<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">%# -*- coding: utf-8 -*-
% tiling.tex
% asymptotebyexample çš„ä¸€ç«&nbsp;ï¼Œæ›²çº¿ç»˜åˆ¶å’Œç¼–ç¨‹å…¥é—¨

\chapter{AndrÃ© Deledicq çš„é“ºç&nbsp;Œæ’ç”»}
\nocite{pstricks}

AndrÃ© æ˜¯ä¸€åå…´è¶£å¹¿æ³›çš„æ³•å›½æ•°å­¦æ•™å¸ˆï¼Œåœ¨ä»–çš„æ–°è‘—ã€ŠLe monde des pavagesã€‹ï¼ˆã€Šé“º
ç&nbsp;Œä¸–ç•Œã€‹ï¼‰ä¸­ï¼Œæ‰“ç®—ç”»ä¸€å¹…æœ‰å…³ç¾Šçš„é“ºç&nbsp;Œæ’ç”»ï¼š
\begin{figure}[H]
  \centering
  \includegraphics{tiling.pdf}
  \caption{AndrÃ© ç†æƒ³ä¸­çš„é“ºç&nbsp;Œå›¾}
  \label{fig:tiling}
\end{figure}

AndrÃ© å¾ˆæ¸…æ¥šä»–è¦ç”»çš„å›¾å½¢çš„æ•°å­¦ç†è®ºï¼Œä½† AndrÃ© çš„æœ‹å‹ Timothy å‘Šè¯‰ä»–è¦ç”»è¿™æ&nbsp;·
çš„å›¾å½¢å¤šå°‘æ˜¯éœ€è¦ä¸€äº›ç¼–ç¨‹çš„çŸ¥è¯†çš„ï¼Œå¯¹äºŽä»–è¿™æ&nbsp;·ä¸€ä½å¾€æ—¥å¯¹è®¡ç®—æœºå¹¶ä¸é€šæ™“çš„äººæ¥è¯´
å¯èƒ½ä¼šæœ‰å›°éš¾ã€‚ä¸è¿‡ AndrÃ© å¹¶ä¸ä»¥ä¸ºæ„ï¼šè¿™ä¸–ä¸Šè¿˜æœ‰ä»€ä¹ˆæ¯”æ•°å­¦æ›´éš¾çš„å‘¢ï¼ŸäºŽæ˜¯ä»–å…´
è‡´å‹ƒå‹ƒçš„å¼€å§‹äº†ã€‚

\section{ä»ŽçŸ©å½¢åˆ°é“ºç&nbsp;Œ}
\label{sec:rect2tiling}

é“ºç&nbsp;Œå›¾ï¼Œé¡¾åæ€ä¹‰ï¼Œå°±æ˜¯åƒé“ºåœ°æ¿ç&nbsp;–ä¸€æ&nbsp;·ï¼ŒæŠŠè®¸å¤šç›¸åŒæ&nbsp;·å¼çš„å›¾å½¢å¹³é“ºå¼€æ¥ã€‚ä¸è¿‡ï¼Œå¹¶
ä¸æ˜¯ä»€ä¹ˆå›¾å½¢éƒ½å¯ä»¥å¹³é“ºå¡«æ»¡æ•´ä¸ªå¹³é¢çš„â€”â€”æ¯”å¦‚åœ†å½¢å°±ä¸è¡Œã€‚æŠŠè®¸å¤šåœ†å½¢ä¸€ä¸ªæŒ¨ä¸€ä¸ª
æŽ’åˆ—èµ·æ¥ï¼Œä¹Ÿåªèƒ½å¾—åˆ°
\begin{figure}[H]
\centering
\begin{asy}
size(0,2cm);
for (int i = 0; i &lt; 3; ++i)
    for (int j = 0; j &lt; 10; ++j)
        filldraw(circle((j,i), 0.5), lightblue);
\end{asy}
\end{figure}
\noindent æˆ–è€…æ˜¯
\begin{figure}[H]
\centering
\begin{asy}
size(0,2cm);
for (int i = 0; i &lt; 3; ++i)
    for (int j = 0; j &lt; 10; ++j)
        filldraw(circle((j+0.5(i%2),sqrt(3)/2*i), 0.5), lightblue);
\end{asy}
\end{figure}
\noindent éƒ½ä¼šç•™ä¸‹è®¸å¤šç©ºéš™ã€‚è€ŒçŸ©å½¢ã€å¹³è¡Œå››è¾¹å½¢ã€å…­è¾¹å½¢ç­‰ç­‰éƒ½å¯ä»¥ä¸ç•™ç©ºéš™åœ°æŠŠå¹³é¢é“ºæ»¡ã€‚

