0,1,0, 1,1,1, 0,1,0),

           B=GSL.Matrix(3,3).set(1,1,1, 1,0,1, 1,1,1);

kfractal(A,4,"vicsek_k.jpg"); kfractal(B,4,"sierpinskiCarpet_k.jpg");</lang>

Output:

vicsek_k.jpg: 243x243 with 3,125 points
sierpinskiCarpet_k.jpg: 243x243 with 32,768 points

Images at Vicsek fractal and Sierpinski Carpet fractal.