% This file was generated from % text5.tex jfig1.ps jfig2.ps jfig3.ps jfig4a.ps jfig4b.ps jfig5.ps jfig6.ps jfig7newer.ps jfig8.ps jfig9.ps jfig10.ps jfig11.ps by /public/bin/texpsinclude. % This file contains included PostScript. % TeX writes out the included PostScript to files. % Here are the macros for writing out the included PostScript: % Macros for dumping included Postscript to files. % Requires Plain TeX. Maybe other flavors will work too? % Jamie Stephens, jamies@math.utexas.edu, 28 Nov 94 % If you're in the UT Math Department, see % /usr/local/doc/tex/texpsinclude.text and /public/bin/texpsinclude. % Parts of these files are attached at the end of this file. % Adapted from Knuth's \answer macro in the TeXbook. \def\endofps{EndOfTheIncludedPostscriptMagicCookie} \chardef\other=12 \newwrite\psdumphandle \outer\def\psdump#1{\par\medbreak \immediate\openout\psdumphandle=#1 \copytoblankline} \def\copytoblankline{\begingroup\setupcopy\copypsline} \def\setupcopy{\def\do##1{\catcode`##1=\other}\dospecials \catcode`\\=\other \obeylines} {\obeylines \gdef\copypsline#1 {\def\next{#1}% \ifx\next\endofps\let\next=\endgroup % \else\immediate\write\psdumphandle{\next} \let\next=\copypsline\fi\next}} \outer\def\closepsdump{ \immediate\closeout\psdumphandle} %% %% Here's a shell script for automating the use of psdump.tex %% % #!/bin/bash % % PSDUMP=/usr/local/tex/macros/local/psdump.tex % TEXPSINCLUDEHELP=/usr/local/doc/tex/texpsinclude.help % % if [ $# -eq 0 ]; then % echo "Usage: $0 file.tex {files.ps} > output.tex" >&2 % echo "See $TEXPSINCLUDEHELP for more information." >&2 % exit 1 % fi % % echo "% This file was generated from" % echo "% $@ by $0." % echo "% This file contains included PostScript." % echo "% TeX writes out the included PostScript to files." % echo % echo "% Here are the macros for writing out the included PostScript:" % cat $PSDUMP % echo % % for argument in $@; do % if [ $argument != $1 ]; then % echo "% Here's the Postscript for $argument:" % echo "\message{Writing file $argument}" % echo -n "\psdump{$argument}" % echo "Including $argument." >&2 % cat $argument % echo "EndOfTheIncludedPostscriptMagicCookie" % echo % echo "\closepsdump" % fi % done % % echo % echo "% Finally, here is $1:" % % cat $1 %% %% Here's some documentation: %% % This file documents how to use the texpsinclude command in /public/bin. % Here's the problem that texpsinclude solves: % You have a TeX file called foo.tex that you want to distribute as a % single TeX file. The problem is that foo.tex needs two Postscript % files, bar1.ps and bar2.ps, for embedded figures. You'd like a single % TeX file which somehow includes bar1.ps and bar2.ps. When TeX % processes foo.tex, TeX should extract bar1.ps and bar2.ps from % foo.tex. % Here's how to do what you want to do: % texpsinclude foo.tex. bar1.ps bar2.ps > bigfoo.tex % (In general: "texpsinclude > ".) If you % enter this command, the result is a new file called bigfoo.tex. The % file bigfoo.tex contains bar1.ps and bar2.ps. If you give bigfoo.tex % to a friend, she can make your document with: % tex bigfoo.tex % This command writes out bar1.ps and bar2.ps, and the command also % TeX's foo.tex. % Note: If *you* run tex on bigfoo.tex in the same directory, then TeX % will write over your .ps files. Be careful when testing your % bigfoo.tex. %% %% Here's another way to use psdump.tex %% % EXAMPLE (remove the leading % signs to make it work): % %\psdump{example.ps}These three lines %are going be dumped "as is" %to the file example.ps %EndOfTheIncludedPostscriptMagicCookie %\closepsdump % Here's the Postscript for jfig1.ps: \message{Writing file jfig1.ps} \psdump{jfig1.ps}%!PS-Adobe-2.0 EPSF-2.0 %%Creator: gnuplot %%DocumentFonts: Courier %%BoundingBox: 50 50 554 554 %%EndComments /gnudict 40 dict def gnudict begin /Color false def /gnulinewidth 5.000 def /vshift -46 def /dl {10 mul} def /hpt 31.5 def /vpt 31.5 def /vpt2 vpt 2 mul def /hpt2 hpt 2 mul def /Lshow { currentpoint stroke moveto 0 vshift rmoveto show } def /Rshow { currentpoint stroke moveto dup stringwidth pop neg vshift rmoveto show } def /Cshow { currentpoint stroke moveto dup stringwidth pop -2 div vshift rmoveto show } def /DL { Color {setrgbcolor [] 0 setdash pop} {pop pop pop 0 setdash} ifelse } def /BL { stroke gnulinewidth 2 mul setlinewidth } def /AL { stroke gnulinewidth 2 div setlinewidth } def /PL { stroke gnulinewidth setlinewidth } def /LTb { BL [] 0 0 0 DL } def /LTa { AL [1 dl 2 dl] 0 setdash 0 0 0 setrgbcolor } def /LT0 { PL [] 0 1 0 DL } def /LT1 { PL [4 dl 2 dl] 0 0 1 DL } def /LT2 { PL [2 dl 3 dl] 1 0 0 DL } def /LT3 { PL [1 dl 1.5 dl] 1 0 1 DL } def /LT4 { PL [5 dl 2 dl 1 dl 2 dl] 0 1 1 DL } def /LT5 { PL [4 dl 3 dl 1 dl 3 dl] 1 1 0 DL } def /LT6 { PL [2 dl 2 dl 2 dl 4 dl] 0 0 0 DL } def /LT7 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 1 0.3 0 DL } def /LT8 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 0.5 0.5 0.5 DL } def /M {moveto} def /L {lineto} def /P { stroke [] 0 setdash currentlinewidth 2 div sub moveto 0 currentlinewidth rlineto stroke } def /D { stroke [] 0 setdash 2 copy vpt add moveto hpt neg vpt neg rlineto hpt vpt neg rlineto hpt vpt rlineto hpt neg vpt rlineto closepath stroke P } def /A { stroke [] 0 setdash vpt sub moveto 0 vpt2 rlineto currentpoint stroke moveto hpt neg vpt neg rmoveto hpt2 0 rlineto stroke } def /B { stroke [] 0 setdash 2 copy exch hpt sub exch vpt add moveto 0 vpt2 neg rlineto hpt2 0 rlineto 0 vpt2 rlineto hpt2 neg 0 rlineto closepath stroke P } def /C { stroke [] 0 setdash exch hpt sub exch vpt add moveto hpt2 vpt2 neg rlineto currentpoint stroke moveto hpt2 neg 0 rmoveto hpt2 vpt2 rlineto stroke } def /T { stroke [] 0 setdash 2 copy vpt 1.12 mul add moveto hpt neg vpt -1.62 mul rlineto hpt 2 mul 0 rlineto hpt neg vpt 1.62 mul rlineto closepath stroke P } def /S { 2 copy A C} def end %%EndProlog gnudict begin gsave 50 50 translate 0.070 0.100 scale 0 setgray /Courier findfont 140 scalefont setfont newpath LTa LTb LTb 3989 -2307 M (Figure 4) Cshow LT0 LT0 1670 491 M 1564 547 L 1882 659 L 2001 491 L 1564 1107 M 1564 1107 L 1564 547 L 1882 659 L 1564 1107 L 1564 547 M 1564 547 L 1564 1107 L 1246 995 L 1564 547 L 1564 547 M 1564 547 L 1008 841 L 1008 911 M 1246 995 L 1564 547 L 1008 957 M 1405 1331 L 1564 1107 L 1008 911 L 1056 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4353 M 3631 4353 L 2995 4689 L 2995 4689 M 2931 4443 L 3631 4353 L 2995 4689 M 2518 4241 L 2868 4196 L 2995 4689 L 2741 3704 M 2741 3704 L 2518 4241 L 2868 4196 L 2741 3704 L 2518 4241 M 2518 4241 L 2741 3704 L 2391 3749 L 2518 4241 L 2518 4241 M 2518 4241 L 2041 3793 L 2391 3749 L 2518 4241 L 2041 3793 M 2041 3793 L currentpoint stroke moveto 2677 3458 L 2741 3704 L 2041 3793 L 2995 4689 M 2518 4241 L 2359 4465 L 2995 4689 L 1723 4241 M 1723 4241 L 2518 4241 L 2359 4465 L 1723 4241 L 2518 4241 M 2518 4241 L 1723 4241 L 1882 4017 L 2518 4241 L 2518 4241 M 2518 4241 L 2041 3793 L 1882 4017 L 2518 4241 L 2041 3793 M 2041 3793 L 1405 4129 L 1723 4241 L 2041 3793 L 1405 4129 M 1405 4129 L 2041 3793 L 1723 3681 L 1405 4129 L 2041 3234 M 2041 3234 L 2041 3793 L 1723 3681 L 2041 3234 L 2041 3793 M 2041 3793 L 2041 3234 L 2359 3346 L 2041 3793 L 2041 3793 M 2041 3793 L 2677 3458 L 2359 3346 L 2041 3793 L 2677 3458 M 2677 3458 L 2200 3010 L 2041 3234 L 2677 3458 L 2200 3010 M 2200 3010 L 1564 3346 L 1882 3458 L 2200 3010 L 1564 3905 M 1564 3905 L 1564 3346 L 1882 3458 L 1564 3905 L 1564 3346 M 1564 3346 L 1564 3905 L 1246 3793 L 1564 3346 L 1564 3346 M 1564 3346 L 1008 3640 L 1008 3709 M 1246 3793 L 1564 3346 L 1008 3756 M 1405 4129 L 1564 3905 L 1008 3709 L 1056 4174 M 1056 4174 L 1008 4164 L 1008 4180 M 1056 4174 L 1008 4164 M 1056 4174 L 1008 3989 L 1008 3756 M 1405 4129 L 1056 4174 L 1008 3989 L 1405 4129 M 1405 4129 L 1008 4339 L 1008 4180 M 1405 4129 L 1405 4129 M 1405 4129 L 1008 4339 L 1008 4549 M 1087 4577 L 1405 4129 L 1008 4549 M 1087 4577 L 1008 4689 L 1564 3346 M 1564 3346 L 1008 3640 L 1008 3417 M 1564 3346 L 1215 3390 M 1215 3390 L 1008 3348 L 1008 3417 M 1215 3390 L 1008 3348 M 1215 3390 L 1151 3144 L 1008 3162 L 1008 2940 M 1087 2898 L 1151 3144 L 1008 3162 L 1087 2898 M 1087 2898 L 1564 3346 L 1215 3390 L 1087 2898 L 1008 2940 M 1087 2898 L 1008 2870 L 1087 2898 M 1087 2898 L 1008 2823 L 1008 2870 M 1087 2898 L 1008 2823 M 1087 2898 L 1246 2674 L 1008 2590 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1405 1890 M 1405 1890 L 1405 2450 L 1723 2338 L 1405 1890 L 1405 2450 M 1405 2450 L 2200 2450 L 2041 2226 L 1405 2450 L 1405 1331 M 1405 1331 L 1405 1890 L 1087 1778 L 1405 1331 L 1008 2100 M 1405 1890 L 1087 1778 L 1008 1890 L 1405 1890 M 1405 1890 L 1008 2100 L 1008 2310 M 1087 2338 L 1405 1890 L 1405 1890 M 1405 1890 L 1405 2450 L 1087 2338 L 1405 1890 L 1405 2450 M 1405 2450 L 1008 2450 L 1008 2310 M 1405 2450 L 1008 2450 M 1405 2450 L 1246 2674 L 1008 2590 L 1882 2898 M 1882 2898 L 1405 2450 L 1246 2674 L 1882 2898 L 1405 2450 M 1405 2450 L 1882 2898 L 2041 2674 L 1405 2450 L 1405 2450 M 1405 2450 L 2200 2450 L 2041 2674 L 1405 2450 L 2200 2450 M 2200 2450 L 2200 3010 L 1882 2898 L 2200 2450 L 2200 3010 M 2200 3010 L 2200 2450 L 2518 2562 L 2200 3010 L 2836 2114 M 2836 2114 L 2200 2450 L 2518 2562 L 2836 2114 L 2200 2450 M 2200 2450 L 2836 2114 L 2518 2002 L 2200 2450 L 2200 2450 M 2200 2450 L 2200 1890 L 2518 2002 L 2200 2450 L 2200 1890 M 2200 1890 L 2995 1890 L 2836 2114 L 2200 1890 L 2200 771 M 2200 771 L 2200 1331 L 2518 1219 L 2200 771 L 2836 1666 M 2836 1666 L 2200 1331 L 2518 1219 L 2836 1666 L 2200 1331 M 2200 1331 L 2836 1666 L 2518 1778 L 2200 1331 L 2200 1331 M 2200 1331 L 2200 1890 L 2518 1778 L 2200 1331 L 2200 1890 M 2200 1890 L 2995 1890 L 2836 1666 L 2200 1890 L 2995 1890 M 2995 1890 L 2995 1331 L 2677 1443 L 2995 1890 L 2359 995 M 2359 995 L 2995 1331 L 2677 1443 L 2359 995 L 2995 1331 M 2995 1331 L 2359 995 L 2677 883 L 2995 1331 L 2995 1331 M 2995 1331 L 2995 771 L 2677 883 L 2995 1331 L 2995 771 M 2995 771 L 2200 771 L 2359 995 L 2995 771 L 2995 1890 M 2995 1890 L 2995 1331 L 3313 1443 L 2995 1890 L 3631 995 M 3631 995 L 2995 1331 L 3313 1443 L 3631 995 L 2995 1331 M 2995 1331 L 3631 995 L 3313 883 L 2995 1331 L 2995 1331 M 2995 1331 L 2995 771 L 3313 883 L 2995 1331 L 2995 771 M 2995 771 L 3790 771 L 3631 995 L 2995 771 L 3790 771 M 3790 771 L 2995 771 L 3154 547 L 3790 771 L 2697 491 M 2995 771 L 3154 547 L 2995 491 L 2995 771 M 2995 771 L 2697 491 L 2399 491 M 2359 547 L 2995 771 L 2995 771 M 2995 771 L 2200 771 L 2359 547 L currentpoint stroke moveto 2995 771 L 2200 771 M 2200 771 L 2200 491 L 2399 491 M 2200 771 L 2200 3010 M 2200 3010 L 2836 2674 L 2518 2562 L 2200 3010 L 2836 2114 M 2836 2114 L 2836 2674 L 2518 2562 L 2836 2114 L 2836 2674 M 2836 2674 L 2836 2114 L 3154 2226 L 2836 2674 L 2836 2674 M 2836 2674 L 3472 2338 L 3154 2226 L 2836 2674 L 3472 2338 M 3472 2338 L 2995 1890 L 2836 2114 L 3472 2338 L 4744 1666 M 4744 1666 L 4108 2002 L 4044 1756 L 4744 1666 L 3345 1846 M 3345 1846 L 4108 2002 L 4044 1756 L 3345 1846 L 4108 2002 M 4108 2002 L 3345 1846 L 3408 2092 L 4108 2002 L 4108 2002 M 4108 2002 L 3472 2338 L 3408 2092 L 4108 2002 L 3472 2338 M 3472 2338 L 2995 1890 L 3345 1846 L 3472 2338 L 2995 1890 M 2995 1890 L 3631 1554 L 3694 1801 L 2995 1890 L 4394 1711 M 4394 1711 L 3631 1554 L 3694 1801 L 4394 1711 L 3631 1554 M 3631 1554 L 4394 1711 L 4330 1465 L 3631 1554 L 3631 1554 M 3631 1554 L 4267 1219 L 4330 1465 L 3631 1554 L 4267 1219 M 4267 1219 L 4744 1666 L 4394 1711 L 4267 1219 L 2995 1890 M 2995 1890 L 3631 1554 L 3313 1443 L 2995 1890 L 3631 995 M 3631 995 L 3631 1554 L 3313 1443 L 3631 995 L 3631 1554 M 3631 1554 L 3631 995 L 3949 1107 L 3631 1554 L 3631 1554 M 3631 1554 L 4267 1219 L 3949 1107 L 3631 1554 L 4267 1219 M 4267 1219 L 3790 771 L 3631 995 L 4267 1219 L 3790 771 M 3790 771 L 4267 1219 L 4426 995 L 3790 771 L 5061 1219 M 5061 1219 L 4267 1219 L 4426 995 L 5061 1219 L 4267 1219 M 4267 1219 L 5061 1219 L 4903 1443 L 4267 1219 L 4267 1219 M 4267 1219 L 4744 1666 L 4903 1443 L 4267 1219 L 4744 1666 M 4744 1666 L 5379 1331 L 5061 1219 L 4744 1666 L 5379 1331 M 5379 1331 L 4903 883 L 4744 1107 L 5379 1331 L 4108 883 M 4108 883 L 4903 883 L 4744 1107 L 4108 883 L 4903 883 M 4903 883 L 4108 883 L 4267 659 L 4903 883 L 4903 883 M 4903 883 L 4485 491 L 4386 491 M 4267 659 L 4903 883 L 4320 491 M 3790 771 L 4108 883 L 4386 491 L 3726 525 M 3726 525 L 3740 491 L 3718 491 M 3726 525 L 3740 491 M 3726 525 L 3988 491 L 4320 491 M 3790 771 L 3726 525 L 3989 491 L 3790 771 M 3790 771 L 3492 491 L 3718 491 M 3790 771 L 3790 771 M 3790 771 L 3492 491 L 3194 491 M 3154 547 L 3790 771 L 3194 491 M 3154 547 L 2995 491 L 4903 883 M 4903 883 L 4485 491 L 4801 491 M 4903 883 L 4839 637 M 4839 637 L 4899 491 L 4801 491 M 4839 637 L 4899 491 M 4839 637 L 5189 592 L 5163 491 L 5479 491 M 5538 547 L 5189 592 L 5163 491 L 5538 547 M 5538 547 L 4903 883 L 4839 637 L 5538 547 L 5479 491 M 5538 547 L 5578 491 L 5538 547 M 5538 547 L 5644 491 L 5578 491 M 5538 547 L 5644 491 M 5538 547 L 5856 659 L 5976 491 L 5538 1107 M 5538 1107 L 5538 547 L 5856 659 L 5538 1107 L 5538 547 M 5538 547 L 5538 1107 L 5220 995 L 5538 547 L 5538 547 M 5538 547 L 4903 883 L 5220 995 L 5538 547 L 4903 883 M 4903 883 L 5379 1331 L 5538 1107 L 4903 883 L 6174 1331 M 6174 1331 L 6969 1331 L 6969 1331 M 6810 1107 L 6174 1331 L 6969 1331 M 6810 1107 L 6969 1051 L 6492 1219 M 6492 1219 L 6969 771 L 6969 1051 M 6492 1219 L 6969 771 M 6492 1219 L 6333 995 L 6969 771 L 6969 771 M 6174 771 L 6333 995 L 6969 771 L 6174 771 M 6174 771 L 6174 1331 L 6492 1219 L 6174 771 L 6373 491 M 6333 547 L 6969 771 L 6969 771 M 6174 771 L 6333 547 L 6969 771 L 6174 771 M 6174 771 L 6174 491 L 6373 491 M 6174 771 L 6174 491 M 6174 771 L 5856 659 L 5976 491 L 5538 1107 M 5538 1107 L 6174 771 L 5856 659 L 5538 1107 L 6174 771 M 6174 771 L 5538 1107 L 5856 1219 L 6174 771 L 6174 771 M 6174 771 L 6174 1331 L 5856 1219 L 6174 771 L 6174 1331 M 6174 1331 L 5379 1331 L 5538 1107 L 6174 1331 L 5379 1331 M 5379 1331 L 6174 1331 L 6015 1554 L 5379 1331 L 6651 1778 M 6651 1778 L 6174 1331 L 6015 1554 L 6651 1778 L 6174 1331 M 6174 1331 L 6651 1778 L 6810 1554 L 6174 1331 L 6174 1331 M 6174 1331 L 6969 1331 L 6969 1331 M 6810 1554 L 6174 1331 L 6969 1890 M 6651 1778 L 6969 1331 L 5379 1331 M 5379 1331 L 5856 1778 L 6015 1554 L 5379 1331 L 6651 1778 M 6651 1778 L 5856 1778 L 6015 1554 L 6651 1778 L 5856 1778 M 5856 1778 L 6651 1778 L 6492 2002 L 5856 1778 L 5856 1778 M 5856 1778 L 6333 2226 L 6492 2002 L 5856 1778 L 6333 2226 M 6333 2226 L 6969 1890 L 6969 1890 M 6651 1778 L 6333 2226 L 6969 2823 M 6810 2674 L 6969 2654 L 6969 2290 M 6810 2674 L 6969 2654 L 6810 2674 M 6810 2674 L 6969 2290 L 6969 2145 M 6683 2181 L 6810 2674 L 6810 2674 M 6810 2674 L 6333 2226 L 6683 2181 L 6810 2674 L 6333 2226 M 6333 2226 L 6969 1890 L 6969 2145 M 6333 2226 L stroke grestore end showpage %%Trailer EndOfTheIncludedPostscriptMagicCookie \closepsdump % Here's the Postscript for jfig2.ps: \message{Writing file jfig2.ps} \psdump{jfig2.ps}%!PS-Adobe-3.0 EPSF-3.0 %%Creator: Adobe Illustrator(TM) 3.0.1. %%For: () () %%Title: (/d1/grad/strauss/Penrose6.ai) %%CreationDate: (6/15/94) (16:04) %%BoundingBox: -46 175 389 646 %%DocumentProcessColors: Black %%DocumentSuppliedResources: procset Adobe_packedarray 2.0 0 %%+ procset Adobe_cmykcolor 1.1 0 %%+ procset Adobe_cshow 1.1 0 %%+ procset Adobe_customcolor 1.0 0 %%+ procset Adobe_IllustratorA_AI3 1.0 1 %AI3_ColorUsage: Black&White %AI3_TemplateBox: 306 396 306 396 %AI3_TileBox: 30 31 582 761 %AI3_DocumentPreview: Header %%Template: %%PageOrigin:30 31 %%AI3_PaperRect:0 792 612 0 %%AI3_Margin:30 31 30 31 %%EndComments %%BeginProlog %%BeginResource: procset Adobe_packedarray 2.0 0 %%Title: (Packed Array Operators) %%Version: 2.0 %%CreationDate: (8/2/90) () %%Copyright: ((C) 1987-1990 Adobe Systems Incorporated All Rights Reserved) userdict /Adobe_packedarray 5 dict dup begin put /initialize % - initialize - { /packedarray where { pop } { Adobe_packedarray begin Adobe_packedarray { dup xcheck { bind } if userdict 3 1 roll put } forall end } ifelse } def /terminate % - terminate - { } def /packedarray % arguments count packedarray array { array astore readonly } def /setpacking % boolean setpacking - { pop } def /currentpacking % - setpacking boolean { false } def currentdict readonly pop end %%EndResource Adobe_packedarray /initialize get exec %%BeginResource: procset Adobe_cmykcolor 1.1 0 %%Title: (CMYK Color Operators) %%Version: 1.1 %%CreationDate: (1/23/89) () %%Copyright: ((C) 1987-1990 Adobe Systems Incorporated All Rights Reserved) currentpacking true setpacking userdict /Adobe_cmykcolor 4 dict dup begin put /initialize % - initialize - { /setcmykcolor where { pop } { userdict /Adobe_cmykcolor_vars 2 dict dup begin put /_setrgbcolor /setrgbcolor load def /_currentrgbcolor /currentrgbcolor load def Adobe_cmykcolor begin Adobe_cmykcolor { dup xcheck { bind } if pop pop } forall end end Adobe_cmykcolor begin } ifelse } def /terminate % - terminate - { currentdict Adobe_cmykcolor eq { end } if } def /setcmykcolor % cyan magenta yellow black setcmykcolor - { 1 sub 4 1 roll 3 { 3 index add neg dup 0 lt { pop 0 } if 3 1 roll } repeat Adobe_cmykcolor_vars /_setrgbcolor get exec pop } def /currentcmykcolor % - currentcmykcolor cyan magenta yellow black { Adobe_cmykcolor_vars /_currentrgbcolor get exec 3 { 1 sub neg 3 1 roll } repeat 0 } def currentdict readonly pop end setpacking %%EndResource %%BeginResource: procset Adobe_cshow 1.1 0 %%Title: (cshow Operator) %%Version: 1.1 %%CreationDate: (1/23/89) () %%Copyright: ((C) 1987-1990 Adobe Systems Incorporated All Rights Reserved) currentpacking true setpacking userdict /Adobe_cshow 3 dict dup begin put /initialize % - initialize - { /cshow where { pop } { userdict /Adobe_cshow_vars 1 dict dup begin put /_cshow % - _cshow proc {} def Adobe_cshow begin Adobe_cshow { dup xcheck { bind } if userdict 3 1 roll put } forall end end } ifelse } def /terminate % - terminate - { } def /cshow % proc string cshow - { exch Adobe_cshow_vars exch /_cshow exch put { 0 0 Adobe_cshow_vars /_cshow get exec } forall } def currentdict readonly pop end setpacking %%EndResource %%BeginResource: procset Adobe_customcolor 1.0 0 %%Title: (Custom Color Operators) %%Version: 1.0 %%CreationDate: (5/9/88) () %%Copyright: ((C) 1987-1990 Adobe Systems Incorporated All Rights Reserved) currentpacking true setpacking userdict /Adobe_customcolor 5 dict dup begin put /initialize % - initialize - { /setcustomcolor where { pop } { Adobe_customcolor begin Adobe_customcolor { dup xcheck { bind } if pop pop } forall end Adobe_customcolor begin } ifelse } def /terminate % - terminate - { currentdict Adobe_customcolor eq { end } if } def /findcmykcustomcolor % cyan magenta yellow black name findcmykcustomcolor object { 5 packedarray } def /setcustomcolor % object tint setcustomcolor - { exch aload pop pop 4 { 4 index mul 4 1 roll } repeat 5 -1 roll pop setcmykcolor } def /setoverprint % boolean setoverprint - { pop } def currentdict readonly pop end setpacking %%EndResource %%BeginResource: procset Adobe_IllustratorA_AI3 1.0 1 %%Title: (Adobe Illustrator (R) Version 3.0 Abbreviated Prolog) %%Version: 1.0 %%CreationDate: (7/22/89) () %%Copyright: ((C) 1987-1990 Adobe Systems Incorporated All Rights Reserved) currentpacking true setpacking userdict /Adobe_IllustratorA_AI3 61 dict dup begin put % initialization /initialize % - initialize - { % 47 vars, but leave slack of 10 entries for custom Postscript fragments userdict /Adobe_IllustratorA_AI3_vars 57 dict dup begin put % paint operands /_lp /none def /_pf {} def /_ps {} def /_psf {} def /_pss {} def /_pjsf {} def /_pjss {} def /_pola 0 def /_doClip 0 def % paint operators /cf currentflat def % - cf flatness % typography operands /_tm matrix def /_renderStart [/e0 /r0 /a0 /o0 /e1 /r1 /a1 /i0] def /_renderEnd [null null null null /i1 /i1 /i1 /i1] def /_render -1 def /_rise 0 def /_ax 0 def % x character spacing (_ax, _ay, _cx, _cy follows awidthshow naming convention) /_ay 0 def % y character spacing /_cx 0 def % x word spacing /_cy 0 def % y word spacing /_leading [0 0] def /_ctm matrix def /_mtx matrix def /_sp 16#020 def /_hyphen (-) def /_fScl 0 def /_cnt 0 def /_hs 1 def /_nativeEncoding 0 def /_useNativeEncoding 0 def /_tempEncode 0 def /_pntr 0 def /_tDict 2 dict def % typography operators /Tx {} def /Tj {} def % compound path operators /CRender {} def % printing /_AI3_savepage {} def % color operands /_gf null def /_cf 4 array def /_if null def /_of false def /_fc {} def /_gs null def /_cs 4 array def /_is null def /_os false def /_sc {} def /_i null def Adobe_IllustratorA_AI3 begin Adobe_IllustratorA_AI3 { dup xcheck { bind } if pop pop } forall end end Adobe_IllustratorA_AI3 begin Adobe_IllustratorA_AI3_vars begin newpath } def /terminate % - terminate - { end end } def % definition operators /_ % - _ null null def /ddef % key value ddef - { Adobe_IllustratorA_AI3_vars 3 1 roll put } def /xput % key value literal xput - { dup load dup length exch maxlength eq { dup dup load dup length 2 mul dict copy def } if load begin def end } def /npop % integer npop - { { pop } repeat } def % marking operators /sw % ax ay string sw x y { dup length exch stringwidth exch 5 -1 roll 3 index 1 sub mul add 4 1 roll 3 1 roll 1 sub mul add } def /swj % cx cy fillchar ax ay string swj x y { dup 4 1 roll dup length exch stringwidth exch 5 -1 roll 3 index 1 sub mul add 4 1 roll 3 1 roll 1 sub mul add 6 2 roll /_cnt 0 ddef {1 index eq {/_cnt _cnt 1 add ddef} if} forall pop exch _cnt mul exch _cnt mul 2 index add 4 1 roll 2 index add 4 1 roll pop pop } def /ss % ax ay string matrix ss - { 4 1 roll { % matrix ax ay char 0 0 {proc} - 2 npop (0) exch 2 copy 0 exch put pop gsave false charpath currentpoint 4 index setmatrix stroke grestore moveto 2 copy rmoveto } exch cshow 3 npop } def /jss % cx cy fillchar ax ay string matrix jss - { 4 1 roll { % cx cy fillchar matrix ax ay char 0 0 {proc} - 2 npop (0) exch 2 copy 0 exch put gsave _sp eq { exch 6 index 6 index 6 index 5 -1 roll widthshow currentpoint } { false charpath currentpoint 4 index setmatrix stroke }ifelse grestore moveto 2 copy rmoveto } exch cshow 6 npop } def % path operators /sp % ax ay string sp - { { 2 npop (0) exch 2 copy 0 exch put pop false charpath 2 copy rmoveto } exch cshow 2 npop } def /jsp % cx cy fillchar ax ay string jsp - { { % cx cy fillchar ax ay char 0 0 {proc} - 2 npop (0) exch 2 copy 0 exch put _sp eq { exch 5 index 5 index 5 index 5 -1 roll widthshow } { false charpath }ifelse 2 copy rmoveto } exch cshow 5 npop } def % path construction operators /pl % x y pl x y { transform 0.25 sub round 0.25 add exch 0.25 sub round 0.25 add exch itransform } def /setstrokeadjust where { pop true setstrokeadjust /c % x1 y1 x2 y2 x3 y3 c - { curveto } def /C /c load def /v % x2 y2 x3 y3 v - { currentpoint 6 2 roll curveto } def /V /v load def /y % x1 y1 x2 y2 y - { 2 copy curveto } def /Y /y load def /l % x y l - { lineto } def /L /l load def /m % x y m - { moveto } def } {%else /c { pl curveto } def /C /c load def /v { currentpoint 6 2 roll pl curveto } def /V /v load def /y { pl 2 copy curveto } def /Y /y load def /l { pl lineto } def /L /l load def /m { pl moveto } def }ifelse % graphic state operators /d % array phase d - { setdash } def /cf {} def % - cf flatness /i % flatness i - { dup 0 eq { pop cf } if setflat } def /j % linejoin j - { setlinejoin } def /J % linecap J - { setlinecap } def /M % miterlimit M - { setmiterlimit } def /w % linewidth w - { setlinewidth } def % path painting operators /H % - H - {} def /h % - h - { closepath } def /N % - N - { _pola 0 eq { _doClip 1 eq {clip /_doClip 0 ddef} if newpath } { /CRender {N} ddef }ifelse } def /n % - n - {N} def /F % - F - { _pola 0 eq { _doClip 1 eq { gsave _pf grestore clip newpath /_lp /none ddef _fc /_doClip 0 ddef } { _pf }ifelse } { /CRender {F} ddef }ifelse } def /f % - f - { closepath F } def /S % - S - { _pola 0 eq { _doClip 1 eq { gsave _ps grestore clip newpath /_lp /none ddef _sc /_doClip 0 ddef } { _ps }ifelse } { /CRender {S} ddef }ifelse } def /s % - s - { closepath S } def /B % - B - { _pola 0 eq { _doClip 1 eq % F clears _doClip gsave F grestore { gsave S grestore clip newpath /_lp /none ddef _sc /_doClip 0 ddef } { S }ifelse } { /CRender {B} ddef }ifelse } def /b % - b - { closepath B } def /W % - W - { /_doClip 1 ddef } def /* % - [string] * - { count 0 ne { dup type (stringtype) eq {pop} if } if _pola 0 eq {newpath} if } def % group operators /u % - u - {} def /U % - U - {} def /q % - q - { _pola 0 eq {gsave} if } def /Q % - Q - { _pola 0 eq {grestore} if } def /*u % - *u - { _pola 1 add /_pola exch ddef } def /*U % - *U - { _pola 1 sub /_pola exch ddef _pola 0 eq {CRender} if } def /D % polarized D - {pop} def /*w % - *w - {} def /*W % - *W - {} def % place operators /` % matrix llx lly urx ury string ` - { /_i save ddef 6 1 roll 4 npop concat userdict begin /showpage {} def false setoverprint pop } def /~ % - ~ - { end _i restore } def % color operators /O % flag O - { 0 ne /_of exch ddef /_lp /none ddef } def /R % flag R - { 0 ne /_os exch ddef /_lp /none ddef } def /g % gray g - { /_gf exch ddef /_fc { _lp /fill ne { _of setoverprint _gf setgray /_lp /fill ddef } if } ddef /_pf { _fc fill } ddef /_psf { _fc ashow } ddef /_pjsf { _fc awidthshow } ddef /_lp /none ddef } def /G % gray G - { /_gs exch ddef /_sc { _lp /stroke ne { _os setoverprint _gs setgray /_lp /stroke ddef } if } ddef /_ps { _sc stroke } ddef /_pss { _sc ss } ddef /_pjss { _sc jss } ddef /_lp /none ddef } def /k % cyan magenta yellow black k - { _cf astore pop /_fc { _lp /fill ne { _of setoverprint _cf aload pop setcmykcolor /_lp /fill ddef } if } ddef /_pf { _fc fill } ddef /_psf { _fc ashow } ddef /_pjsf { _fc awidthshow } ddef /_lp /none ddef } def /K % cyan magenta yellow black K - { _cs astore pop /_sc { _lp /stroke ne { _os setoverprint _cs aload pop setcmykcolor /_lp /stroke ddef } if } ddef /_ps { _sc stroke } ddef /_pss { _sc ss } ddef /_pjss { _sc jss } ddef /_lp /none ddef } def /x % cyan magenta yellow black name gray x - { /_gf exch ddef findcmykcustomcolor /_if exch ddef /_fc { _lp /fill ne { _of setoverprint _if _gf 1 exch sub setcustomcolor /_lp /fill ddef } if } ddef /_pf { _fc fill } ddef /_psf { _fc ashow } ddef /_pjsf { _fc awidthshow } ddef /_lp /none ddef } def /X % cyan magenta yellow black name gray X - { /_gs exch ddef findcmykcustomcolor /_is exch ddef /_sc { _lp /stroke ne { _os setoverprint _is _gs 1 exch sub setcustomcolor /_lp /stroke ddef } if } ddef /_ps { _sc stroke } ddef /_pss { _sc ss } ddef /_pjss { _sc jss } ddef /_lp /none ddef } def % locked object operator /A % value A - { pop } def currentdict readonly pop end setpacking % annotate page operator /annotatepage { } def %%EndResource %AI3-Grid.0 18 18 3 0 0 0 3 %%EndProlog %%BeginSetup Adobe_cmykcolor /initialize get exec Adobe_cshow /initialize get exec Adobe_customcolor /initialize get exec Adobe_IllustratorA_AI3 /initialize get exec %%EndSetup 0 A u u u u 0 R 0 G 0 i 0 J 0 j 1 w 4 M []0 d %AI3_Note: 0 D 103.8711 412.8662 m 134.8572 390.3529 L 103.8706 367.8406 L 111.1857 390.3537 L 103.8706 412.8666 L 103.8711 412.8662 L s U u 65.5686 367.8405 m 34.5839 390.3542 L 22.7471 353.9286 L 46.4184 353.9279 L 65.5689 367.8411 L 65.5686 367.8405 L s U u 84.7177 308.9012 m 115.7044 331.4135 L 84.7182 353.9268 L 77.4031 331.4137 L 84.7181 308.9008 L 84.7177 308.9012 L s U u 84.7197 353.9272 m 115.7044 331.4135 L 127.5412 367.8391 L 103.8699 367.8398 L 84.7194 353.9266 L 84.7197 353.9272 L s U u 46.4183 353.9272 m 84.7197 353.9272 L 72.8838 390.3536 L 65.5686 367.8405 L 46.4178 353.9271 L 46.4183 353.9272 L s U u 111.1856 390.3527 m 72.8849 390.3538 L 84.7197 353.9272 L 103.8706 367.8406 L 111.1855 390.3533 L 111.1856 390.3527 L s U u 84.7197 353.9272 m 46.4189 353.9283 L 58.2537 317.5017 L 77.4046 331.4151 L 84.7196 353.9278 L 84.7197 353.9272 L s U u 103.8685 294.9884 m 65.5678 294.9895 L 77.4046 331.4151 L 84.7189 308.9021 L 103.8691 294.9883 L 103.8685 294.9884 L s U U u s U U u u u 1 w 127.5397 367.8398 m 115.7026 331.4149 L 154.0028 331.4132 L 146.6889 353.926 L 127.5391 367.8399 L 127.5397 367.8398 L s U u 103.8682 367.8407 m 134.8548 390.3523 L 146.6889 353.926 L 127.5388 367.8401 L 103.8677 367.8404 L 103.8682 367.8407 L s U u 158.5257 390.3513 m 196.8266 390.3507 L 184.9913 426.7769 L 177.676 404.2642 L 158.5251 390.3512 L 158.5257 390.3513 L s U u 134.8548 390.3523 m 146.6889 353.926 L 177.6755 376.4375 L 158.5253 390.3517 L 134.8543 390.352 L 134.8548 390.3523 L s U u 177.6759 404.2634 m 146.692 426.7772 L 134.8548 390.3523 L 158.5257 390.3513 L 177.6762 404.264 L 177.6759 404.2634 L s U U u s U U u u u 1 w 77.4019 258.5626 m 65.5666 294.989 L 103.8672 294.9884 L 84.7166 281.0748 L 77.402 258.562 L 77.4019 258.5626 L s U u 134.8538 272.4756 m 146.6877 236.0498 L 177.6739 258.561 L 158.524 272.475 L 134.8533 272.4753 L 134.8538 272.4756 L s U u 154.0052 331.4132 m 115.7046 331.4138 L 127.5398 294.9875 L 146.6905 308.9011 L 154.0051 331.4139 L 154.0052 331.4132 L s U u 127.5384 294.9881 m 115.7046 331.4138 L 84.7184 308.9027 L 103.8683 294.9887 L 127.539 294.9883 L 127.5384 294.9881 L s U u 158.5245 272.4755 m 127.5384 294.9881 L 115.7032 258.5619 L 134.8538 272.4756 L 158.525 272.4752 L 158.5245 272.4755 L s U u 84.7173 281.0756 m 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ax ay string sw x y { dup length exch stringwidth exch 5 -1 roll 3 index 1 sub mul add 4 1 roll 3 1 roll 1 sub mul add } def /swj % cx cy fillchar ax ay string swj x y { dup 4 1 roll dup length exch stringwidth exch 5 -1 roll 3 index 1 sub mul add 4 1 roll 3 1 roll 1 sub mul add 6 2 roll /_cnt 0 ddef {1 index eq {/_cnt _cnt 1 add ddef} if} forall pop exch _cnt mul exch _cnt mul 2 index add 4 1 roll 2 index add 4 1 roll pop pop } def /ss % ax ay string matrix ss - { 4 1 roll { % matrix ax ay char 0 0 {proc} - 2 npop (0) exch 2 copy 0 exch put pop gsave false charpath currentpoint 4 index setmatrix stroke grestore moveto 2 copy rmoveto } exch cshow 3 npop } def /jss % cx cy fillchar ax ay string matrix jss - { 4 1 roll { % cx cy fillchar matrix ax ay char 0 0 {proc} - 2 npop (0) exch 2 copy 0 exch put gsave _sp eq { exch 6 index 6 index 6 index 5 -1 roll widthshow currentpoint } { false charpath currentpoint 4 index setmatrix stroke }ifelse grestore moveto 2 copy rmoveto } exch cshow 6 npop } def % path operators /sp % ax ay string sp - { { 2 npop (0) exch 2 copy 0 exch put pop false charpath 2 copy rmoveto } exch cshow 2 npop } def /jsp % cx cy fillchar ax ay string jsp - { { % cx cy fillchar ax ay char 0 0 {proc} - 2 npop (0) exch 2 copy 0 exch put _sp eq { exch 5 index 5 index 5 index 5 -1 roll widthshow } { false charpath }ifelse 2 copy rmoveto } exch cshow 5 npop } def % path construction operators /pl % x y pl x y { transform 0.25 sub round 0.25 add exch 0.25 sub round 0.25 add exch itransform } def /setstrokeadjust where { pop true setstrokeadjust /c % x1 y1 x2 y2 x3 y3 c - { curveto } def /C /c load def /v % x2 y2 x3 y3 v - { currentpoint 6 2 roll curveto } def /V /v load def /y % x1 y1 x2 y2 y - { 2 copy curveto } def /Y /y load def /l % x y l - { lineto } def /L /l load def /m % x y m - { moveto } def } {%else /c { pl curveto } def /C /c load def /v { currentpoint 6 2 roll pl curveto } def /V /v load def /y { pl 2 copy curveto } def /Y /y load def /l { pl 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_doClip gsave F grestore { gsave S grestore clip newpath /_lp /none ddef _sc /_doClip 0 ddef } { S }ifelse } { /CRender {B} ddef }ifelse } def /b % - b - { closepath B } def /W % - W - { /_doClip 1 ddef } def /* % - [string] * - { count 0 ne { dup type (stringtype) eq {pop} if } if _pola 0 eq {newpath} if } def % group operators /u % - u - {} def /U % - U - {} def /q % - q - { _pola 0 eq {gsave} if } def /Q % - Q - { _pola 0 eq {grestore} if } def /*u % - *u - { _pola 1 add /_pola exch ddef } def /*U % - *U - { _pola 1 sub /_pola exch ddef _pola 0 eq {CRender} if } def /D % polarized D - {pop} def /*w % - *w - {} def /*W % - *W - {} def % place operators /` % matrix llx lly urx ury string ` - { /_i save ddef 6 1 roll 4 npop concat userdict begin /showpage {} def false setoverprint pop } def /~ % - ~ - { end _i restore } def % color operators /O % flag O - { 0 ne /_of exch ddef /_lp /none ddef } def /R % flag R - { 0 ne /_os exch ddef /_lp /none ddef } def /g % gray g - { /_gf exch ddef /_fc { _lp /fill ne { _of setoverprint _gf setgray /_lp /fill ddef } if } ddef /_pf { _fc fill } ddef /_psf { _fc ashow } ddef /_pjsf { _fc awidthshow } ddef /_lp /none ddef } def /G % gray G - { /_gs exch ddef /_sc { _lp /stroke ne { _os setoverprint _gs setgray /_lp /stroke ddef } if } ddef /_ps { _sc stroke } ddef /_pss { _sc ss } ddef /_pjss { _sc jss } ddef /_lp /none ddef } def /k % cyan magenta yellow black k - { _cf astore pop /_fc { _lp /fill ne { _of setoverprint _cf aload pop setcmykcolor /_lp /fill ddef } if } ddef /_pf { _fc fill } ddef /_psf { _fc ashow } ddef /_pjsf { _fc awidthshow } ddef /_lp /none ddef } def /K % cyan magenta yellow black K - { _cs astore pop /_sc { _lp /stroke ne { _os setoverprint _cs aload pop setcmykcolor /_lp /stroke ddef } if } ddef /_ps { _sc stroke } ddef /_pss { _sc ss } ddef /_pjss { _sc jss } ddef /_lp /none ddef } def /x % cyan magenta yellow black name gray x - { /_gf exch ddef findcmykcustomcolor /_if exch ddef /_fc { _lp /fill ne { _of 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4822 a @beginspecial 173 @llx 273 @lly 418 @urx 528 @ury 5400 @rwi @setspecial %%BeginDocument: junk7.ps %AI3_ColorUsage: Black&White %AI3_TemplateBox: 306 396 306 396 %AI3_TileBox: 30 31 582 761 %AI3_DocumentPreview: Header userdict /Adobe_packedarray 5 dict dup begin put /initialize % - initialize - { /packedarray where { pop } { Adobe_packedarray begin Adobe_packedarray { dup xcheck { bind } if userdict 3 1 roll put } forall end } ifelse } def /terminate % - terminate - { } def /packedarray % arguments count packedarray array { array astore readonly } def /setpacking % boolean setpacking - { pop } def /currentpacking % - setpacking boolean { false } def currentdict readonly pop end Adobe_packedarray /initialize get exec currentpacking true setpacking userdict /Adobe_cmykcolor 4 dict dup begin put /initialize % - initialize - { /setcmykcolor where { pop } { userdict /Adobe_cmykcolor_vars 2 dict dup begin put /_setrgbcolor /setrgbcolor load def /_currentrgbcolor /currentrgbcolor load def Adobe_cmykcolor begin Adobe_cmykcolor { dup xcheck { bind } if pop pop } forall end end Adobe_cmykcolor begin } ifelse } def /terminate % - terminate - { currentdict Adobe_cmykcolor eq { end } if } def /setcmykcolor % cyan magenta yellow black setcmykcolor - { 1 sub 4 1 roll 3 { 3 index add neg dup 0 lt { pop 0 } if 3 1 roll } repeat Adobe_cmykcolor_vars /_setrgbcolor get exec pop } def /currentcmykcolor % - currentcmykcolor cyan magenta yellow black { Adobe_cmykcolor_vars /_currentrgbcolor get exec 3 { 1 sub neg 3 1 roll } repeat 0 } def currentdict readonly pop end setpacking currentpacking true setpacking userdict /Adobe_cshow 3 dict dup begin put /initialize % - initialize - { /cshow where { pop } { userdict /Adobe_cshow_vars 1 dict dup begin put /_cshow % - _cshow proc {} def Adobe_cshow begin Adobe_cshow { dup xcheck { bind } if userdict 3 1 roll put } forall end end } ifelse } def /terminate % - terminate - { } def /cshow % proc string cshow - { exch Adobe_cshow_vars exch /_cshow exch put { 0 0 Adobe_cshow_vars /_cshow get exec } forall } def currentdict readonly pop end setpacking currentpacking true setpacking userdict /Adobe_customcolor 5 dict dup begin put /initialize % - initialize - { /setcustomcolor where { pop } { Adobe_customcolor begin Adobe_customcolor { dup xcheck { bind } if pop pop } forall end Adobe_customcolor begin } ifelse } def /terminate % - terminate - { currentdict Adobe_customcolor eq { end } if } def /findcmykcustomcolor % cyan magenta yellow black name findcmykcustomcolor object { 5 packedarray } def /setcustomcolor % object tint setcustomcolor - { exch aload pop pop 4 { 4 index mul 4 1 roll } repeat 5 -1 roll pop setcmykcolor } def /setoverprint % boolean setoverprint - { pop } def currentdict readonly pop end setpacking currentpacking true setpacking userdict /Adobe_IllustratorA_AI3 61 dict dup begin put % initialization /initialize % - initialize - { % 47 vars, but leave slack of 10 entries for custom Postscript fragments userdict /Adobe_IllustratorA_AI3_vars 57 dict dup begin put % paint operands /_lp /none def /_pf {} def /_ps {} def /_psf {} def /_pss {} def /_pjsf {} def /_pjss {} def /_pola 0 def /_doClip 0 def % paint operators /cf currentflat def % - cf flatness % typography operands /_tm matrix def /_renderStart [/e0 /r0 /a0 /o0 /e1 /r1 /a1 /i0] def /_renderEnd [null null null null /i1 /i1 /i1 /i1] def /_render -1 def /_rise 0 def /_ax 0 def % x character spacing (_ax, _ay, _cx, _cy follows awidthshow naming convention) /_ay 0 def % y character spacing /_cx 0 def % x word spacing /_cy 0 def % y word spacing /_leading [0 0] def /_ctm matrix def /_mtx matrix def /_sp 16#020 def /_hyphen (-) def /_fScl 0 def /_cnt 0 def /_hs 1 def /_nativeEncoding 0 def /_useNativeEncoding 0 def /_tempEncode 0 def /_pntr 0 def /_tDict 2 dict def % typography operators /Tx {} def /Tj {} def % compound path operators /CRender {} def % printing /_AI3_savepage {} def % color operands /_gf null def /_cf 4 array def /_if null def /_of false def /_fc {} def /_gs null def /_cs 4 array def /_is null def /_os false def /_sc {} def /_i null def Adobe_IllustratorA_AI3 begin Adobe_IllustratorA_AI3 { dup xcheck { bind } if pop pop } forall end end Adobe_IllustratorA_AI3 begin Adobe_IllustratorA_AI3_vars begin newpath } def /terminate % - terminate - { end end } def % definition operators /_ % - _ null null def /ddef % key value ddef - { Adobe_IllustratorA_AI3_vars 3 1 roll put } def /xput % key value literal xput - { dup load dup length exch maxlength eq { dup dup load dup length 2 mul dict copy def } if load begin def end } def /npop % integer npop - { { pop } repeat } def % marking operators /sw % ax ay string sw x y { dup length exch stringwidth exch 5 -1 roll 3 index 1 sub mul add 4 1 roll 3 1 roll 1 sub mul add } def /swj % cx cy fillchar ax ay string swj x y { dup 4 1 roll dup length exch stringwidth exch 5 -1 roll 3 index 1 sub mul add 4 1 roll 3 1 roll 1 sub mul add 6 2 roll /_cnt 0 ddef {1 index eq {/_cnt _cnt 1 add ddef} if} forall pop exch _cnt mul exch _cnt mul 2 index add 4 1 roll 2 index add 4 1 roll pop pop } def /ss % ax ay string matrix ss - { 4 1 roll { % matrix ax ay char 0 0 {proc} - 2 npop (0) exch 2 copy 0 exch put pop gsave false charpath currentpoint 4 index setmatrix stroke grestore moveto 2 copy rmoveto } exch cshow 3 npop } def /jss % cx cy fillchar ax ay string matrix jss - { 4 1 roll { % cx cy fillchar matrix ax ay char 0 0 {proc} - 2 npop (0) exch 2 copy 0 exch put gsave _sp eq { exch 6 index 6 index 6 index 5 -1 roll widthshow currentpoint } { false charpath currentpoint 4 index setmatrix stroke }ifelse grestore moveto 2 copy rmoveto } exch cshow 6 npop } def % path operators /sp % ax ay string sp - { { 2 npop (0) exch 2 copy 0 exch put pop false charpath 2 copy rmoveto } exch cshow 2 npop } def /jsp % cx cy fillchar ax ay string jsp - { { % cx cy fillchar ax ay char 0 0 {proc} - 2 npop (0) exch 2 copy 0 exch put _sp eq { exch 5 index 5 index 5 index 5 -1 roll widthshow } { false charpath }ifelse 2 copy rmoveto } exch cshow 5 npop } def % path construction operators /pl % x y pl x y { transform 0.25 sub round 0.25 add exch 0.25 sub round 0.25 add exch itransform } def /setstrokeadjust where { pop true setstrokeadjust /c % x1 y1 x2 y2 x3 y3 c - { curveto } def /C /c load def /v % x2 y2 x3 y3 v - { currentpoint 6 2 roll curveto } def /V /v load def /y % x1 y1 x2 y2 y - { 2 copy curveto } def /Y /y load def /l % x y l - { lineto } def /L /l load def /m % x y m - { moveto } def } {%else /c { pl curveto } def /C /c load def /v { currentpoint 6 2 roll pl curveto } def /V /v load def /y { pl 2 copy curveto } def /Y /y load def /l { pl lineto } def /L /l load def /m { pl moveto } def }ifelse % graphic state operators /d % array phase d - { setdash } def /cf {} def % - cf flatness /i % flatness i - { dup 0 eq { pop cf } if setflat } def /j % linejoin j - { setlinejoin } def /J % linecap J - { setlinecap } def /M % miterlimit M - { setmiterlimit } def /w % linewidth w - { setlinewidth } def % path painting operators /H % - H - {} def /h % - h - { closepath } def /N % - N - { _pola 0 eq { _doClip 1 eq {clip /_doClip 0 ddef} if newpath } { /CRender {N} ddef }ifelse } def /n % - n - {N} def /F % - F - { _pola 0 eq { _doClip 1 eq { gsave _pf grestore clip newpath /_lp /none ddef _fc /_doClip 0 ddef } { _pf }ifelse } { /CRender {F} ddef }ifelse } def /f % - f - { closepath F } def /S % - S - { _pola 0 eq { _doClip 1 eq { gsave _ps grestore clip newpath /_lp /none ddef _sc /_doClip 0 ddef } { _ps }ifelse } { /CRender {S} ddef }ifelse } def /s % - s - { closepath S } def /B % - B - { _pola 0 eq { _doClip 1 eq % F clears _doClip gsave F grestore { gsave S grestore clip newpath /_lp /none ddef _sc /_doClip 0 ddef } { S }ifelse } { /CRender {B} ddef }ifelse } def /b % - b - { closepath B } def /W % - W - { /_doClip 1 ddef } def /* % - [string] * - { count 0 ne { dup type (stringtype) eq {pop} if } if _pola 0 eq {newpath} if } def % group operators /u % - u - {} def /U % - U - {} def /q % - q - { _pola 0 eq {gsave} if } def /Q % - Q - { _pola 0 eq {grestore} if } def /*u % - *u - { _pola 1 add /_pola exch ddef } def /*U % - *U - { _pola 1 sub /_pola exch ddef _pola 0 eq {CRender} if } def /D % polarized D - {pop} def /*w % - *w - {} def /*W % - *W - {} def % place operators /` % matrix llx lly urx ury string ` - { /_i save ddef 6 1 roll 4 npop concat userdict begin /showpage {} def false setoverprint pop } def /~ % - ~ - { end _i restore } def % color operators /O % flag O - { 0 ne /_of exch ddef /_lp /none ddef } def /R % flag R - { 0 ne /_os exch ddef /_lp /none ddef } def /g % gray g - { /_gf exch ddef /_fc { _lp /fill ne { _of setoverprint _gf setgray /_lp /fill ddef } if } ddef /_pf { _fc fill } ddef /_psf { _fc ashow } ddef /_pjsf { _fc awidthshow } ddef /_lp /none ddef } def /G % gray G - { /_gs exch ddef /_sc { _lp /stroke ne { _os setoverprint _gs setgray /_lp /stroke ddef } if } ddef /_ps { _sc stroke } ddef /_pss { _sc ss } ddef /_pjss { _sc jss } ddef /_lp /none ddef } def /k % cyan magenta yellow black k - { _cf astore pop /_fc { _lp /fill ne { _of setoverprint _cf aload pop setcmykcolor /_lp /fill ddef } if } ddef /_pf { _fc fill } ddef /_psf { _fc ashow } ddef /_pjsf { _fc awidthshow } ddef /_lp /none ddef } def /K % cyan magenta yellow black K - { _cs astore pop /_sc { _lp /stroke ne { _os setoverprint _cs aload pop setcmykcolor /_lp /stroke ddef } if } ddef /_ps { _sc stroke } ddef /_pss { _sc ss } ddef /_pjss { _sc jss } ddef /_lp /none ddef } def /x % cyan magenta yellow black name gray x - { /_gf exch ddef findcmykcustomcolor /_if exch ddef /_fc { _lp /fill ne { _of setoverprint _if _gf 1 exch sub setcustomcolor /_lp /fill ddef } if } ddef /_pf { _fc fill } ddef /_psf { _fc ashow } ddef /_pjsf { _fc awidthshow } ddef /_lp /none ddef } def /X % cyan magenta yellow black name gray X - { /_gs exch ddef findcmykcustomcolor /_is exch ddef /_sc { _lp /stroke ne { _os setoverprint 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3518 445 L 76 104 V 25 -52 V 3518 445 L 76 104 R 101 -78 V -51 -26 V -50 104 V 3191 860 M 0 -130 V 50 26 V -50 104 V 3292 652 M -101 78 V 50 26 V 51 -104 V -101 78 R 101 -78 V -51 -26 V -50 104 V 0 -129 V 50 25 V -50 104 V 0 -129 R 126 0 V -25 51 V 3191 601 L 0 -260 R 0 130 V 50 -26 V 3191 341 L 101 208 R 3191 471 L 50 -26 V 51 104 V 3191 471 M 101 78 V -51 26 V 3191 471 L 0 130 V 50 -26 V 3191 471 L 0 130 R 126 0 V -25 -52 V -101 52 V 126 0 R 0 -130 V -51 26 V 51 104 V 3216 393 M 101 78 V -51 26 V 3216 393 L 101 78 R 3216 393 L 50 -26 V 51 104 V 0 -130 V -51 26 V 51 104 V 0 -130 R -126 0 V 25 52 V 101 -52 V 0 260 R 0 -130 V 50 26 V -50 104 V 3417 393 M -100 78 V 50 26 V 50 -104 V -100 78 R 100 -78 V -50 -26 V -50 104 V 0 -130 V 50 26 V -50 104 V 0 -130 R 126 0 V -26 52 V 3317 341 L 126 0 R -126 0 V 25 -52 V 101 52 V 3241 237 M 76 104 V 25 -52 V 3241 237 L 76 104 R 3241 237 L -25 52 V 101 52 V -126 0 V 25 -52 V 101 52 V -126 0 R 0 -130 V 50 26 V -50 104 V 0 -130 R 126 0 V -25 52 V 3191 211 L 201 104 R 3317 211 L -25 52 V 100 52 V 3317 211 M 75 104 V 25 -52 V 3317 211 L 126 0 V -26 52 V 3317 211 L 126 0 R 0 130 V -51 -26 V 51 -104 V 252 0 R -126 0 V 25 52 V 101 -52 V 3493 315 M 76 -104 V 25 52 V -101 52 V 76 -104 R -76 104 V -25 -52 V 101 -52 V -126 0 V 25 52 V 101 -52 V -126 0 R 0 130 V 50 -26 V 3443 211 L 0 130 R 126 0 V -26 -52 V -100 52 V 3644 237 M -75 104 V -26 -52 V 101 -52 V -75 104 R 75 -104 V 25 52 V -100 52 V 126 0 V -26 -52 V -100 52 V 126 0 R 0 -130 V -51 26 V 51 104 V -252 0 R 126 0 V -26 52 V 3443 341 L 201 104 R 3569 341 L -26 52 V 101 52 V 3569 341 M 75 104 V 25 -52 V 3569 341 L 126 0 V -26 52 V 3569 341 L 126 0 R 0 130 V -51 -26 V 51 -104 V 0 130 R 0 -130 V 50 26 V -50 104 V 3795 263 M -100 78 V 50 26 V 50 -104 V -100 78 R 100 -78 V -50 -26 V -50 104 V 0 -130 V 50 26 V -50 104 V 0 -130 R 126 0 V -26 52 V 3695 211 L 5710 2808 M -76 -104 V -25 52 V 101 52 V 5508 2704 M 126 0 V -25 52 V -101 -52 V 126 0 R -126 0 V 25 -52 V 101 52 V -76 -104 V -25 52 V 101 52 V -76 -104 R -100 78 V 50 26 V 50 -104 V 5407 2392 M 76 104 V -56 11 V -20 -115 V 41 229 R 35 -125 V -56 11 V 21 114 V currentpoint stroke M 35 -125 R -35 125 V 55 -11 V -20 -114 V 75 104 V -55 10 V -20 -114 V 75 104 R -100 78 V -10 -57 V 110 -21 V -100 78 R -76 -104 V 56 -10 V 20 114 V -41 -229 R -35 125 V 56 -10 V -21 -115 V -35 125 R 35 -125 V -55 11 V 20 114 V -75 -104 V 55 -10 V 20 114 V -75 -104 R 100 -78 V 10 57 V -110 21 V 151 208 R -76 -104 V -25 52 V 101 52 V 5256 2574 M 126 0 V -25 52 V -101 -52 V 126 0 R -126 0 V 25 -52 V 101 52 V -75 -104 V -26 52 V 101 52 V -75 -104 R -101 78 V 50 26 V 51 -104 V -101 78 R 101 -78 V -51 -26 V -50 104 V 101 -208 R 0 130 V -51 -26 V 51 -104 V 0 130 R 0 -130 V 50 26 V -50 104 V 100 -78 V -50 -26 V -50 104 V 100 -78 R -75 -104 V -25 52 V 100 52 V 4702 2288 M 126 0 V -25 52 V -101 -52 V 202 104 R -76 -104 V -25 52 V 101 52 V -76 -104 R 76 104 V 25 -52 V -101 -52 V 126 0 V -25 52 V -101 -52 V 126 0 R 0 130 V -50 -26 V 50 -104 V 252 0 R 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126 0 V -25 -51 V -101 51 V 126 0 R 0 -129 V -50 26 V 50 103 V 0 -129 R -126 0 V 25 52 V 101 -52 V -202 103 R 76 -103 V 25 52 V -101 51 V 76 -103 R -76 103 V -25 -51 V 101 -52 V -126 0 V 25 52 V 101 -52 V -126 0 R 0 129 V 50 -26 V -50 -103 V 252 0 R -126 0 V 25 -52 V 101 52 V 4878 2055 M 76 104 V 25 -52 V -101 -52 V 76 104 R -76 -104 V -25 52 V 101 52 V -126 0 V 25 -52 V 101 52 V -126 0 R 0 -130 V 50 26 V -50 104 V 0 -130 R 0 130 V -50 -26 V 50 -104 V -101 208 R 101 -78 V -50 -26 V -51 104 V 101 -78 R -101 78 V 51 25 V 50 -103 V 0 129 V -50 -26 V 50 -103 V 0 129 R -126 0 V 25 -51 V 101 51 V 504 0 R -76 -103 V -25 52 V 101 51 V 5130 2185 M 126 0 V -25 52 V -101 -52 V 126 0 R -126 0 V 25 -52 V 101 52 V -75 -104 V -26 52 V 101 52 V -75 -104 R -101 78 V 50 26 V 51 -104 V 5030 1873 M 75 104 V -55 10 V -20 -114 V 40 228 R 35 -124 V -55 10 V 20 114 V 35 -124 R -35 124 V 55 -10 V -20 -114 V 76 104 V -56 10 V -20 -114 V 76 104 R -101 78 V -10 -58 V 111 -20 V -101 78 R -76 -104 V 56 -11 V 20 115 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10 57 V 111 -20 V -100 78 V -11 -58 V 111 -20 V -100 78 R -76 -104 V 55 -11 V 21 115 V 201 -156 R -101 78 V 51 26 V 50 -104 V -101 208 R 0 -130 V 51 26 V -51 104 V 0 -130 R 0 130 V -50 -26 V 50 -104 V -100 78 V 50 26 V 50 -104 V -100 78 R 75 103 V 25 -51 V -100 -52 V 75 103 R -75 -103 V -26 52 V 101 51 V 4501 2185 M 126 0 V -26 52 V -100 -52 V 126 0 R -126 0 V 25 -52 V 101 52 V -76 -104 V -25 52 V 101 52 V -76 -104 R -101 78 V 51 26 V 50 -104 V 4198 730 M 76 104 V -56 11 V 4198 730 L 41 229 R 35 -125 V -56 11 V 21 114 V 35 -125 R -35 125 V 55 -11 V 4274 834 L 75 104 V -55 10 V 4274 834 L 75 104 R -100 78 V -10 -57 V 110 -21 V 152 208 R -76 -104 V -25 52 V 101 52 V 4299 1042 M 126 0 V -25 52 V -101 -52 V 126 0 R -126 0 V 25 -52 V 101 52 V 4349 938 L -25 52 V 101 52 V 4349 938 M -100 78 V 50 26 V 50 -104 V -100 78 R 75 104 V 25 -52 V -100 -52 V 201 104 R -126 0 V 25 -52 V 101 52 V -126 0 R 126 0 V -25 52 V -101 -52 V 76 104 V 25 -52 V -101 -52 V 76 104 R 101 -78 V -51 -26 V -50 104 V 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65781.76 div N}N /startTexFig{/psf$SavedState save N userdict maxlength dict begin /magscale false def normalscale currentpoint TR /psf$ury psfts /psf$urx psfts /psf$lly psfts /psf$llx psfts /psf$y psfts /psf$x psfts currentpoint /psf$cy X /psf$cx X /psf$sx psf$x psf$urx psf$llx sub div N /psf$sy psf$y psf$ury psf$lly sub div N psf$sx psf$sy scale psf$cx psf$sx div psf$llx sub psf$cy psf$sy div psf$ury sub TR /showpage{}N /erasepage{}N /copypage{}N /p 3 def @MacSetUp}N /doclip{ psf$llx psf$lly psf$urx psf$ury currentpoint 6 2 roll newpath 4 copy 4 2 roll moveto 6 -1 roll S lineto S lineto S lineto closepath clip newpath moveto}N /endTexFig{end psf$SavedState restore}N /@beginspecial{SDict begin /SpecialSave save N gsave normalscale currentpoint TR @SpecialDefaults count /ocount X /dcount countdictstack N}N /@setspecial {CLIP 1 eq{newpath 0 0 moveto hs 0 rlineto 0 vs rlineto hs neg 0 rlineto closepath clip}if ho vo TR hsc vsc scale ang rotate rwiSeen{rwi urx llx sub div rhiSeen{rhi ury lly sub div}{dup}ifelse scale llx neg lly neg TR }{rhiSeen{rhi ury lly sub div dup scale llx neg lly neg TR}if}ifelse CLIP 2 eq{newpath llx lly moveto urx lly lineto urx ury lineto llx ury lineto closepath clip}if /showpage{}N /erasepage{}N /copypage{}N newpath }N /@endspecial{count ocount sub{pop}repeat countdictstack dcount sub{ end}repeat grestore SpecialSave restore end}N /@defspecial{SDict begin} N /@fedspecial{end}B /li{lineto}B /rl{rlineto}B /rc{rcurveto}B /np{ /SaveX currentpoint /SaveY X N 1 setlinecap newpath}N /st{stroke SaveX SaveY moveto}N /fil{fill SaveX SaveY moveto}N /ellipse{/endangle X /startangle X /yrad X /xrad X /savematrix matrix currentmatrix N TR xrad yrad scale 0 0 1 startangle endangle arc savematrix setmatrix}N end %%EndProcSet TeXDict begin 40258431 52099146 1200 400 400 (/tmp_mnt/d6/faculty/radin/graphics/fault2.dvi) @start end %%EndProlog %%BeginSetup %%Feature: *Resolution 400dpi TeXDict begin %%EndSetup %%Page: 2 1 2 0 bop 332 2933 a @beginspecial 0 @llx 0 @lly 460 @urx 727 @ury 2784 @rwi @setspecial %%BeginDocument: fault2.eps /$F2psDict 200 dict def $F2psDict begin $F2psDict /mtrx matrix put /l {lineto} bind def /m {moveto} bind def /s {stroke} bind def /n {newpath} bind def /gs {gsave} bind def /gr {grestore} bind def /clp {closepath} bind def /graycol {dup dup currentrgbcolor 4 -2 roll mul 4 -2 roll mul 4 -2 roll mul setrgbcolor} bind def /col-1 {} def /col0 {0 0 0 setrgbcolor} bind def /col1 {0 0 1 setrgbcolor} bind def /col2 {0 1 0 setrgbcolor} bind def /col3 {0 1 1 setrgbcolor} bind def /col4 {1 0 0 setrgbcolor} bind def /col5 {1 0 1 setrgbcolor} bind def /col6 {1 1 0 setrgbcolor} bind def /col7 {1 1 1 setrgbcolor} bind def /DrawEllipse { /endangle exch def /startangle exch def /yrad exch def /xrad exch def /y exch def /x exch def /savematrix mtrx currentmatrix def x y translate xrad yrad scale 0 0 1 startangle endangle arc savematrix setmatrix } def end /$F2psBegin {$F2psDict begin /$F2psEnteredState save def} def /$F2psEnd 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col-1 s gr % Ellipse n 369 249 5 5 0 360 DrawEllipse gs 0.00 setgray fill gr gs col-1 s gr % Ellipse n 339 199 5 5 0 360 DrawEllipse gs 0.00 setgray fill gr gs col-1 s gr % Ellipse n 279 199 5 5 0 360 DrawEllipse gs 0.00 setgray fill gr gs col-1 s gr % Ellipse n 189 149 5 5 0 360 DrawEllipse gs 0.00 setgray fill gr gs col-1 s gr % Ellipse n 189 49 5 5 0 360 DrawEllipse gs 0.00 setgray fill gr gs col-1 s gr % Ellipse n 99 99 5 5 0 360 DrawEllipse gs 0.00 setgray fill gr gs col-1 s gr % Ellipse n 159 99 5 5 0 360 DrawEllipse gs 0.00 setgray fill gr gs col-1 s gr % Ellipse n 129 49 5 5 0 360 DrawEllipse gs 0.00 setgray fill gr gs col-1 s gr % Ellipse n 69 49 5 5 0 360 DrawEllipse gs 0.00 setgray fill gr gs col-1 s gr % Ellipse n 69 149 5 5 0 360 DrawEllipse gs 0.00 setgray fill gr gs col-1 s gr % Ellipse n 129 149 5 5 0 360 DrawEllipse gs 0.00 setgray fill gr gs col-1 s gr % Ellipse n 279 99 5 5 0 360 DrawEllipse gs 0.00 setgray fill gr gs col-1 s gr % Ellipse n 219 99 5 5 0 360 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% Ellipse n 429 149 5 5 0 360 DrawEllipse gs 0.00 setgray fill gr gs col-1 s gr % Ellipse n 489 149 5 5 0 360 DrawEllipse gs 0.00 setgray fill gr gs col-1 s gr % Ellipse n 639 99 5 5 0 360 DrawEllipse gs 0.00 setgray fill gr gs col-1 s gr % Ellipse n 579 99 5 5 0 360 DrawEllipse gs 0.00 setgray fill gr gs col-1 s gr % Ellipse n 699 99 5 5 0 360 DrawEllipse gs 0.00 setgray fill gr gs col-1 s gr % Ellipse n 669 49 5 5 0 360 DrawEllipse gs 0.00 setgray fill gr gs col-1 s gr % Ellipse n 609 49 5 5 0 360 DrawEllipse gs 0.00 setgray fill gr gs col-1 s gr % Ellipse n 609 149 5 5 0 360 DrawEllipse gs 0.00 setgray fill gr gs col-1 s gr % Ellipse n 669 149 5 5 0 360 DrawEllipse gs 0.00 setgray fill gr gs col-1 s gr % Ellipse n 579 199 5 5 0 360 DrawEllipse gs 0.00 setgray fill gr gs col-1 s gr % Ellipse n 489 249 5 5 0 360 DrawEllipse gs 0.00 setgray fill gr gs col-1 s gr % Ellipse n 429 249 5 5 0 360 DrawEllipse gs 0.00 setgray fill gr gs col-1 s gr % Ellipse n 549 249 5 5 0 360 DrawEllipse gs 0.00 setgray fill gr gs col-1 s gr % Ellipse n 519 199 5 5 0 360 DrawEllipse gs 0.00 setgray fill gr gs col-1 s gr % Ellipse n 459 199 5 5 0 360 DrawEllipse gs 0.00 setgray fill gr gs col-1 s gr % Ellipse n 669 249 5 5 0 360 DrawEllipse gs 0.00 setgray fill gr gs col-1 s gr % Ellipse n 609 249 5 5 0 360 DrawEllipse gs 0.00 setgray fill gr gs col-1 s gr % Ellipse n 699 199 5 5 0 360 DrawEllipse gs 0.00 setgray fill gr gs col-1 s gr % Ellipse n 639 199 5 5 0 360 DrawEllipse gs 0.00 setgray fill gr gs col-1 s gr % Ellipse n 369 349 5 5 0 360 DrawEllipse gs 0.00 setgray fill gr gs col-1 s gr % Ellipse n 369 449 5 5 0 360 DrawEllipse gs 0.00 setgray fill gr gs col-1 s gr % Ellipse n 189 449 5 5 0 360 DrawEllipse gs 0.00 setgray fill gr gs col-1 s gr % Ellipse n 189 349 5 5 0 360 DrawEllipse gs 0.00 setgray fill gr gs col-1 s gr % Ellipse n 99 399 5 5 0 360 DrawEllipse gs 0.00 setgray fill gr gs col-1 s gr % Ellipse n 159 399 5 5 0 360 DrawEllipse gs 0.00 setgray fill gr gs col-1 s gr % Ellipse n 129 349 5 5 0 360 DrawEllipse gs 0.00 setgray fill gr gs col-1 s gr % Ellipse n 69 349 5 5 0 360 DrawEllipse gs 0.00 setgray fill gr gs col-1 s gr % Ellipse n 69 449 5 5 0 360 DrawEllipse gs 0.00 setgray fill gr gs col-1 s gr % Ellipse n 129 449 5 5 0 360 DrawEllipse gs 0.00 setgray fill gr gs col-1 s gr % Ellipse n 279 399 5 5 0 360 DrawEllipse gs 0.00 setgray fill gr gs col-1 s gr % Ellipse n 219 399 5 5 0 360 DrawEllipse gs 0.00 setgray fill gr gs col-1 s gr % Ellipse n 339 399 5 5 0 360 DrawEllipse gs 0.00 setgray fill gr gs col-1 s gr % Ellipse n 309 349 5 5 0 360 DrawEllipse gs 0.00 setgray fill gr gs col-1 s gr % Ellipse n 249 349 5 5 0 360 DrawEllipse gs 0.00 setgray fill gr gs col-1 s gr % Ellipse n 249 449 5 5 0 360 DrawEllipse gs 0.00 setgray fill gr gs col-1 s gr % Ellipse n 309 449 5 5 0 360 DrawEllipse gs 0.00 setgray fill gr gs col-1 s gr % Ellipse n 399 299 5 5 0 360 DrawEllipse gs 0.00 setgray fill gr gs col-1 s gr % Ellipse n 219 299 5 5 0 360 DrawEllipse gs 0.00 setgray fill gr gs col-1 s gr % Ellipse n 99 299 5 5 0 360 DrawEllipse gs 0.00 setgray fill gr gs col-1 s gr % Ellipse n 159 299 5 5 0 360 DrawEllipse gs 0.00 setgray fill gr gs col-1 s gr % Ellipse n 279 299 5 5 0 360 DrawEllipse gs 0.00 setgray fill gr gs col-1 s gr % Ellipse n 339 299 5 5 0 360 DrawEllipse gs 0.00 setgray fill gr gs col-1 s gr % Ellipse n 579 299 5 5 0 360 DrawEllipse gs 0.00 setgray fill gr gs col-1 s gr % Ellipse n 459 299 5 5 0 360 DrawEllipse gs 0.00 setgray fill gr gs col-1 s gr % Ellipse n 519 299 5 5 0 360 DrawEllipse gs 0.00 setgray fill gr gs col-1 s gr % Ellipse n 639 299 5 5 0 360 DrawEllipse gs 0.00 setgray fill gr gs col-1 s gr % Ellipse n 699 299 5 5 0 360 DrawEllipse gs 0.00 setgray fill gr gs col-1 s gr % Ellipse n 549 449 5 5 0 360 DrawEllipse gs 0.00 setgray fill gr gs col-1 s gr % Ellipse n 549 349 5 5 0 360 DrawEllipse gs 0.00 setgray fill gr gs col-1 s gr % Ellipse n 459 399 5 5 0 360 DrawEllipse gs 0.00 setgray fill gr gs col-1 s gr % Ellipse n 399 399 5 5 0 360 DrawEllipse gs 0.00 setgray fill gr gs col-1 s gr % Ellipse n 519 399 5 5 0 360 DrawEllipse gs 0.00 setgray fill gr gs col-1 s gr % Ellipse n 489 349 5 5 0 360 DrawEllipse gs 0.00 setgray fill gr gs col-1 s gr % Ellipse n 429 349 5 5 0 360 DrawEllipse gs 0.00 setgray fill gr gs col-1 s gr % Ellipse n 429 449 5 5 0 360 DrawEllipse gs 0.00 setgray fill gr gs col-1 s gr % Ellipse n 489 449 5 5 0 360 DrawEllipse gs 0.00 setgray fill gr gs col-1 s gr % Ellipse n 639 399 5 5 0 360 DrawEllipse gs 0.00 setgray fill gr gs col-1 s gr % Ellipse n 579 399 5 5 0 360 DrawEllipse gs 0.00 setgray fill gr gs col-1 s gr % Ellipse n 699 399 5 5 0 360 DrawEllipse gs 0.00 setgray fill gr gs col-1 s gr % Ellipse n 669 349 5 5 0 360 DrawEllipse gs 0.00 setgray fill gr gs col-1 s gr % Ellipse n 609 349 5 5 0 360 DrawEllipse gs 0.00 setgray fill gr gs col-1 s gr % Ellipse n 609 449 5 5 0 360 DrawEllipse gs 0.00 setgray fill gr gs col-1 s gr % Ellipse n 669 449 5 5 0 360 DrawEllipse gs 0.00 setgray fill gr gs col-1 s gr % Ellipse n 369 689 5 5 0 360 DrawEllipse gs 0.00 setgray fill gr gs col-1 s gr % Ellipse n 369 589 5 5 0 360 DrawEllipse gs 0.00 setgray fill gr gs col-1 s gr % Ellipse n 399 739 5 5 0 360 DrawEllipse gs 0.00 setgray fill gr gs col-1 s gr % Ellipse n 219 739 5 5 0 360 DrawEllipse gs 0.00 setgray fill gr gs col-1 s gr % Ellipse n 129 789 5 5 0 360 DrawEllipse gs 0.00 setgray fill gr gs col-1 s gr % Ellipse n 69 789 5 5 0 360 DrawEllipse gs 0.00 setgray fill gr gs col-1 s gr % Ellipse n 189 789 5 5 0 360 DrawEllipse gs 0.00 setgray fill gr gs col-1 s gr % Ellipse n 159 739 5 5 0 360 DrawEllipse gs 0.00 setgray fill gr gs col-1 s gr % Ellipse n 99 739 5 5 0 360 DrawEllipse gs 0.00 setgray fill gr gs col-1 s gr % Ellipse n 309 789 5 5 0 360 DrawEllipse gs 0.00 setgray fill gr gs col-1 s gr % Ellipse n 249 789 5 5 0 360 DrawEllipse gs 0.00 setgray fill gr gs col-1 s gr % Ellipse n 369 789 5 5 0 360 DrawEllipse gs 0.00 setgray fill gr gs col-1 s gr % 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0.00 setgray fill gr gs col-1 s gr % Ellipse n 309 589 5 5 0 360 DrawEllipse gs 0.00 setgray fill gr gs col-1 s gr % Ellipse n 249 589 5 5 0 360 DrawEllipse gs 0.00 setgray fill gr gs col-1 s gr % Ellipse n 249 689 5 5 0 360 DrawEllipse gs 0.00 setgray fill gr gs col-1 s gr % Ellipse n 309 689 5 5 0 360 DrawEllipse gs 0.00 setgray fill gr gs col-1 s gr % Ellipse n 549 689 5 5 0 360 DrawEllipse gs 0.00 setgray fill gr gs col-1 s gr % Ellipse n 549 589 5 5 0 360 DrawEllipse gs 0.00 setgray fill gr gs col-1 s gr % Ellipse n 459 639 5 5 0 360 DrawEllipse gs 0.00 setgray fill gr gs col-1 s gr % Ellipse n 399 639 5 5 0 360 DrawEllipse gs 0.00 setgray fill gr gs col-1 s gr % Ellipse n 519 639 5 5 0 360 DrawEllipse gs 0.00 setgray fill gr gs col-1 s gr % Ellipse n 489 589 5 5 0 360 DrawEllipse gs 0.00 setgray fill gr gs col-1 s gr % Ellipse n 429 589 5 5 0 360 DrawEllipse gs 0.00 setgray fill gr gs col-1 s gr % Ellipse n 429 689 5 5 0 360 DrawEllipse gs 0.00 setgray fill gr gs col-1 s gr % 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\closepsdump % Finally, here is text5.tex: \magnification 1200 \global\let\figures\relax % % ??