ä½†é—®é¢˜æ˜¯ï¼Œå¦‚ä½•è®¾è®¡å‡º AndrÃ© ç†æƒ³ä¸­çš„é‚£ç§çœ‹èµ·æ¥å½¢çŠ¶ä¸è§„åˆ™çš„é“ºç&nbsp;Œå›¾æ¡ˆå‘¢ï¼Ÿ

èº«ä¸ºæ•°å­¦æ•™å¸ˆçš„ AndrÃ© å½“ç„¶æœ‰åŠžæ³•ã€‚å…¶å®žä¸è§„åˆ™é“ºç&nbsp;Œå›¾æ¡ˆè¿˜æ˜¯è§„åˆ™å›¾æ¡ˆçš„å˜å½¢ã€‚
AndrÃ© è¦ç”»çš„ç¾Šå½¢é“ºç&nbsp;Œå›¾ï¼Œå…¶å®žå°±æ˜¯ä»ŽçŸ©å½¢é“ºç&nbsp;Œå˜åŒ–è€Œæ¥çš„ã€‚åªè¦æŠŠä¸€ä¸ªçŸ©å½¢å›¾æ¡ˆçš„ä¸Š
ä¸‹ä¸¤è¾¹ã€å·¦å³ä¸¤è¾¹åˆ†åˆ«å˜å½¢ï¼Œä½¿å¾—å˜å½¢åŽçš„ä¸Šè¾¹ä¸Žä¸‹è¾¹ã€å·¦è¾¹ä¸Žå³è¾¹è¿˜å¯¹åº”é‡åˆï¼Œå°±ä¾
ç„¶å¯ä»¥å®Œç¾Žåœ°æ‹¼åˆèµ·æ¥ã€‚è¿™æ­£æ˜¯é“ºç&nbsp;Œå›¾æ¡ˆæœ€åŸºæœ¬çš„æž„æˆæ–¹å¼ï¼š
\begin{figure}[H]
\centering
\begin{asy}
size(0,5cm);
defaultpen(linewidth(1mm));
path rec = box((0,0), (2,1));
draw(rec);

draw(shift(3,0)*rec, gray+0.5mm);
guide left = (0,1) -- (-0.2,0.8) -- (0,0.6) -- (0.1,0.1) -- (0,0);
guide bot = (0,0) -- (0.2,0.1) -- (1.6,-0.1) -- (2,0);
draw(shift(3,0) * (left ^^ shift(2,0)*left), heavyblue);
draw(shift(3,0) * (bot ^^ shift(0,1)*bot), heavygreen);

guide shape = left &amp; bot &amp; shift(2,0)*reverse(left)
    &amp; shift(0,1)*reverse(bot) &amp; cycle;
for (int i = 0 ; i &lt; 2; ++i)
    for (int j = 0; j &lt; 4; ++j)
	filldraw(shift(0.5+j,-1.5+0.5i)*scale(0.5)*shape, lightblue);
\end{asy}
\end{figure}

æœ‰äº†è¿™ä¸ªæ–¹æ³•ï¼Œå¯¹å¤æ‚çš„é“ºç&nbsp;Œå›¾ï¼Œä¹Ÿåªè¦ä»Žä¸€ä¸ªåŸºæœ¬å½¢çŠ¶ï¼ˆæ¯”å¦‚çŸ©å½¢ã€æ­£å…­è¾¹å½¢ï¼‰å¼€å§‹
å˜å½¢ï¼Œå°±ç­‰åˆ°é“ºç&nbsp;Œæ‰€éœ€è¦çš„ä¸€å—â€œç&nbsp;–â€ã€‚

