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## Summary

Description
English: Illustration for the paradoxical decomposition of ${\displaystyle F_{2}}$ used in the proof of the Banach-Tarski paradox.
Date
Source Own work, made using MetaPost program.
Author David Benbennick
Permission
(Reusing this file)
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.
 This file is licensed under the Creative Commons Attribution-Share Alike 2.0 Generic license. You are free: to share – to copy, distribute and transmit the work to remix – to adapt the work Under the following conditions: attribution – You must attribute the work in the manner specified by the author or licensor (but not in any way that suggests that they endorse you or your use of the work). share alike – If you alter, transform, or build upon this work, you may distribute the resulting work only under the same or similar license to this one.
 This file is licensed under the Creative Commons Attribution-Share Alike 2.5 Generic license. You are free: to share – to copy, distribute and transmit the work to remix – to adapt the work Under the following conditions: attribution – You must attribute the work in the manner specified by the author or licensor (but not in any way that suggests that they endorse you or your use of the work). share alike – If you alter, transform, or build upon this work, you may distribute the resulting work only under the same or similar license to this one.
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;

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;

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}
\end{document}


I then ran the following commands:

mpost Paradoxical_decomposition_F2
dvips -mode ljfzzz -D 1200 Paradoxical_decomposition_F2 -o