File:Paradoxical decomposition F 2.svg

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Summary

Description

English: Illustration for the paradoxical decomposition used in the proof of the Banach–Tarski paradox.

Date

2 November 2014

Source

Paradoxical decomposition F2.svg

Author

Qef

SVG development

The SVG code is valid.

This vector image was created with MetaPost.

Source code

PostScript code

metapost-source

Linethickness = 0.5pt;
 
beginfig(1);
  recursion_depth := 6;
  const := 2.2;
 
  picture reds, blues, greens;
  blues := reds := greens := currentpicture;
 
  vardef orbitdots(expr iter, maxiter, xoffset, yoffset, lastg) =
   save offlen;
   offlen := 1.3in/(const**iter);
   if lastg = "":
        addto blues doublepath (xoffset, yoffset) withpen pencircle scaled 2bp withcolor blue;
     if iter <= maxiter:
        orbitdots(iter+1, maxiter, xoffset + offlen, yoffset, "a");
        orbitdots(iter+1, maxiter, xoffset - offlen, yoffset, "a-1"); 
        orbitdots(iter+1, maxiter, xoffset, yoffset + offlen, "b");
        orbitdots(iter+1, maxiter, xoffset, yoffset - offlen, "b-1");
     fi;
   fi;
   if lastg = "a":
        addto greens doublepath (xoffset, yoffset) withpen pencircle scaled 2bp withcolor 0.8green;
     if iter <= maxiter:
        orbitdots(iter+1, maxiter, xoffset + offlen, yoffset, "a");
        orbitdots(iter+1, maxiter, xoffset, yoffset + offlen, "b");
        orbitdots(iter+1, maxiter, xoffset, yoffset - offlen, "b-1");
     fi;
   fi;
   if lastg = "a-1":
        addto reds doublepath (xoffset, yoffset) withpen pencircle scaled 1.2bp withcolor red;
        addto blues doublepath (xoffset, yoffset) withpen pencircle scaled 2bp withcolor blue;
     if iter <= maxiter:
        orbitdots(iter+1, maxiter, xoffset, yoffset + offlen, "b"); 
        orbitdots(iter+1, maxiter, xoffset - offlen, yoffset, "a-1");
        orbitdots(iter+1, maxiter, xoffset, yoffset - offlen, "b-1");
     fi;
   fi;
   if lastg = "b":
        addto blues doublepath (xoffset, yoffset) withpen pencircle scaled 2bp withcolor blue;
     if iter <= maxiter:
        orbitdots(iter+1, maxiter, xoffset + offlen, yoffset, "a");
        orbitdots(iter+1, maxiter, xoffset, yoffset + offlen, "b"); 
        orbitdots(iter+1, maxiter, xoffset - offlen, yoffset, "a-1");
     fi;
   fi;
   if lastg = "b-1":
        addto blues doublepath (xoffset, yoffset) withpen pencircle scaled 2bp withcolor blue;
     if iter <= maxiter:
        orbitdots(iter+1, maxiter, xoffset + offlen, yoffset, "a");
        orbitdots(iter+1, maxiter, xoffset - offlen, yoffset, "a-1");
        orbitdots(iter+1, maxiter, xoffset, yoffset - offlen, "b-1");
     fi;
   fi;
  enddef;
 
  numeric length;
  length := 1.3in / (const**recursion_depth);
 
  pickup pencircle scaled Linethickness;
  picture p, q, r;
 
  for i = 1 upto recursion_depth - 2:
    draw (0,0) -- (length,0);
    draw (0,-length) -- (0,length);
 
    length := length * const;
 
    currentpicture := currentpicture shifted (length,0);
    p := currentpicture rotated 90;
    q := p yscaled -1;
 
    addto currentpicture also p;
    addto currentpicture also q;
  endfor;
 
  draw (0,0) -- (length,0);
  draw (0,-length) -- (0,length);
 
  length := length * const;
 
  currentpicture := currentpicture shifted (length,0);
  p := currentpicture rotated 90;
  q := p yscaled -1;
  r := currentpicture xscaled -1;
 
  addto currentpicture also p;
  addto currentpicture also q;
  addto currentpicture also r;
 
  draw (-length,0) -- (length,0);
  draw (0,-length) -- (0,length);
 
  dotlabel.urt(btex$e$etex, (0,0));
  dotlabel.ulft(btex$a$etex, (length,0));
  dotlabel.lrt(btex$b$etex, (0,length));
 
  label.top(btex$S(a^{-1})$etex, (-1.2length, 0.9length)) withcolor red;
 
  label.bot(btex$aS(a^{-1})$etex, (-1.3length,-0.9length)) withcolor blue;

  label.rt(btex$S(a)$etex, (1.1length,1.1length)) withcolor 0.8green;
 
  orbitdots(1, recursion_depth-2, 0, 0, "");
  addto currentpicture also greens;
  addto currentpicture also blues;
  addto currentpicture also reds;
  currentpicture := currentpicture shifted (10cm,10cm);
  currentpicture := currentpicture scaled 3.5;
endfig;
end;

Data

mptopdf fig.mp
pdf2svg fig-1.pdf fig-1.svg

Licensing

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to share – to copy, distribute and transmit the work
to remix – to adapt the work

Under the following conditions:

attribution – You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
share alike – If you remix, transform, or build upon the material, you must distribute your contributions under the same or compatible license as the original.

File history

Click on a date/time to view the file as it appeared at that time.

	Date/Time	Dimensions	User	Comment
current	22:54, 4 October 2015	679 × 679 (308 KB)	Damluk	Add inverse labels
	07:52, 4 November 2014	679 × 679 (288 KB)	Damluk	Move blue label a little to the left.
	07:48, 4 November 2014	679 × 679 (288 KB)	Damluk	Add third disjoint set S(a)
	20:16, 2 November 2014	713 × 679 (277 KB)	Damluk	preserve pretty font.
	18:20, 2 November 2014	763 × 729 (261 KB)	Damluk	Corrections regarding shown subsets.
	13:18, 2 November 2014	685 × 655 (271 KB)	Damluk	User created page with UploadWizard