å›&nbsp;æ­¤ï¼Œè¦ç”»å‡ºç¾Šå¤´å½¢çŠ¶é“ºç&nbsp;Œå›¾ï¼Œåªè¦æŠŠä¸€ä¸ªçŸ©å½¢æŒ‰ç…§ä¸Šé¢çš„è¦æ±‚å˜å½¢ä¸ºä¸€ä¸ªç¾Šå¤´å½¢çŠ¶ï¼Œ
åœ¨ä¸åŒçš„ä½ç½®é‡å¤ç”»å‡ºå°±å¯ä»¥äº†ã€‚

\section{å˜é‡ä¸Žæ›²çº¿}

ä¸‹é¢çš„é—®é¢˜å°±æ˜¯ï¼Œæ€Žä¹ˆç”»ä¸€ä¸ªç¾Šå¤´å‘¢ï¼Ÿæ›´å…·ä½“åœ°è¯´ï¼Œæ€Žä¹ˆç”»å‡ºç¾Šå¤´çš„æ›²çº¿å‘¢ï¼Ÿ

é‚£ä¹ˆï¼Œé¦–å…ˆè¦äº†è§£å¦‚ä½•åœ¨ \Asy{} ä¸­æè¿°æ›²çº¿ã€‚\ref{sec:linedraw} èŠ‚ä¸­æåˆ° |--| è¿ž
ç»“ä¸€ç»„åæ&nbsp;‡å°±æˆä¸ºç›´ï¼ˆæŠ˜ï¼‰çº¿æ®µï¼›ç±»ä¼¼åœ°ï¼Œç”¨ |..| è¿žç»“åæ&nbsp;‡å°±å¾—åˆ°ç»è¿‡è¿™äº›åæ&nbsp;‡ç‚¹çš„
æ›²çº¿ï¼š
\begin{lstlisting}
size(5cm,0);
pair z1 = (0,1), z2 = (1,1), z3 = (2,1),
     z4 = (0,0), z5 = (1,0), z6 = (2,0);
path p = z4 .. z1 .. z2 .. z6;
draw(p, gray+2mm);
\end{lstlisting}
\begin{figure}[H]
  \centering
\begin{asy}
size(5cm,0);
pair z1 = (0,1), z2 = (1,1), z3 = (2,1),
     z4 = (0,0), z5 = (1,0), z6 = (2,0);
path p = z4 .. z1 .. z2 .. z6;
draw(p, gray+2mm);
dot(Label("1", align=NW), z1);
dot(Label("2", align=NE), z2);
dot("3", z3);
dot("4", z4);
dot("5", z5);
dot("6", z6);
\end{asy}
\end{figure}

åœ¨è¿™é‡Œï¼Œæˆ‘ä»¬å®šä¹‰äº†ä¸€äº›å˜é‡\index{å˜é‡}ä»¥ä½¿ä»£ç&nbsp;æ¸…æ™°ï¼ˆè¿™é‡Œç•¥åŽ»äº†ç”»ç‚¹å’Œæ&nbsp;‡ç­¾çš„ä»£
ç&nbsp;ï¼‰ã€‚|pair|\index{pair@\lstinline=pair=} ç±»åž‹çš„å˜é‡ |z1|, \ldots, |z6| ä¿å­˜
å…­ä¸ªåæ&nbsp;‡\index{åæ&nbsp;‡}ï¼Œ|path|\index{path@\lstinline=path=} ç±»åž‹çš„å˜é‡ |p| ä¿å­˜
ä¸€æ¡æ›²çº¿è·¯å¾„\index{è·¯å¾„}ã€‚å›&nbsp;è€Œä¸Šé¢ |size| ä¹‹åŽçš„ç»˜å›¾ä»£ç&nbsp;å°±ç›¸å½“äºŽ
\begin{lstlisting}
draw( (0,0) .. (0,1) .. (1,1) .. (2,0), gray+2mm );
\end{lstlisting}
å…¶ä¸­å‰é¢çš„ä¸€å¥ |size(5cm,0)|\index{size@\lstinline=size=} è¡¨ç¤ºä»£ç&nbsp;ä¸­çš„åæ&nbsp;‡åª
æ˜¯ç›¸å¯¹ä½ç½®ï¼Œæœ€åŽå°†æ•´ä¸ªå›¾å½¢æŒ‰æ¯”ä¾‹æ”¾ç¼©ä¸º $5$\,cm å®½\footnote{æ³¨æ„åæ&nbsp;‡ã€å›¾å½¢ä¼šè¢«
æ”¾ç¼©ï¼Œä½†ç”»ç¬”çš„å®½åº¦ä¸ä¼šæ”¾ç¼©ã€‚}ã€‚ç±»ä¼¼åœ°ï¼Œä¹Ÿå¯ä»¥ä½¿ç”¨ |size(0,4cm)| æŠŠå›¾å½¢æ”¾ç¼©åˆ°
$4$\,cm é«˜ã€‚

