Image:Paradoxical decomposition F2.png

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Illustration for the paradoxical decomposition of F₂ used in the proof of the Banach-Tarski paradox.

David Benbennick made it with the following MetaPost program:

Linethickness = 1pt;
beginfig(1);
  recursion_depth := 6;
  const := 2.2;

  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));

  pair P, Q, R;
  P := (-0.5length, 0.5length);
  Q := (-length, 0.9length);

  draw P .. Q .. (-2length,0) .. Q yscaled -1 .. P yscaled -1
    .. cycle withcolor red;

  label.top(btex$S(a^{-1})$etex, (-1.2length, 0.9length)) withcolor red;

  Q := (length, length);
  R := (0, 2length);
  draw P xscaled -1 --- Q .. R .. (-2.2length, 0) .. R yscaled -1
    .. Q yscaled -1 --- P scaled -1 .. cycle withcolor blue;

  label.rt(btex$aS(a^{-1})$etex, (1.1length,1.1length)) withcolor blue;

  currentpicture := currentpicture shifted (10cm,10cm);
endfig;
end;

I saved the above as Paradoxical_decomposition_F2.mp, and the following LaTeX program as Paradoxical_decomposition_F2.tex:

%&latex
\documentclass[12pt]{article}
 \usepackage{graphicx}
 \pagestyle{empty}
\begin{document}
 \includegraphics{Paradoxical_decomposition_F2.1}
\end{document}

I then ran the following commands:

mpost Paradoxical_decomposition_F2
tex Paradoxical_decomposition_F2
dvips -mode ljfzzz -D 1200 Paradoxical_decomposition_F2 -o
convert -density 1200 Paradoxical_decomposition_F2.ps Paradoxical_decomposition_F2.pnm
pnmcrop Paradoxical_decomposition_F2.pnm | pnmtopng > Paradoxical_decomposition_F2.png
advpng -z4 Paradoxical_decomposition_F2.png

I, the copyright holder of this work, hereby publish it under the following licenses: Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation license, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled "GNU Free Documentation license". Aragonés | العربية | Asturianu | Български | বাংলা | ইমার ঠার/বিষ্ণুপ্রিয়া মণিপুরী | Brezhoneg | Bosanski | Català | Cebuano | Česky | Dansk | Deutsch | Ελληνικά | English | Esperanto | Español | Eesti | Euskara | فارسی | Suomi | Français | Gaeilge | Galego | עברית | Hrvatski | Magyar | Bahasa Indonesia | Ido | Íslenska | Italiano | 日本語 | ქართული | ភាសាខ្មែរ | 한국어 | Kurdî / كوردی | Latina | Lëtzebuergesch | Lietuvių | Bahasa Melayu | Nnapulitano | Nederlands | ‪Norsk (nynorsk)‬ | ‪Norsk (bokmål)‬ | Occitan | Polski | Português | Română | Русский | Slovenčina | Slovenščina | Shqip | Српски / Srpski | Svenska | తెలుగు | ไทย | Türkçe | Українська | اردو | Tiếng Việt | Volapük | Yorùbá | ‪中文(中国大陆)‬ | ‪中文(台灣)‬ | +/- This file is licensed under Creative Commons Attribution ShareAlike 2.0 License (cc-by-sa-2.0). In short: you are free to share and make derivative works of the file under the conditions that you appropriately attribute it, and that you distribute it under this or a similar cc-by-sa license. This file is licensed under the Creative Commons Attribution ShareAlike 2.5 License. In short: you are free to share and make derivative works of the file under the conditions that you appropriately attribute it, and that you distribute it only under a license identical to this one. Official license You may select the license of your choice.

• Banach–Tarski paradox