/epsf.tex (written by Radical Eye Software and copied below) % defines the macro \epsfbox with one argument, % the encapsulated PostScript file to include. % Invoking it causes a \vbox with the natural size of the drawing % to be inserted at the point of invocation. % Usually figures are meant to be centered and set off, and possibly % to have a title and/or a figure number. The macros below do that. % You should assign values that please you to the variables % \abovefigskip, \belowfigskip, figtitleskip, figtitlefont,... % AFTER \inputting the present file: \input /u2/kbi/tex/epsfig % and BEFORE the first invocation of any of the macros below. % RESET AT YOUR PLEASURE THE VARIABLES AT THE VERY BOTTOM! % % For convenience we make a dimension for figures: \newdimen\FigSize \FigSize=.9\hsize % alter at your convenience % % For a SCALED HORIZONTALLY CENTERED FIGURE use \epsfig. % First argument is the horizontal width of the figure, given % in any way TeX can understand. % The second argument is an encapsulated PostScript file name (filnam.eps). % Note the mandatory semicolons between arguments in this example!: % \epsfig .8\hsize; example.ps; % will put a centered scaled \vbox of width .8\hsize suitably offset % at the point of invocation \newskip\abovefigskip \newskip\belowfigskip \gdef\epsfig#1;#2;{\par\vskip\abovefigskip\penalty -500 {\everypar={}\epsfxsize=#1\noindent \centerline{\epsfbox{#2}}}% \vskip\belowfigskip}% % % SCALED TITLED EPSFIG HORIZONTALLY CENTERED: \tepsfig. % First argument is the horizontal width of the figure, % second an encapsulated PostScript file name, % third a title for the figure. % Note the mandatory semicolons between arguments! % example: \tepsfig5truein; example.ps;{This is a figure} \newskip\figtitleskip \gdef\tepsfig#1;#2;#3{\par\vskip\abovefigskip\penalty -500 {\everypar={}\epsfxsize=#1\noindent \vbox {\centerline{\epsfbox{#2}}\vskip\figtitleskip \centerline{\figtitlefont#3}}}% \vskip\belowfigskip}% % % SCALED NUMBERED TITLED EPSFIG HORIZONTALLY CENTERED: \nepsfig % The figure number is automatically increased for every % invocation of \nepsfig or \nipsfig. \newcount\FigNr \global\FigNr=0 \gdef\nepsfig#1;#2;#3{\global\advance\FigNr by 1 \tepsfig#1;#2;{Figure\space\the\FigNr.\space#3}}% % % % Often you would rather have TeX decide where to put the figure % by using \midinsert. Here are macros that do that % (mnemonics ``ipsfig'' is for ``midInsert PS FIGure'') % % TeX-PLACED SCALED EPSFIG HORIZONTALLY CENTERED: \ipsfig \gdef\ipsfig#1;#2;{%\goodbreak ?? \midinsert{\everypar={}\epsfxsize=#1\noindent \centerline{\epsfbox{#2}}}% \endinsert}% % % TeX-PLACED SCALED TITLED EPSFIG HORIZONTALLY CENTERED: \tipsfig \gdef\tipsfig#1;#2;#3{\midinsert {\everypar={}\epsfxsize=#1\noindent \vbox{\centerline{\epsfbox{#2}}% \vskip\figtitleskip \centerline{\figtitlefont#3}}}\endinsert}% % % TeX-PLACED SCALED NUMBERED TITLED EPSFIG HORIZONTALLY CENTERED: \nipsfig % example: \nipsfigd.9\hsize;example.ps;{This is an example figure} \gdef\nipsfig#1;#2;#3{\global\advance\FigNr by1% \tipsfig#1;#2;{Figure\space\the\FigNr.\space#3}}% % % ================================================================ % old: % \ifx\Input\undefined\let\next\input\else\let\next\Input\fi % \next /usr/local/lib/tex/inputs/epsf.tex %On Suns %% \next /usr/lib/tex/inputs/epsf.tex %On NexTs % % to make sure no new version of epsf.tex corrupts the above, % I copy here that file in toto: % % EPSF.TEX macro file: % Written by Tomas Rokicki of Radical Eye Software, 29 Mar 1989. % Revised by Don Knuth, 3 Jan 1990. % Revised by Tomas Rokicki to accept bounding boxes with no % space after the colon, 18 Jul 1990. % % TeX macros to include an Encapsulated PostScript graphic. % Works by finding the bounding box comment, % calculating the correct scale values, and inserting a vbox % of the appropriate size at the current position in the TeX document. % % To use with the center environment of LaTeX, preface the \epsffile % call with a \leavevmode. (LaTeX should probably supply this itself % for the center environment.) % % To use, simply say % \input epsf % somewhere early on in your TeX file % \epsfbox{filename.ps} % where you want to insert a vbox for a figure % % Alternatively, you can type % % \epsfbox[0 0 30 50]{filename.ps} % to supply your own BB % % which will not read in the file, and will instead use the bounding % box you specify. % % The effect will be to typeset the figure as a TeX box, at the % point of your \epsfbox command. By default, the graphic will have its % `natural' width (namely the width of its bounding box, as described % in filename.ps). The TeX box will have depth zero. % % You can enlarge or reduce the figure by saying % \epsfxsize= \epsfbox{filename.ps} % (or % \epsfysize= \epsfbox{filename.ps}) % instead. Then the width of the TeX box will be \epsfxsize and its % height will be scaled proportionately (or the height will be % \epsfysize and its width will be scaled proportiontally). The % width (and height) is restored to zero after each use. % % A more general facility for sizing is available by defining the % \epsfsize macro. Normally you can redefine this macro % to do almost anything. The first parameter is the natural x size of % the PostScript graphic, the second parameter is the natural y size % of the PostScript graphic. It must return the xsize to use, or 0 if % natural scaling is to be used. Common uses include: % % \epsfxsize % just leave the old value alone % 0pt % use the natural sizes % #1 % use the natural sizes % \hsize % scale to full width % 0.5#1 % scale to 50% of natural size % \ifnum#1>\hsize\hsize\else#1\fi % smaller of natural, hsize % % If you want TeX to report the size of the figure (as a message % on your terminal when it processes each figure), say `\epsfverbosetrue'. % \newread\epsffilein % file to \read \newif\ifepsffileok % continue looking for the bounding box? \newif\ifepsfbbfound % success? \newif\ifepsfverbose % report what you're making? \newdimen\epsfxsize % horizontal size after scaling \newdimen\epsfysize % vertical size after scaling \newdimen\epsftsize % horizontal size before scaling \newdimen\epsfrsize % vertical size before scaling \newdimen\epsftmp % register for arithmetic manipulation \newdimen\pspoints % conversion factor % \pspoints=1bp % Adobe points are `big' \epsfxsize=0pt % Default value, means `use natural size' \epsfysize=0pt % ditto % \def\epsfbox#1{\global\def\epsfllx{72}\global\def\epsflly{72}% \global\def\epsfurx{540}\global\def\epsfury{720}% \def\lbracket{[}\def\testit{#1}\ifx\testit\lbracket \let\next=\epsfgetlitbb\else\let\next=\epsfnormal\fi\next{#1}}% % \def\epsfgetlitbb#1#2 #3 #4 #5]#6{\epsfgrab #2 #3 #4 #5 .\\% \epsfsetgraph{#6}}% % \def\epsfnormal#1{\epsfgetbb{#1}\epsfsetgraph{#1}}% % \def\epsfgetbb#1{% % % The first thing we need to do is to open the % PostScript file, if possible. % \openin\epsffilein=#1 \ifeof\epsffilein\errmessage{I couldn't open #1, will ignore it}\else % % Okay, we got it. Now we'll scan lines until we find one that doesn't % start with %. We're looking for the bounding box comment. % {\epsffileoktrue \chardef\other=12 \def\do##1{\catcode`##1=\other}\dospecials \catcode`\ =10 \loop \read\epsffilein to \epsffileline \ifeof\epsffilein\epsffileokfalse\else % % We check to see if the first character is a % sign; % if not, we stop reading (unless the line was entirely blank); % if so, we look further and stop only if the line begins with % `%%BoundingBox:'. % \expandafter\epsfaux\epsffileline:. \\% \fi \ifepsffileok\repeat \ifepsfbbfound\else \ifepsfverbose\message{No bounding box comment in #1; using defaults}\fi\fi }\closein\epsffilein\fi}% % % Now we have to calculate the scale and offset values to use. % First we compute the natural sizes. % \def\epsfsetgraph#1{% \epsfrsize=\epsfury\pspoints \advance\epsfrsize by-\epsflly\pspoints \epsftsize=\epsfurx\pspoints \advance\epsftsize by-\epsfllx\pspoints % % If `epsfxsize' is 0, we default to the natural size of the picture. % Otherwise we scale the graph to be \epsfxsize wide. % \epsfxsize\epsfsize\epsftsize\epsfrsize \ifnum\epsfxsize=0 \ifnum\epsfysize=0 \epsfxsize=\epsftsize \epsfysize=\epsfrsize % % We have a sticky problem here: TeX doesn't do floating point arithmetic! % Our goal is to compute y = rx/t. The following loop does this reasonably % fast, with an error of at most about 16 sp (about 1/4000 pt). % \else\epsftmp=\epsftsize \divide\epsftmp\epsfrsize \epsfxsize=\epsfysize \multiply\epsfxsize\epsftmp \multiply\epsftmp\epsfrsize \advance\epsftsize-\epsftmp \epsftmp=\epsfysize \loop \advance\epsftsize\epsftsize \divide\epsftmp 2 \ifnum\epsftmp>0 \ifnum\epsftsize<\epsfrsize\else \advance\epsftsize-\epsfrsize \advance\epsfxsize\epsftmp \fi \repeat \fi \else\epsftmp=\epsfrsize \divide\epsftmp\epsftsize \epsfysize=\epsfxsize \multiply\epsfysize\epsftmp \multiply\epsftmp\epsftsize \advance\epsfrsize-\epsftmp \epsftmp=\epsfxsize \loop \advance\epsfrsize\epsfrsize \divide\epsftmp 2 \ifnum\epsftmp>0 \ifnum\epsfrsize<\epsftsize\else \advance\epsfrsize-\epsftsize \advance\epsfysize\epsftmp \fi \repeat \fi % % Finally, we make the vbox and stick in a \special that dvips can parse. % \ifepsfverbose\message{#1: width=\the\epsfxsize, height=\the\epsfysize}\fi \epsftmp=10\epsfxsize \divide\epsftmp\pspoints \vbox to\epsfysize{\vfil\hbox to\epsfxsize{% \special{PSfile=#1 llx=\epsfllx\space lly=\epsflly\space urx=\epsfurx\space ury=\epsfury\space rwi=\number\epsftmp}% \hfil}}% \epsfxsize=0pt\epsfysize=0pt}% % % We still need to define the tricky \epsfaux macro. This requires % a couple of magic constants for comparison purposes. % {\catcode`\%=12 \global\let\epsfpercent=%\global\def\epsfbblit{%BoundingBox}}% % % So we're ready to check for `%BoundingBox:' and to grab the % values if they are found. % \long\def\epsfaux#1#2:#3\\{\ifx#1\epsfpercent \def\testit{#2}\ifx\testit\epsfbblit \epsfgrab #3 . . . \\% \epsffileokfalse \global\epsfbbfoundtrue \fi\else\ifx#1\par\else\epsffileokfalse\fi\fi}% % % Here we grab the values and stuff them in the appropriate definitions. % \def\epsfgrab #1 #2 #3 #4 #5\\{% \global\def\epsfllx{#1}\ifx\epsfllx\empty \epsfgrab #2 #3 #4 #5 .\\\else \global\def\epsflly{#2}% \global\def\epsfurx{#3}\global\def\epsfury{#4}\fi}% % % We default the epsfsize macro. % \def\epsfsize#1#2{\epsfxsize}% % % Finally, another definition for compatibility with older macros. % \let\epsffile=\epsfbox % ================================================================ % execution: why not set \epsfverbosetrue % reset at your pleasure \abovefigskip=\baselineskip % reset at your pleasure \belowfigskip=\baselineskip % reset at your pleasure \global\let\figtitlefont\bf % reset at your pleasure \global\figtitleskip=.5\baselineskip % reset at your pleasure \let\nd\noindent % NOINDENT \def\NL{\hfill\break}% NEWLINE \def\qed{\hbox{\hskip 6pt\vrule width6pt height7pt depth1pt \hskip1pt}} \def\natural{{\rm I\kern-.18em N}} \def\N{{\bf N}} \def\integer{{\rm Z\kern-.32em Z}} \def\B{{\cal B}} \def\A{{\cal A}} \def\EE{{\mit E}} \def\GG{{\mit G}} \def\HH{{\mit H}} \def\jj{{\mit j}} \def\kk{{\mit k}} \def\mm{{\mit m}} \def\nn{{\mit n}} \def\Z{{\bf Z}} \def\real{{\rm I\kern-.2em R}} \def\R{{\bf R}} \def\complex{\kern.1em{\raise.47ex\hbox{ $\scriptscriptstyle |$}}\kern-.40em{\rm C}} \def\C{{\bf C}} \def\vs#1 {\vskip#1truein} \def\hs#1 {\hskip#1truein} \def\signed{\vskip1truein\hskip 2.5truein{Charles Radin}} \def\Sinc{\vskip .5truein\hskip 2.5truein} \def\Indent{\vskip 0truein\hskip 2.5truein} \def\Month{\ifcase\number\month \relax\or January \or February \or March \or April \or May \or June \or July \or August \or September \or October \or November \or December \else \relax\fi } \def\date{\Month \the\day, \the\year} \def\lett{\nopagenumbers\hbox{}\vskip 1truein \hskip 2.5truein{\date}\vskip.5truein} \hsize=6truein \hoffset=.25truein %was \hoffset=1.2truein \vsize=8.8truein %\voffset=1truein \pageno=1 \baselineskip=12pt \parskip=0 pt \parindent=20pt \overfullrule=0pt \lineskip=0pt \lineskiplimit=0pt \hbadness=10000 \vbadness=10000 % REPORT ONLY BEYOND THIS BADNESS %\nopagenumbers %\lett %\nd \pageno=0 \font\bit=cmbxti10 \footline{\ifnum\pageno=0\hss\else\hss\tenrm\folio\hss\fi} \hbox{} \vskip 1truein\centerline{{\bf Miles of Tiles}*} \vskip .5truein\centerline{by} \centerline{Charles Radin} \footnote{}{*\ Research supported in part by NSF Grant No. DMS-9304269 and \vs0 \hs.15 Texas ARP Grant 003658-113\hfil} \vskip .25truein\centerline{Mathematics Department} \centerline{University of Texas} \centerline{Austin, TX\ \ 78712} \vs.125 \centerline{radin@math.utexas.edu} \vs1 \centerline{August, 1994} \vfill \eject \pageno=1 \nd {\bf Introduction.}\ \ This article corresponds closely to four lectures given in conjunction with the July 1994 workshop at Warwick; the informal format of those lectures is maintained. Our main goal is to discuss the generalization of subshifts with $\Z^d$ actions, especially those of finite-type, to tiling systems with $\R^d$ actions. Our justification for this generalization is the natural action of the Euclidean group, in particular rotations, on tiling systems. The new ``statistical'' form of rotational symmetries which appear are a significant addition to the mathematics of subshifts, and of decided value in applications, which we discuss. There are four sections to this article. We begin slowly, warming up within the familiar context of subshifts. The two themes in section I) are first the contrast between subshifts of finite-type and substitution subshifts, and second the entropy of uniquely ergodic subshifts of finite-type. Sections II) and III) generalize the subject in two ways. In II) we make the leap from subshifts to tiling systems, and discuss further the connection between finite-type and substitution systems. The main examples used for illustration are the pinwheel and Penrose tilings of the plane; see Figures 1 and 2. Section III) is devoted to the use of intuition from physics (statistical mechanics) to finite-type systems, and in particular leads into the subject of rotational symmetries in such systems. All the previous discussion becomes focused in section IV) with a discussion of statistical symmetries and their use in ergodic theory and in the analysis of patterns in space. Again the main illustrations are the pinwheel and Penrose tilings of the plane. \vs.5 \centerline{I.\ \ The Wang/Berger phenomenon} \vs.2 First some notation. Throughout this section $\A$ will denote a fixed, finite alphabet of cardinality at least 2, $T^t$ will denote translation by $t\in \Z^d$ on $\A^{\Z^d}$ (compact in the product topology), and $X$ will be a subshift: that is, a closed, translation invariant (nonempty) subset of $\A^{\Z^d}$. An element of $\A$ will be called a letter, and an element of $X$ will be called a configuration. We will be concerned with two types of subshifts, substitution and finite-type, and we consider them first in the classical regime, $d=1$. \vs.1 \nd {\bf a)} $d=1$. We begin with substitution subshifts. We define $\B$, the set of ``words'', as $\bigcup_{K\ge 1}\A^K$. To define a substitution subshift one must have a substitution function, $F:\A\longrightarrow \B$. One then extends $F$ to be a map from $\B$ to itself. (If $b=a_1a_2\cdots a_K,\ F(b)\equiv F(a_1)F(a_2)\cdots F(a_K)$.) $\B_F$ is then defined by $$\B_F\equiv \{b\in \B\, :\, b=F^k(a),\hbox{ for some }a\in \A\}\eqno 1)$$ \nd If $b=F^k(a)$ we call $b$ a letter of ``level $k$''. Finally, we define $X_F$, the substitution subshift associated with $F$, by $$X_F\equiv\{x\in\A^{\Z}:\hbox{ for each }t,j\in \Z,\ (x_{t+1},x_{t+2},\cdots x_{t+j})\subset b,\hbox{ for some }b\in \B_F\}\hfil\eqno 2)$$ We now record some simple facts about substitution subshifts. \vs.1 \nd {\bf Lemma 1.} Each $x\in X_F$ ``decomposes'' into adjacent letters of level 1, and therefore also into adjacent letters of level $k$, for any fixed $k$. \vs0 \nd (The proof is a simple exercise using compactness.) \vs.1 \nd {\bf Lemma 2.