æœ€é‡è¦çš„å½“ç„¶è¿˜æ˜¯æ›²çº¿çš„è¡¨ç¤ºã€‚ä»¥ |..| è¿žç»“çš„åæ&nbsp;‡ä¼šä»¥ä¸€ç§å°½é‡æŽ¥è¿‘åœ†å¼§çš„æ–¹å¼è¿žä¸º
ç»è¿‡è¿™äº›ç‚¹çš„å…‰æ»‘æ›²çº¿ã€‚ä¸Žç”»ç›´çº¿ç±»ä¼¼ï¼Œ|cycle|\index{cycle@\lstinline=cycle=} å¯
ä»¥ä½œä¸ºä¸€ä¸ªç‰¹æ®Šçš„åæ&nbsp;‡äº§ç”Ÿé—­åˆæ›²çº¿ï¼Œå³ä¸€æ¡é—­è·¯å¾„\index{è·¯å¾„!é—­è·¯å¾„}ï¼š
\begin{lstlisting}
path q = z4 .. z1 .. z2 .. z6 .. cycle;
draw(q, gray+2mm);
\end{lstlisting}
\begin{figure}[H]
  \centering
\begin{asy}
size(5cm,0);
pair z1 = (0,1), z2 = (1,1), z3 = (2,1),
     z4 = (0,0), z5 = (1,0), z6 = (2,0);
path q = z4 .. z1 .. z2 .. z6 .. cycle;
draw(q, gray+2mm);
dot(Label("1", align=NW), z1);
dot(Label("2", align=NE), z2);
dot("3", z3);
dot("4", z4);
dot("5", z5);
dot("6", z6);
\end{asy}
\end{figure}

å˜é‡ä¸ä»…ä»…æ˜¯ç»™äº†åæ&nbsp;‡ã€è·¯å¾„ç­‰å¯¹è±¡ä¸€ä¸ªç®€æ´çš„åå­—ï¼Œå®ƒä¹Ÿä½¿å¾—å¯¹åŒä¸€ä¸ªå¯¹è±¡é‡å¤ä½¿ç”¨
å¹¶è¿›è¡Œä¸åŒçš„æ“ä½œå˜å¾—ååˆ†æ–¹ä¾¿ï¼š
\begin{lstlisting}
fill(q, lightblue);
draw(q, gray+2mm);
\end{lstlisting}
\begin{figure}[H]
  \centering
\begin{asy}
size(5cm,0);
pair z1 = (0,1), z2 = (1,1), z3 = (2,1),
     z4 = (0,0), z5 = (1,0), z6 = (2,0);
path q = z4 .. z1 .. z2 .. z6 .. cycle;
fill(q, lightblue);
draw(q, gray+2mm);
dot(Label("1", align=NW), z1);
dot(Label("2", align=NE), z2);
dot("3", z3);
dot("4", z4);
dot("5", z5);
dot("6", z6);
\end{asy}
\end{figure}

å°±åƒä½¿ç”¨ |box| å¯ä»¥ç›´æŽ¥å¾—åˆ°çŸ©å½¢ä¸€æ&nbsp;·ï¼Œæœ€å¸¸ç”¨çš„æ›²çº¿ï¼šåœ†ã€æ¤­åœ†å’Œåœ†å¼§ï¼Œä¹Ÿå¯ä»¥ä½¿ç”¨
çŽ°æˆçš„å‘½ä»¤å¾—åˆ°ï¼š
\begin{table}[H]
\noindent
\begin{tabular}{ll}
|circle(c, r)| &amp; åœ†å¿ƒ |c|ï¼ŒåŠå¾„ |r| çš„åœ†ï¼Œè¿™æ˜¯é€†æ—¶é’ˆæ–¹å‘çš„é—­æ›²çº¿ï¼› \\
|ellipse(c, a, b)| &amp; ä¸­å¿ƒä¸º |c|ï¼Œé•¿åŠè½´ |a|ï¼ŒçŸ­åŠè½´ |b| çš„æ¤­åœ†ï¼Œè¿™ä¹Ÿæ˜¯é€†æ—¶é’ˆ
æ–¹å‘çš„é—­æ›²çº¿ï¼› \\
|arc(c, r, angle1, angle2)| &amp; åœ†å¿ƒ |c|ï¼ŒåŠå¾„ |r|ï¼Œè§’åº¦ä»Ž |angle1| åˆ° |angle2|
çš„åœ†å¼§ã€‚
\end{tabular}
\end{table}

ä¾‹å¦‚ï¼š
\begin{lstlisting}
filldraw( circle((0,0), 1cm), lightblue, gray+2mm );
draw( arc((5cm,0), 1cm, 45, 135), gray+2mm );
\end{lstlisting}
\begin{figure}[H]
\centering
\begin{asy}
filldraw( circle((0,0), 1cm), lightblue, gray+2mm );
draw( arc((5cm,0), 1cm, 45, 135), gray+2mm );
\end{asy}
\end{figure}

ä¸€æ¡ç”¨ |cycle| äº§ç”Ÿçš„é—­è·¯å¾„å’Œç®€å•åœ°æŠŠé¦–å°¾ç»“ç‚¹é‡åˆçš„è·¯å¾„æ˜¯éžå¸¸ä¸åŒçš„ã€‚é¦–å…ˆï¼Œåª
æœ‰é—­è·¯å¾„å¯ä»¥å¡«å……é¢œè‰²ï¼›å…¶æ¬¡ï¼Œä½¿ç”¨ |cycle| è¿žç»“çš„æ›²çº¿åœ¨èµ·ç‚¹å¤„æ˜¯å…‰æ»‘è¿žæŽ¥çš„ï¼Œè€Œå¦‚
æžœåªæ˜¯é¦–å°¾ç»“ç‚¹é‡åˆåˆ™ä¸ä¼šå…‰æ»‘è¿žæŽ¥ã€‚è¯•å°†ä¸‹é¢çš„æ›²çº¿ |q2| ä¸Žä¸Šé¢çš„æ›²çº¿ |q| æ¯”è¾ƒï¼š
\begin{lstlisting}
path q2 = z4 .. z1 .. z2 .. z6 .. z4;
draw(q2, gray+2mm);
\end{lstlisting}
\begin{figure}[H]
\centering
\begin{asy}
size(5cm,0);
pair z1 = (0,1), z2 = (1,1), z3 = (2,1),
     z4 = (0,0), z5 = (1,0), z6 = (2,0);
path q2 = z4 .. z1 .. z2 .. z6 .. z4;
draw(q2, gray+2mm);
dot(Label("1", align=NW), z1);
dot(Label("2", align=NE), z2);
dot("3", z3);
dot("4", z4);
dot("5", z5);
dot("6", z6);
\end{asy}
\end{figure}