} Assume there exists $p$ such that for all $a\in \A$, $F^p(a)$ contains all $a'\in \A$. Then $X_F$ is uniquely ergodic -- that is, there is one and only one translation invariant Borel probability measure on $X_F$. \vs0 \nd (We will sketch the proof in the last section.) \vs.1 \nd {\bf Lemma 3.} Assume the decomposition of Lemma 1 is unique for each $x\in X_F$. Then for all $x\in X_F$, $T^tx=x$ implies $t=0$. \vs0 \nd (We will give the proof in the last section.) \vs.1 To clarify the uniqueness assumption of Lemma 3 we leave it as an exercise to see that there is not uniqueness for the substitution: $$\hfil\A=\{0,1\},\ F(0)=101,\ F(1)=010\hfil\eqno 3)$$ \vs.2 We now consider finite-type systems. For fixed positive integer $K$ and nonempty subset $\phi\subset \A^K$ we define the finite-type subshift $X_\phi$ associated with $\phi$ by $$X_\phi\equiv\{x\in \A^{\Z}\, :\, \hbox{ for all }t\in \Z,\ (x_{t+1},x_{t+2},\cdots x_{t+K})\in \phi\}\eqno 4)$$ \nd (Note the difference from substitution systems for which restrictions are made for subwords of all sizes.) \vs.1 \nd {\bf Lemma 4.} If $X_\phi$ is nonempty there exists $\tilde x\in X_\phi$ and $\tilde t\ne 0$ such that $T^{\tilde t}\tilde x= \tilde x$. \vs.1 \nd Note the difference between Lemmas 3 and 4. Since the assumptions in these lemmas are very weak, we make the intuitive conclusion that these two classes of subshifts, the substitutions and the finite-type subshifts, are essentially disjoint. \vs.1 \nd {\bf b)} $d\ge 2$. We now define $\B$, the set of ``words'', as $\bigcup_{K\subset \Z^d} \A^K$. And again, to define a substitution subshift one must have a substitution function, $F:\A\longrightarrow \B$. But now there can be a problem extending $F$ to a function from $\B$ to itself. This problem will naturally disappear when we extend our format to tilings. (We will see then that basically substitutions are representations of similarity transformations of Euclidean space, and the discreteness of $\Z^d$ creates artificial difficulties such as this extension problem.) As an example consider the following. Let $d=2$, $\A = \{a,b,c,d\}$, and define the substitution function $F$ by $$F(a)= \matrix{a&c\cr b&d\cr};\ \ F(b)= \matrix{b&d\cr a&c\cr};\ \ F(c)= \matrix{c&a\cr d&b\cr};\ \ F(d)= \matrix{d&b\cr c&a\cr}\eqno 5)$$ \nd This $F$ has a natural extension from $\B$ to itself. And then we can define the subshift associated with $F$ as for $d=1$. It is easy to check that Lemmas 1, 2 and 3 are still true in this more general setting. For subshifts of finite-type there is less of a problem; we just fix some bounded $K\subset \Z^d$ and nonempty $\phi \subset \A^K$, and define the subshift $X_\phi$ associated with $\phi$ as before. The big difference is that {\bf Lemma 4 is not true for $d\ge 2$.} Specifically, in 1966 Robert Berger [Ber] answered a question of his thesis advisor, the philosopher Hao Wang, by giving an example with $d=2$ and an alphabet $\A$ with over 20,000 letters, of a subshift $X_\phi$ of finite-type such that for all $x\in X_\phi$, $T^tx=x$ implies $t=0$. After much intermediate research, for example by Raphael Robinson [Rob], this situation was clarified by Shahar Mozes in 1989 [Moz]. \vs.1 \nd {\bf Theorem 1.} Given a substitution subshift $X_F\subset \A^{\Z^2}$ which is uniquely ergodic with unique measure $\mu_F$, (and satisfies some weak conditions we do not list), there exists an alphabet $\tilde \A$ and some $\phi\subset \tilde \A^K$ such that \vs0 i) $X_\phi$ is uniquely ergodic, with unique measure $\mu_\phi$; \vs0 ii) $(X_\phi,\Z^2,\mu_\phi)$ and $(X_F,\Z^2,\mu_F)$ are metrically conjugate. \vs.1 Considering the correspondence of this result with Lemmas 3 and 4 we conclude that although for $d=1$ the class of substitution subshifts and the class of subshifts of finite-type are basically disjoint, for $d\ge 2$ the former is basically a subset of the latter! We call this the Wang/Berger phenomenon, the starting point for research in what is now called aperiodic tilings, or forced tilings. We will follow up this path in later sections, but first we want to add one fundamental fact about subshifts of finite-type, a result by Jacek Mi\c ekisz and the author [Rad1]. \vs.1 \nd {\bf Theorem 2.} If the subshift of finite-type $(X_\phi,\Z^d)$ is uniquely ergodic then it has zero entropy. \vs.1 \nd {\it sketch of the proof.} The case of $d=1$ is immediate from Lemma 4. Assume for simplicity of notation that $d=2$ and $K$ is a $2\times 2$ subset of $\Z^2$. Let $S_N$ be the square set of $N$ lattice points of $\Z^2$ centered at the origin. Consider the cylinder sets based on $S_N$, that is, sets of configurations with fixed values in $S_N$. Think of the letters of the alphabet as colors, and the fixed values of such configurations as (colored) ``pictures''. We will assume the entropy is $\alpha > 0$, and obtain a contradiction. From Birkhoff's theorem, as a function of $N$ there are roughly $e^{\alpha N}$ pictures on $S_N$ which appear with positive frequency in all configurations. Consider a picture frame of unit thickness around $S_N$; there are at most $e^{\beta \sqrt N}$ different ways to color in the frame. So for large $N$ there is some colored frame $D$ ``compatible with'' more than $N^2$ different pictures on $S_N$, that is, such that each of those pictures and the colored frame $D$ together appear with positive frequency in all configurations. Fix some configuration $x$ and large $N$ with some picture $C$ on $S_N$ inside the colored frame $D$. Consider $S_{9N}$. It ``contains'' at most roughly $N$ different pictures of shape $\sqrt N\times \sqrt N$, so there is some picture $C'$, compatible with $D$, which doesn't appear in $S_{9N}$ in $x$. If necessary we ignore some appearances of $C$ inside $D$ in $S_{9N}$ so the remaining ones do not overlap. The remainder still occur with positive frequency. Change the configuration $x$ to a new configuration $x'$ by replacing each of these remaining appearances of $C$ by $C'$. It is easy to see that $x'$ still belongs to our subshift $X_\phi$. On the other hand it is also clear that the frequency of $C'$ in $x'$ is higher than it is in $x$, and this is incompatible with the assumption that $X_\phi$ is uniquely ergodic. This contradiction proves that the entropy is zero. \qed \vs.1 We conclude this section with some comments on the assumption of unique ergodicity. This assumption is unusual when considering subshifts of finite-type, because in the classical regime of $d=1$ it follows from Lemma 4 that only trivial subshifts would have the property. Now the classical regime of $d=1$ corresponds to, or originated from, considerations of time evolution, wherein configurations can correspond to very different conditions, and be very different one from another. We assert that the regime $d\ge 2$ corresponds to a different paradigm, namely space rather than time translation. And there the assumption of unique ergodicity is much more natural, corresponding as it does to ``essentially'' only one global pattern -- that is, locally unique if not globally. So the study of subshifts of finite-type with $\Z^d$ action is the study of spatial patterns, and the assumption of unique ergodicity is a natural one of nondegeneracy [Rad3]. \vs.5 \centerline{II.\ \ The pinwheel and Penrose tilings} \vs.2 One of the key ideas in the last section was that interesting subshifts of finite-type could be constructed which are metrically conjugate to given substitution subshifts. And in fact Mozes has given a rather general prescription for this. Before we discuss this further, it is appropriate to make the transition to tilings. First some notation. We will consider (finite) alphabets $\A$ whose letters are polyhedra in $\R^d$. A ``tiling (from $\A$)'' will then be a countable collection of ``tiles'' (sets which are congruent to letters of $\A$), which is simultaneously a covering and a packing of $\R^d$. Now although in the last section I presented the work of Berger {\it et al} as results about subshifts, in fact this early work was formulated in terms of tilings of the plane. The letters in $\A$ are there all basically unit squares, but each has a different pattern of bumps and dents on the edges so that, like jigsaw puzzle pieces, they only fit together in certain ways. This is easily seen to determine a set $\phi\subset \A^K$, where $K$ is a $2\times 2$ subset of $\Z^2$, and thus a subshift $X_\phi$ of finite-type. We will refer to such tilings as ``square-ish''. In order to study symmetry properties it is necessary to generalize the above mathematics to include polyhedra which are not basically squares or cubes. We will introduce the ideas through two examples. In both cases there is a substitution system and an associated finite-type system. As with subshifts, a substitution tiling system will be a special set of tilings associated with a substitution function, and a finite-type tiling system will consist of {\bf all} tilings by the letters in the given alphabet, the finite-type condition consisting of the geometric restriction that the tiles fit together and cover space. We will begin by describing the two substitution systems. For the Penrose tilings the alphabet $\A$ for the substitution system consists of two letters, called the ``kite'' and the ``dart''; see Figure 3. The family $\B$ of words is the set of all finite collections of tiles (that is, sets congruent to letters of $\A$), whose union is connected and simply connected, and in which tiles have pairwise disjoint interiors. The substitution function $F:\A\longrightarrow \B$ should be thought of as obtained by first associating with each letter (shown by dotted lines in Figure 4) a collection of ``shrunken tiles'' (shown by solid lines in Figure 4), small by a factor $(1+\sqrt 5)/2$, and then expanding the collection about the origin by $(1+\sqrt 5)/2$. With this convention we define the family $\B_F$ of words of the form $F^k(a)$, $a\in \A$, $k\ge 1$. Finally we define the substitution system $X_F$ associated with $F$ to be the set of all tilings $x$ of the plane such that every subword of $x$ is congruent to a subword of some word in $\B_F$. For the pinwheel tilings the alphabet for the substitution system consists of a $1, 2, \sqrt 5$ right triangle and its reflection, as in Figure 5. The substitution function, indicated in Figure 6, should again be considered a composite of two maps. The substitution system $X_F$ is defined as above. We have thus specified two substitution systems. The main question in this section is how to associate finite-type systems with such substitution systems, in analogy with the special case of square-ish tilings ({\it i.e.\ subshifts}) discussed in the last section. The vague idea is that one wants a new alphabet similar to the given one, but altered so that the tiles can {\it only} fit together in the special way the original letters do in the given substitution tilings. For the Penrose system the answer is easy. By examining small words in any tiling one sees that the kite and dart can be prevented from abutting incorrectly (for example from forming a rhombus, copies of which could then tile periodically) by adding bumps and dents to them as in Figure 7. And with these letters one can only tile the plane as in the substitution. So for the Penrose substitution system it is easy, by examining small words in the tilings, to create a new alphabet which defines a finite-type system conjugate to the original. It is then natural to ask whether a similar analysis can be made for the pinwheel tilings. The answer is no, and we will present a simplified version of an argument proving this, an argument due to Ludwig Danzer. Before we do this however, we consider the following generalization of the above analysis. Given a tiling $x$ let $W$ be the set of all words in $x$ with upper bound $D$ on the diameter. We use $W$ to create the dynamical system $X_W$, defined to contain all tilings $y$ by the letters in $x$ such that all words of diameter at most $D$ in $y$ are congruent to words in $W$. It is then not hard to construct a new alphabet of polyhedra such that the dynamical system consisting of all tilings by these letters is conjugate to $X_W$. One aspect of this generalized method should be emphasized. For the Penrose system we examined words of two letters in the substitution system, and ended up with an alphabet for the finite-type system of the same cardinality as the original, just distortions of the original letters. But in this generalized method, where we examine larger words in the substitution system, we may end up with a much larger alphabet than for the substitution system. We now return to the pinwheel substitution system. Consider the periodic tiling $P$ of which the unit cell is either of the two words called $B$ and $C$ in \hbox{Figure 8.} $F^5(a)$ is shown in Figure 9. In this word one sees several copies of $B$ and $C$. Now given words of diameter $D$ in a pinwheel tiling, apply the substitution at least $2D\sqrt 5$ times to $P$, creating a new periodic tiling $P'$, with $B$ and $C$ leading to two unit cells of $P'$ which we call $B'$ and $C'$. It is easy to see that all words of diameter at most $D$ in $P'$ are interior to one of these two unit cells, and {\it so must appear in the pinwheel}. This proves the above assertion since any finite-type rule, based on a set of words of a certain size appearing in the pinwheel, will be compatible with the periodic tiling $P'$ -- that is, $P'$ would belong to the space of tilings, in contradiction to Lemma 3. \vs.1 Now as we noted there is a finite-type pinwheel system conjugate to the pinwheel substitution, although it cannot be obtained directly from words of fixed size by the above method. Furthermore, the technique used by Mozes for square-ish tiles has also proved to be too simple. Next we will sketch a method which works for the pinwheel [Rad4]. First an overview. The letters in the new alphabet $\A'$ come in two families, one for each of the two triangles in the original $\A$. The letters in each family can be thought of as perturbed versions of the original triangle, that is, the original triangle with some pattern of bumps and dents on each edge. But this is not the best way to think of them; it is preferable to view each letter as a triangle with certain information associated with each vertex. (It is proven in [Rad4] that the information can transformed to bumps and dents.) The finite-type system is then defined by requiring that the triangles may abut if and only if the information associated with vertices of abutting triangles is ``consistent'' in a certain precise sense. Roughly speaking, the information can be thought of as a picture of what ``should be'' the environment of that vertex, referring implicitely to any triangles abutting the given one at that vertex. (By ``consistent'' we mean that their pictures agree.) We write ``should be'' because the information can only be thought of as a wish list, and it is necessary to prove that in any tiling with such marked triangles all the wish lists are automatically satisfied. Next we want to indicate what information is included at vertices. Before this can be done we will pick out one tiling from the substitution system and define a certain hierarchical structure in it. The substitution function (Figure 6) associates five triangles with each letter of $\A$. We give them labels, $A$ through $E$, as in Figure 10. Next we construct a certain tiling of the plane, called ``the tesselation'', as follows. Start with a letter sitting in the plane and use the substitution. Reorient the word obtained so the ``$C$'' triangle is sitting over the original letter. Apply the substitution to this word, reorienting again so the ``$C$'' associated with the first ``$C$'' again lies over the original letter. Repeating this process indefinitely produces the tesselation. We introduce the following notation with reference to the above process. We say the original letter is a letter of level 0. The first application of the substitution produces a word of five letters, a letter of level 1, containing the five letters of \hbox{level 0.