çŽ°åœ¨æœ‰äº†ç»˜åˆ¶æ›²çº¿çš„æ–¹æ³•ï¼Œç”»å‡ºä¸€ä¸ªç¾Šå¤´å°±åªæ˜¯æŠŠè‰ç¨¿ä¸Šçš„åæ&nbsp;‡è¿žæŽ¥èµ·æ¥è€Œå·²ã€‚AndrÃ©
æœ‰ä¸€ä¸ªçº¸ä¸Šçš„è‰å›¾ï¼ŒäºŽæ˜¯åœ¨æå‡ºå‡&nbsp;ä¸ªç‚¹ä»¥åŽï¼Œä»–å¾ˆå¿«å¾—åˆ°è¿™æ&nbsp;·çš„ç»“æžœï¼ˆè¿™é‡Œç»™å›¾å½¢å¢žåŠ&nbsp;
äº†è¾…åŠ©ç½‘æ&nbsp;¼ï¼‰ï¼š
\begin{lstlisting}
size(0,4cm);
pen outline = black+1mm;
// `\color{comment}å¤´`
path head = (0.5,-0.2) .. (0.6,0.5) .. (0.2,1.3) .. (0,1.5) .. (0,1.5)
    .. (0.4,1.3) .. (0.8,1.5) .. (2.2,1.9) .. (3,1.5) .. (3.2,1.3)
    .. (3.6,0.5) .. (3.4,-0.3) .. (3,0) .. (2.2,0.4) .. (0.5,-0.2) .. cycle;
filldraw(head, cyan, outline);
dot(head, red+1mm); // `\color{comment}ç”»å‡ºç¾Šå¤´è·¯å¾„ä¸Šçš„ç»“ç‚¹`
// `\color{comment}äº”å®˜`
fill( circle((2.65,1.25), 0.12), outline );
fill( (3.5,0.3) .. (3.35,0.45) .. (3.5,0.6) .. (3.6,0.4) .. cycle, outline );
draw( (3,0.35) .. (3.3,0.1) .. (3.6,0.05), outline );
draw( (2.3,1.3) .. (2.1, 1.5) .. (2.15,1.7), outline );
draw( (2.1,1.7) .. (2.35,1.6) .. (2.45,1.4), outline );
\end{lstlisting}
\begin{figure}[H]
  \centering
\begin{asy}
size(0,4cm);
import math;
add(scale(1/2)*shift(0,-1)*grid(8,5,gray));
dot(Label("$O$",align=left), 0);
label("$1$", (0,1), align=W); label("$2$", (0,2), align=W);
label("$1$", (1,-0.5), align=S); label("$2$", (2,-0.5), align=S);
label("$3$", (3,-0.5), align=S); label("$4$", (4,-0.5), align=S);
pen outline = black+1mm;
path head = (0.5,-0.2) .. (0.6,0.5) .. (0.2,1.3) .. (0,1.5) .. (0,1.5)
    .. (0.4,1.3) .. (0.8,1.5) .. (2.2,1.9) .. (3,1.5) .. (3.2,1.3)
    .. (3.6,0.5) .. (3.4,-0.3) .. (3,0) .. (2.2,0.4) .. (0.5,-0.2) .. cycle;
filldraw(head, cyan, outline);
dot(head, red+1mm);
fill( circle((2.65,1.25), 0.12), outline );
fill( (3.5,0.3) .. (3.35,0.45) .. (3.5,0.6) .. (3.6,0.4) .. cycle, outline );
draw( (3,0.35) .. (3.3,0.1) .. (3.6,0.05), outline );
draw( (2.3,1.3) .. (2.1, 1.5) .. (2.15,1.7), outline );
draw( (2.1,1.7) .. (2.35,1.6) .. (2.45,1.4), outline );
\end{asy}
\end{figure}

åœ¨ä¸€å¼€å§‹ï¼ŒAndrÃ© ä½¿ç”¨
\begin{lstlisting}
pen outline = black+1mm;
\end{lstlisting}
å®šä¹‰ä¸€ä¸ª |pen|\index{pen@\lstinline=pen=} ç±»åž‹çš„å˜é‡ |outline| è¡¨ç¤ºç”¨æ¥ç”»ç¾Šå¤´
è½®å»“çš„ç”»ç¬”\index{ç”»ç¬”}ï¼Œä»¥å¤‡ä½¿ç”¨ã€‚

ç„¶åŽï¼ŒAndrÃ© ç›´æŽ¥ç”¨ |..| è¿žç»“ä¸€ç»„åæ&nbsp;‡æ¥å®šä¹‰ç¾Šçš„å¤´éƒ¨è½®å»“ï¼š
\begin{lstlisting}
path head = (0.5,-0.2) .. (0.6,0.5) .. (0.2,1.3) .. (0,1.5) .. (0,1.5)
    .. (0.4,1.3) .. (0.8,1.5) .. (2.2,1.9) .. (3,1.5) .. (3.2,1.3)
    .. (3.6,0.5) .. (3.4,-0.3) .. (3,0) .. (2.2,0.4) .. (0.5,-0.2) .. cycle;
\end{lstlisting}
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