} The five letters of level 0 are given labels $A,B,C,D$ or $E$. When the substitution is applied again, the letter of level 1 is (by choice) seen to be a letter with label $C$ inside a letter of level 2, {\it etc.} So the tesselation can be understood to be a tiling of the plane by letters of all levels, each labelled by $A$ through $E$. Finally we introduce the notion of ``complete edges'' for these letters. Referring to the three edges of the letters as ``small'', ``medium'' and ``large'' (or $S$, $M$ and $L$ respectively) in the obvious way, we define the complete edges to be the small edges of letters (of any level) with label $A$, $B$, $C$ and $D$, the medium edges of letters with label $C$, $D$ and $E$, and the large edges of letters with label $B$ and $C$. These happen to be the ``internal'' edges in the letters of level $n$ as these sit in a letter of level $n+1$; see Figure 10. It is easy to see from the construction that in the tesselation each edge of each letter of level 0 is geometrically part of precisely one complete edge. Next we show how this fact is used in the information associated with the letters of the alphabet $\A'$ of the finite-type system. The information associated with each edge of the letters of $\A'$ is incorporated in a tree-like structure. The highest level in the tree is the variable $\mu$. With each of the three vertices of the letter, labelled $S$, $M$ and $L$ referring to the size of the angle of the vertex, there is the variable $\mu_S$ or $\mu_M$ or $\mu_L$. These variables each contain other variables, some of which contain other variables {\it etc.} A complete list of the variables involved is: \vs.2 $\mu =(\mu_{S},\mu_{M},\mu_{L})$ \vs.1 $\mu_j = (A_1,\EE^1_1,\EE^2_1;A_2,\EE^1_2,\EE^2_2;\cdots,A_e,\EE^1_e,\EE^2_e)$ \vs.1 $3\le e\le 8$ \vs.4 $A_j\in (S,M,L,\pi)$ \vs.4 $\EE^j_k = (J,N,P,F,G,H)$ \vs.1 $J\in (S,M,L)$ \vs.1 $N\in (A,B,C,D,E)$ \vs.1 $P\in (A,B,C,D,E)$ \vs.1 $F\in (S,M,L,Z,R)$ \vs.4 $\GG=(\GG^{\alpha1M}_A, \GG^{\alpha1S}_E, \GG^{\alpha1S}_B, \GG^{\alpha1S}_D, \GG^{\alpha1M}_D, \GG^{\alpha2M}_A, \GG^{\alpha2S}_E, \GG^{\alpha2S}_B, \GG^{\alpha2S}_D, \GG^{\alpha2M}_D, \GG^{\epsilon1M}_A,$ \vs0 \hs1.75 $\GG^{\epsilon1S}_E, \GG^{\epsilon1S}_B, \GG^{\epsilon1S}_D, \GG^{\epsilon1M}_D, \GG^{\epsilon2M}_A, \GG^{\epsilon2S}_E, \GG^{\epsilon2S}_B, \GG^{\epsilon2S}_D, \GG^{\epsilon2M}_D)$ \vs.1 $\GG^{\jj\kk\nn}_\mm$ $\in$ $\{+1,-1\}$ \vs.3 $\HH=(\HH^{1M}_A$, $\HH^{1S}_E, \HH^{1S}_B, \HH^{1S}_D, \HH^{1M}_D, \HH^{2M}_A, \HH^{2S}_E, \HH^{2S}_B, \HH^{2S}_D, \HH^{2M}_D)$ \vs.1 $H^{kn}_m$ $\in$ $\{+1,-1\}$ \vs.2 There do not exist letters with all possible values of these variables; there are restrictions, forty two of them, which spell out explicitly which combinations of values may occur. It would be hard to keep track of the variables if they did not have simple geometric meaning, so we next indicate this interpretation of the variables. The basic structure of a variable such as $\mu_S$ is a sequence of triples: an $A$ followed by two $E$'s. This refers to the letters of $\A'$ which ``should'' abut the given letter, and gives their geometric relationships. The $A$'s refer to the sizes of the angles of each of the abutting letters at the vertex in question. Each edge which ``should'' emanate from the vertex is split into two to enable reference to each of the two letters sharing that edge, and this is what the pair of $E$'s refer to in each triple. Now each $E^j_k$ is a complicated quantity, but essentially all the information in it relates to certain complete edges. For example the variable $J$ gives the size (small, medium or large) of the complete edge that the letter edge is part of; $N$ gives the label of the letter, ($A$ through $E$), of whatever level, that that complete edge is part of; and $P$ gives the label of the letter ($A$ through $E$) that that letter, of whatever level, is part of. The variable $F$ describes how the complete edge which contains the given level 0 letter behaves as it meets the vertex in question; it might end at an angle of size $S$, $M$ or $L$, or it might not end in which case $F$ has the value $z$, or it might not end and not even have a vertex there (value $R$.) This takes care of all the variables except for $G$ and $H$. The ``gun'' variables $G$ ``shoot'' information out of vertices. The information is oscillatory -- that is, the value of $+1$ or $-1$ does not carry intrinsic meaning, but is like a phase, and what is significant is if two variables meet at a vertex in phase or out of phase. The variables travel along channels prepared, along certain complete edges, by the $H$ variables. The gun variables are needed in the induction proof which shows that in any tiling with the letters from $\A'$ each of these letters is contained in a unique letter of level $k$ for every $k\ge 1$. This means that letters of level 0 must group together uniquely in groups of 5, then letters of level 1 must also, {\it etc.} The gun variables are needed to force this grouping. Roughly, the label which a given letter of level $k$ is supposed to have is kept in a certain geometric place, and that information has to travel around in a reliable way to ensure that the neighboring letters of that level are correct. So much for our sketch of the proof that the pinwheel is finite-type; for the full story one must read [Rad4]. We end this section with the hope that just as the simpler technique of Mozes works quite generally for square-ish substitutions, some version of the pinwheel technique will work for a large class of substitutions of general shapes, thereby giving an understanding for the way hierarchical (``fractal'') structures could originate from local, finite-type, rules. \vs.5 \centerline{III.\ \ Statistical mechanics and tilings} \vs.2 This section is concerned with certain ideas from statistical physics which give an unconventional and useful viewpoint to the general area of subshifts of finite-type with $\Z^d$ action, or finite-type tiling systems with $\R^d$ action. We begin with a short, elementary introduction to (classical) statistical mechanics. Heuristically, assume we wish to analyze a piece of matter composed of many atoms (for simplicity we assume they are all of the same element), at various temperatures $\tau>0$ and chemical potentials $c$. Assume there are $N$ of these atoms constrained in a cube $L$ of volume $|L|$, which means their positions constitute $N$ variable points in $L$, or a point in $L^N$. We somehow assign a ``(potential) energy'' $E^L(x;c)$ to all possible configurations $x$ of $L^N$. (This will be discussed further below. We are ignoring the kinetic energy as its contribution to the following is relatively minor and easy to insert if desired.) Then for each $\tau>0$ we consider $$f_L(x)\equiv {\exp[-E^L(x;c)/\tau] \over \int_{L^N}\exp[-E^L(x;c)/\tau]\,d^Nx}\eqno 6)$$ \nd as a probability density for $x$. For fixed $c$ we take the (weak-$\ast$) limit of this probability measure as $L\to \R^d$, getting a Gibbs measure $\mu_{\tau,c}$ on configurations in $\R^d$. We are mainly interested in the weak-$\ast$ limit $\tau\downarrow 0$, giving $\mu_{0,c}$. Let $X$ be the collection of all countable subsets of $\R^d$ such that no two points are less than unit distance apart. Let $H$ be the subcollection of finite subsets in $X$ which contain the origin. Define an ``interaction'' as a function $e\,:\,H\longrightarrow \R$ such that $e(h)=0$ if $|q| \ge r$ for some $q\in h$. (Here $r$ is some fixed positive number, the ``range of the many-body interaction''.) For each cube $L$ define $E^L$ on $X$: $$E^L(x;c)\equiv \sum_{q\in x\cap L}\{e[T^{-q}x\cap D_{r,0}]-c\}\eqno 7)$$ \nd where $D_{r,0}$ is the open ball centered at the origin 0 of radius $r$, and as usual $T$ denotes translation. Finally, define $X_{GS}\subset X$ by: $x\in X_{GS}$ if for every $L$, and every $x'\in X$ such that $x\cap(\R^d\backslash L)= x'\cap(\R^d\backslash L)$, it follows that $E^L(x;c)\le E^L(x';c)$. $GS$ stands for ``ground state'', and the notation ignores the dependence on $c$. Note: it can be proven [Rad2] under rather general conditions on $E^L$ that $\mu_{0,c}(X_{GS})=1$. It is not hard to check that $X_{GS}$ is always nonempty, and closed under translation. We will postpone until the next section the definition of the natural topology for spaces like $X_{GS}$; suffice it to say that two configurations are ``close'' if they differ in some large neighborhood of the origin, and then only by a small rigid motion. We note that $X_{GS}$ is compact and metrizable in that topology [R-W]. Consider the following example. Let $d=2$, $c=0$ and $$e(h)\equiv \cases {-25+24|h_1-h_2|,&if $h=\{h_1,\ h_2\}$ and $1\le |h_1-h_2|\le 25/24$;\cr 0,&otherwise\cr}\eqno 8)$$ \nd Then if $x$ is the triangular lattice as in Figure 11 a) then $x\in X_{GS}$. Also, if $x'$ is the configuration in Figure 11 b) then still $x'\in X_{GS}$. Notice the horizontal ``fault line'' in $x'$. The ergodic measures on $X_{GS}$ each have as support the translations of $x$ rotated by some fixed angle. We now discuss one way in which configurations enter into physical theory -- how they respond to beams of waves directed at them. We will assume the points $\{t_j\}$ in a configuration $x\in X_{GS}$ represent the centers of atoms, all with the same electron density. Let the electron density at $t\in\R^3$ of an atom centered at $t_j$ be $$\rho_j(t)=k(t-t_j)=\cases {1/10-|t-t_j|,&$|t-t_j|\le 1/10$\cr 0,&$|t-t_j|>1/10$\cr}\eqno 9)$$ \nd Define $\psi\,:\,X_{GS}\longrightarrow \R$ by: $$\psi(x)=\sum_{t_j\in x}k(-2t_j)\eqno 10)$$ \nd Assume $X_{GS}$ is uniquely ergodic, with measure $\mu_{GS}$; this is a nondegeneracy assumption, essentially equivalent to assuming that the value of $c$ is not that of a phase transition [Rad2]. Fix $x=\{t_j\}\in X_{GS}$ and an increasing sequence of cubes $L_i$. Steven Dworkin has shown [Dwo1] that if a plane wave of wavelength $s$ in the direction $W_0$ is scattered off the atoms in $L_i$ the intensity $I_i$ scattered in the direction $W$ satisfies: $$ {I_i \over |L_i|}\ \ \lower1ex\hbox{${\hbox to 50pt{\rightarrowfill}\atop {|L_i|\to \infty}}$}\ \ d\negthinspace <\negthinspace E_\lambda \psi,\,\psi\negthinspace >\eqno 11)$$ \nd where $\lambda=(W-W_0)/s$ and $\{E_\lambda\,:\,\lambda\in \R^3\}$ is the spectral family associated with translations on $X_{GS}$. (That is, $\{E_\lambda\}$ is the spectral family, guaranteed by Stone's theorem [R-N], of the unitary representation of the translations on the complex Hilbert space of complex valued functions on $X_{GS}$ square integrable with respect to the unique invariant measure on $X_{GS}$.) This is intimately connected with the history of research on tilings as follows. Assume we have some finite-type tiling system $X_F$ such as the Penrose or pinwheel. With each of the letters in the alphabet we associate one or more ``atoms'', enough to avoid accidental symmetries (in a sense to be clarified below). Then for each tiling of the tiling system we can associate a configuration of points in $\R^d$, the centers of the atoms associated with each tile. If the atoms are placed generically in the letters, the system of configurations $\tilde X_F$ thus produced is naturally isomorphic to the tiling system $X_F$, including of course the action $\R^d$ on it. Now $\tilde X_F$ is intuitively like a ground state system $X_{GS}$ in that configurations belong to $\tilde X_F$ if and only if they satisfy the local rules of the tiles, which is very much like the local rules of a ground state system. This idea occurred to physicists and is the basis of their interest in Penrose tilings to model quasicrystals. They use 3 dimensional versions of the Penrose system to which they associate ``atoms'' as above, creating a space $\tilde X_F$ of configurations of ``atoms''. And they do this precisely because the dynamical spectrum of the 2 or 3 dimensional Penrose systems has unusual rotational symmetries, namely symmetry under rotations by $2\pi/10$ about certain axes. Such symmetries are impossible for the scattering intensities of ordinary crystals, but are in fact the hallmark of certain real materials called quasicrystals. To summarize, tilings such as those of Penrose are of interest in the modeling of quasicrystals because of the symmetries of their dynamical spectra. (There is also interest in the {\it smoothness} of their spectra; see [Rue], [B-R] and [Rad2]. And another related direction suggested by physics is that of ``deceptions'' of tilings; see [D-S] and [Dwo2].) Before ending this section we make one more observation. Consider the dynamical system of configurations under space translations together with a Gibbs measure $\mu_{\tau,c}$. We have been concerned with the case $\tau =0$ and have been led to interesting uniquely ergodic finite-type systems, with zero entropy. (The zero entropy result of the first section has been generalized to tiling systems by Jiunn-I Shieh [Shi].) It can be proven that entropy increases with $\tau$, and that at high temperature or entropy the positions become statistically independent (the particles become ``free''). The high entropy regime is the one best understood in ergodic theory and physics, with results on Bernoulli systems {\it etc}. But the low or zero entropy regime seems to contain a great deal of new and exciting possibilities, such as these tiling systems; just as the physics is more interesting in that regime, so too seems to be the mathematics. \vs.5 \centerline{IV.\ \ A new form of symmetry} \vs.2 We will now draw upon elements from the last two sections to elucidate a new form of geometrical symmetry and its relevance for ergodic theory [Rad6]. We will illustrate the idea by tilings of $d$ dimensional Euclidean space by polyhedral tiles. One usually says that a tiling (or other pattern) in space has a certain symmetry if it is invariant under the corresponding rigid motion of the space. The new form of symmetry we will describe is different, and is of a statistical nature. We will use the Penrose and pinwheel tilings as motivation, but first we need some notation. Let $X$ be some tiling dynamical system made from some finite alphabet $\A$ of polyhedra in $\R^d$. (We do not require at this stage that $X$ be either a finite-type or substitution system, but only require that it be locally finite in the following sense. Given any ``radius'' $r>0$, look at all the clusters of tiles that are a distance less than $r$ from the center of mass of one of the tiles in any tiling in $X$. We require that there be only finitely many possible such clusters up to congruence. This is automatic for substitution systems.) Since we distinguish tilings which are rigid motions of one another there is a natural representation on $X$ of $G$, the connected part of the Euclidean group, which we denote $T^g\,:\,X\longrightarrow X$ for $g\in G$. We put a topology on $X$ using a neighborhood basis with three parameters: $x\in X$, a neighborhood $\Theta$ of the identity of $G$, and a neighborhood $R$ of the origin of $\R^d$. The corresponding open set of tilings is: $$C_{x,\Theta,R}\equiv \{x'\in X\,:\, T^gx'|_R = x|_R,\hbox{ for some }g\in\Theta\}\eqno 12)$$ \nd where ``$|_R$'' refers to restriction to $R$. It is not hard to show that $X$ is compact and metrizable in this topology [R-W]. We noted that the Euclidean group is naturally represented on $X$. Now we assume another feature of the Penrose and pinwheel tilings, the representation of a similarity. That is, we assume that for each $x\in X$ there is a {\it unique} $y(x)\in X$ associated, made from tiles which are larger by some factor $\gamma >1$. Let $\sigma$ be the similarity on $\R^d$ which takes $t$ to $t/\gamma$, and let $T^\sigma\,:\,X\longrightarrow X$ be defined by $T^\sigma x=(1/\gamma) y(x)$. This homeomorphism is thought of as a representation of the similarity $\sigma$ since it has the correct relations with the representation of $G$, for example: $$(T^\sigma)T^t(T^\sigma)^{-1}=T^{\sigma(t)}=T^{t/\gamma}\eqno 13)$$ \nd for all translations $t$. One use of this is the following. Assume for some tiling $x_0$ that $T^{t_0}x_0=x_0$. Then for all $n$, $T^{\sigma^{n}(t_0)}[T^{\sigma^{n}}x_0]=T^{\sigma^{n}}x_0$. But $T^{\sigma^{n}(t_0)}=T^{t_0/\gamma^n}$. And this implies that $t_0=0$ since otherwise $T^{\sigma^n}x_0$ would be invariant under a translation by an amount smaller than the size of any tile. We need one more set of ideas before we can consider the new symmetry. We must examine the content of Birkhoff's theorem in this tiling context. Assume $\mu$ is an ergodic Borel probability measure on $X$. Then for each basic open set $C=C_{x,\Theta,R}$ we have: $$\mu(C)=\int_X\raise.5ex\hbox{$\chi$}_C(x')\,d\mu = \lim_{L\to \R^d}{1\over |L|}\int_L\raise.5ex\hbox{$\chi$}_C(T^tx')\,dt\eqno 14)$$ \nd where $\raise.5ex\hbox{$\chi$}_C$ is the characteristic function for $C$. Consider the fraction $${1\over |L|}\int_L\raise.5ex\hbox{$\chi$}_C(T^tx')\,dt\eqno 15)$$ \nd It is roughly $${\epsilon^d \times (\hbox{the number of times the ``picture of $C$ appears'' in }x')\over |L|}\eqno 16)$$ \nd where the ``picture of $C$ '' refers to the part of the tiling in the definition of $C$, and the $\epsilon^d$ refers to the amount of translation and rotation the picture is allowed to move, in the definition of $C$. In summary, this fraction is, up to a factor, just the {\it frequency} with which the picture of $C$ appears in $x'$. Now although in general Birkhoff's theorem only holds for $\mu$ almost all $x'$, if $X$ is uniquely ergodic under translations then the convergence is not only valid for all $x'$ but is in fact uniform in $x'$ [Fur]. This is relevant because of the following, in which $X_{pin}$ refers to the pinwheel system and $X_{Pen}$ refers to the Penrose system. \vs.1 \nd {\bf Theorem 3.} $X_{pin}$ is uniquely ergodic. $X_{Pen}=\bigcup_{\alpha\in[0,2\pi/10)}X_\alpha$, where the $X_\alpha$ are uniquely ergodic and rotations of one another. \vs.1 We will give a general technique for proving such results but first we want to discuss its use. Note that any rotation on $X$ can be lifted to a rotation on the set of translation invariant probability measures on $X$. So if there is only one such measure, as the theorem claims is the case for the pinwheel, then that measure must be invariant under all rotations of $X$. A similar argument shows for the Penrose system that all translation invariant measures are invariant under rotations by multiples of $2\pi/10$. And from the above analysis of the geometric meaning of such measures we obtain the following statistical interpretation of these symmetries. For the Penrose tilings, any finite cluster of tiles appears in each tiling with the {\it same frequency} in each of 10 equally separated orientations. For the pinwheel, given any two equal-size intervals $I_1$ and $I_2$ of orientations, and any finite cluster of tiles in a tiling, the cluster will appear in that (or any other) tiling having an orientation in $I_1$ with the {\it same frequency} as it does with an orientation in $I_2$. In other words, the rotational symmetries are not leaving invariant tilings, but translation invariant measures on a space of tilings; and the invariance of all such measures is equivalent to the invariance of the above ``frequencies'' of finite portions of the tilings. Notice that this is a fundamentally new meaning of symmetry for a tiling or pattern, justified at least by its application in the structural study of materials [Rad6]. \vs.1 We now sketch the proof of a general argument [Rad5] for proving unique ergodicity of 2 dimensional tilings, which implies Theorem 3. We assume a general tiling system $X$ as above, and will make a few assumptions as needed. To prove unique ergodicity of $X$, given 15), we need to prove that for each finite cluster of tiles and interval of orientation of that cluster (with respect to some arbitrary standard), there is an associated frequency which is independent of the tiling and independent of rotation of the interval. We begin by ignoring the orientations, or what is the same thing, only considering the case where the interval of orientations is the full circle. One key idea is a refinement of the requirement concerning the similarity $\sigma$. We further require that this representation of $\sigma$ be effected by a substitution in the usual way. Namely, the ``small size'' tiling $x$ is obtained from the ``large size'' tiling $y(x)$ by simultaneously replacing each of the tiles in $y(x)$ by an appropriate word of small size letters. When such a word is expanded about the origin by the factor $\gamma$ one gets a tile or letter ``of level 1'', {\it etc.} (We note that in simple examples like the pinwheel the letters of all levels are geometrically similar; this is not the case for the Penrose tilings, and is not necessary for the method.) If the finite alphabet of $X$ is $\A=\{a_1,a_2,\ldots,a_K\}$ we introduce the notation that the letters of all levels, obtained by starting with $a_k$ at level 0, are tiles or letters of ``type $k$''. The above hierarchical assumption on $X$ is useful in that all we need prove now is that each given finite cluster of tiles has the same frequency of occurance {\it within each tile of level $N>>1$}, and the frequency is independent of the type of the level $N$ tile. (The reasoning is that we are just neglecting the occurances of the cluster on the edges between the tiles of level $N$, and for large $N$ this is negligible.) We now introduce the $K\times K$ matrix $M$ for which $M_{jk}$ is the number of type $j$ tiles (level 0) in a type $k$ tile of level 1. It follows that $M^N_{jk}$ is the number of type $j$ tiles in a type $k$ tile of level $N$. For the pinwheel, $$M= \pmatrix{2&3\cr 3&2\cr}\eqno 17)$$ \nd Next we assume there is some $p$ such that $M^p_{jk}>0$ for all $j,k$. It then follows from the Perron-Frobenius theorem that there is some $\delta >0$, and functions $g,h$, such that for all $j,k$: $${M^N_{jk} \over \delta^N}\ \ \lower1ex\hbox{${\hbox to 50pt{\rightarrowfill}\atop {N\to \infty}}$}\ \ g(j)h(k)>0\eqno 18)$$ \nd Therefore $${M^N_{jk} \over M^N_{j'k}}\ \ \lower1ex\hbox{${\hbox to 50pt{\rightarrowfill}\atop {N\to \infty}}$}\ \ {g(j)\over g(j')}>0\eqno 19)$$ \nd The important points are that the limits in 19) exist, and that they are independent of $k$, which are precisely what was needed to prove the result for clusters consisting of single tiles. (The generalization to larger clusters is routine and we refer to [Rad5] for the details.) This is a well-known argument for substitution subshifts [Que], and suffices to prove the result on Penrose systems in Theorem 3; one just enlarges the Penrose alphabet to include the kite and dart rotated by multiples of $2\pi/10$ and only considers tiles which are {\it translates} of letters, not {\it congruent} to letters. But to deal with the pinwheel, or other tilings in which rotations play an essential role, we must now deal with the orientations. To do this we generalize the matrix $M$ above to a family of matrices parametrized by $m\in \Z$: $$M(m)_{jk}\equiv \sum_\ell \exp(im\alpha_{\ell k})\eqno 20)$$ \nd where $\alpha_{\ell k}$ is the angle of the $\ell^{th}$ copy of the tile of type $j$ in a tile of type $k$ of level 1. Note that $M(0)$ is just the previous matrix $M$ and that $M(1)$ keeps track of the angles of the constituents of the level 1 tiles. We now make the further assumption that there is some tile of level $q$ which contains two tiles of level 0 with irrational relative orientation. (For the pinwheel $q=2$ suffices.) With this assumption we can use the Weyl criterion on equidistribution of points on a circle, and an old matrix result of Wielandt, to prove that the orientations of tiles (and then finite clusters of tiles) are equally distributed in every tiling in $X$; see [Rad5] for details.\qed \vs.1 At this point we would like to mention three open problems that are suggested by the above. First, although the method of this section makes sense in any dimension $d\ge 2$, we do not have a tool as effective as Weyl's criterion in higher dimensions. And yet tiling in higher dimensions could well bring new phenomena since their rotation groups are more interesting. This suggests the study of some 3 dimensional version of the pinwheel, in which the added richness of rotations in 3 dimensions comes into play. And finally there is the obvious need to generalize to a wide class of substitutions the method used to solve the pinwheel. \vs.1 \nd {\bf Summary.}\ \ The main theme of this article has been the generalization of subshifts of finite-type (with $\Z^d$ actions) to tiling systems (with $\R^d$ actions), and in particular the new role played by rotations. This new marriage of groups of rotations with ergodic theory should prove beneficial in many ways. It clearly enriches ergodic theory, with at least new ``symmetry'' invariants, but also provides new lines of research. And in the other direction ergodic theory seems to provide a new way to analyze the geometric symmetries of patterns in space, through ``statistical'' symmetries, in which the invariance of a pattern is replaced by the invariance of the frequencies of all its finite subpatterns [Rad6]. \vs.5 \centerline{Bibliography} \vs.1 \nd [B-R] D.\ Berend and C.\ Radin, Are there chaotic tilings? {\it Comm.\ Math.\ Phys.} 152 (1993), 215-219. \vs.1 \nd [Ber] R.\ Berger, The undecidability of the domino problem, {\it Mem.\ Amer.\ Math.\ Soc. no.\ 66}, (1966). \vs.1 \nd [Dwo1] S.\ Dworkin, Spectral theory and x-ray diffraction, {\it J.\ Math.\ Phys.} 34 (1993), 2965-2967. \vs.1 \nd [Dwo2] S.\ Dworkin, Deceptions and choice in tiling systems of finite type, University of Texas preprint, 1994. \vs.1 \nd [D-S] S.\ Dworkin and J.-I.\ Shieh, Deceptions in quasicrystal growth, {\it Comm.\ Math.\ Phys.} to appear. \vs.1 \nd [Fur] H.\ Furstenberg, {\it Recurrence in Ergodic Theory and Combinatorial Number Theory}, Princeton University Press, Princeton, 1981. \vs.1 \nd [Moz] S.\ Mozes, Tilings, substitution systems and dynamical systems generated by them, {\it J. d'Analyse Math.}\ 53 (1989), 139-186. \vs.1 \nd [Que] M.\ Queff\'elec, {\it Substitution Dynamical Systems - Spectral Analysis}, Lecture Notes in Mathematics, Vol.\ 1294, Springer-Verlag, Berlin, 1987. \vs.1 \nd [Rad1] C.\ Radin, Disordered Ground States of Classical Lattice Models, {\it Revs.\ Math.\ Phys.} 3 (1991), 125-135. \vs.1 \nd [Rad2] C.\ Radin, Global Order from Local Sources, {\it Bull.\ Amer.\ Math.\ Soc.} 25 (1991), 335-364. \vs.1 \nd [Rad3] C.\ Radin, ${\bf Z}^n$ versus ${\bf Z}$ actions for systems of finite type, {\it Contemporary Mathematics} 135 (1992), 339-342. \vs.1 \nd [Rad4] C.\ Radin, The pinwheel tilings of the plane, {\it Annals of Math.} 139 (1994), 661-702. \vs.1 \nd [Rad5] C.\ Radin, Space tilings and substitutions, {\it Geometriae Dedicata} to appear. \vs.1 \nd [Rad6] C.\ Radin, Symmetry and tilings, {\it Notices Amer.\ Math.\ Soc.} 42 (1995), 26-31. \vs.1 \nd [R-W] C.\ Radin and M.\ Wolff, Space tilings and local isomorphism, {\it Geometriae Dedicata} 42 (1992), 355-360. \vs.1 \nd [R-N] F.\ Riesz and B.\ Sz-Nagy, {\it Functional Analysis}, tr. by L.F.\ Boron, Frederick Ungar, New York, 1955, 392-393. \vs.1 \nd [Rob] R.M.\ Robinson, Undecidability and nonperiodicity for tilings of the plane, {\it Invent.\ Math.}\ 12 (1971), 177-209. \vs.1 \nd [Rue] D.\ Ruelle, Do turbulent crystals exist?, {\it Physica} 113A (1982), 619-623. \vs.1 \nd [Shi] J.-I.\ Shieh, The entropy of uniquely ergodic tiling systems, University of Texas preprint, 1994. \vfill\eject \hbox{} \vs.1 \hbox{}\hs-1.5 \vbox{\epsfxsize=7.5truein\epsfbox{jfig1.ps}} \vs.1 \centerline{Figure 1.\ \ \ A pinwheel tiling} \vfill \eject \hbox{} \epsfig 1.15\hsize; jfig2.ps; \vs-.5 \centerline{Figure 2.\ \ \ A Penrose tiling} \vfill \eject \hbox{}\vs2 \hs.5 \vbox{\epsfxsize=5truein\epsfbox{jfig3.ps}} \vs.5 \centerline{\hs1.3 \hbox{kite}\hs2.2 \hbox{dart}\hfil} \vs.3 \centerline{Figure 3.\ \ \ Penrose letters for substitution} \vfill\eject \hbox{}\vs-1 \vbox{\epsfxsize=6.5truein\epsfbox{jfig4a.ps}} \vs-1.5 \centerline{Figure 4a.\ \ \ Penrose substitution for dart} \vfill\eject \hbox{}\vbox{\epsfxsize=6.5truein\epsfbox{jfig4b.ps}} \vs-2 \centerline{Figure 4b.\ \ \ Penrose substitution for kite} \vfill\eject \hbox{}\vbox{\epsfxsize=6.5truein\epsfbox{jfig5.ps}} \vs-3 \centerline{Figure 5.\ \ \ Pinwheel letters for substitution} \vfill\eject \hbox{} \vbox{\epsfxsize=6.5truein\epsfbox{jfig6.ps}} \vs-3 \centerline{Figure 6.\ \ \ Pinwheel letters of level one for substitution} \vfill\eject \hbox{}\vs2 \hbox{}\hs.2 \vbox{\epsfxsize=4.5truein\epsfbox{jfig7newer.ps}} \vs.3 \centerline{\hs.45 \hbox{kite}\hs2 \hbox{dart}\hfil} \vs.4 \centerline{\hs-.75 Figure 7.\ \ \ Penrose letters for finite-type} \vfill\eject \hbox{}\vs2 \vbox{\epsfxsize=5truein\epsfbox{jfig8.ps}} \vs.2 \centerline{\hs-.1 \hbox{unit cell B}\hs2.2 \hbox{unit cell C}\hfil} \vs.3 \centerline{Figure 8.} \vfill\eject \hbox{}\vbox{\epsfxsize=4truein\epsfbox{jfig9.ps}} \centerline{Figure 9.\ \ \ Pinwheel letter of level 5 for substitution} \vfill\eject \hbox{}\vs2 \vbox{\epsfxsize=5truein\epsfbox{jfig10.ps}} \vs1 \centerline{Figure 10.\ \ \ Tiles with labels A through E in tesselation} \vfill\eject \hbox{}\vbox{\epsfxsize=6.5truein\epsfbox{jfig11.ps}} \vs-5.3 \centerline{a)\ \ no fault} \vs3.2 \centerline{b)\ \ fault} \vs1 \centerline{Figure 11.\ \ \ Two ground state configurations} \vfill \bye