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Marc Bezem
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Merge branch 'master' of github.com:UniMath/SymmetryBook
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‎geometry.tex‎

Lines changed: 254 additions & 65 deletions
Original file line numberDiff line numberDiff line change
@@ -367,73 +367,262 @@ \section{The icosahedron}
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\end{figure}
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\section{Frieze patterns}
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\label{sec:friezes}
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371-
\begin{center}
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\begin{tikzpicture}[thick]
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\draw[dashed,thin] (0.25,-1) rectangle (8.25,1);
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\clip (0.25,-1) rectangle (8.25,1);
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\draw (-1,0) -- (9,0);
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\foreach \i in {0,1,...,8} {
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\draw (\i,.5) -- (.5+\i,0);
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}
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\end{tikzpicture}\\[10pt]
380-
\begin{tikzpicture}[thick]
381-
\draw[dashed,thin] (0.25,-1) rectangle (8.25,1);
382-
\clip (0.25,-1) rectangle (8.25,1);
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\draw (-1,0) -- (9,0);
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\foreach \i in {0,2,...,8} {
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\draw (\i,.5) -- (.5+\i,0);
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}
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\foreach \i in {1,3,...,7} {
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\draw (\i,-.5) -- (.5+\i,0);
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}
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\end{tikzpicture}\\[10pt]
391-
\begin{tikzpicture}[thick]
392-
\draw[dashed,thin] (0.25,-1) rectangle (8.25,1);
393-
\clip (0.25,-1) rectangle (8.25,1);
394-
\draw (-1,0) -- (9,0);
395-
\foreach \i in {0,1,...,8} {
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\draw (\i,.5) -- (\i,0);
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}
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\end{tikzpicture}\\[10pt]
399-
\begin{tikzpicture}[thick]
400-
\draw[dashed,thin] (0.25,-1) rectangle (8.25,1);
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\clip (0.25,-1) rectangle (8.25,1);
402-
\draw (-1,0) -- (9,0);
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\foreach \i in {0,1,...,8} {
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\draw (\i,.5) -- (1+\i,-.5);
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}
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\end{tikzpicture}\\[10pt]
407-
\begin{tikzpicture}[thick]
408-
\draw[dashed,thin] (0.25,-1) rectangle (8.25,1);
409-
\clip (0.25,-1) rectangle (8.25,1);
410-
\draw (-1,0) -- (9,0);
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\foreach \i in {0,2,...,8} {
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\draw (\i,.5) -- (1+\i,-.5);
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}
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\foreach \i in {1,3,...,7} {
415-
\draw (\i,-.5) -- (1+\i,.5);
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}
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\end{tikzpicture}\\[10pt]
418-
\begin{tikzpicture}[thick]
419-
\draw[dashed,thin] (0.25,-1) rectangle (8.25,1);
420-
\clip (0.25,-1) rectangle (8.25,1);
421-
\draw (-1,0) -- (9,0);
422-
\foreach \i in {0,1,...,8} {
423-
\draw (\i,.5) -- (.5+\i,0);
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\draw (\i,-.5) -- (.5+\i,0);
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}
426-
\end{tikzpicture}\\[10pt]
427-
\begin{tikzpicture}[thick]
428-
\draw[dashed,thin] (0.25,-1) rectangle (8.25,1);
429-
\clip (0.25,-1) rectangle (8.25,1);
430-
\draw (-1,0) -- (9,0);
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\foreach \i in {0,1,...,8} {
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\draw (\i,.5) -- (\i,-.5);
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}
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\end{tikzpicture}
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\begin{figure}
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\newcommand*{\friezesep}{\hskip2pt\rule[-9cm]{\pgflinewidth}{9.2cm}\hskip2pt}%
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\begin{sidecaption}%
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{The seven frieze patterns up to isometry,
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with their orbifold symbols.}[fig:friezes]
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\begin{tikzpicture}[baseline=(desc.base)]%\infty\infty
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\node (desc) at (0,8.5) {${\infty}{\infty}$};
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\foreach \n in {1,...,6} {
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\pic at (0,\n) {tendril={1.5}{1.5}{70}{black!20}};
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}
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\node[inner sep=0pt] (a7) at (0,7) {\tvdots};
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\node[inner sep=0pt] (a0) at (0,0) {\tvdots};
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\end{tikzpicture}\friezesep
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%
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\begin{tikzpicture}[baseline=(desc.base)]%\infty\times
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\node (desc) at (0.5,9) {${\infty}{\times}$};
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\foreach \n in {1,...,4} {
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\pic at (0,-.5+1.7*\n) {tendril={1.5}{1.5}{70}{black!20}};
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\pic at (1,.35+1.7*\n) {tendril={-1.5}{1.5}{70}{black!20}};
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}
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\node[inner sep=0pt] (a5) at (0,8) {\tvdots};
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\node[inner sep=0pt] (b0) at (1,.35) {\tvdots};
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\end{tikzpicture}\friezesep
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%
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\begin{tikzpicture}[baseline=(desc.base)]%22\infty
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\node (desc) at (0.5,8.5) {${2}{2}{\infty}$};
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\foreach \n in {1,...,6} {
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\pic at (0,\n) {tendril={1.5}{1.5}{70}{black!20}};
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\pic at (1,\n) {tendril={1.5}{1.5}{250}{black!20}};
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}
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\node[inner sep=0pt] (x0) at (0,0) {\tvdots};
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\node[inner sep=0pt] (x7) at (0,7) {\tvdots};
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\node[inner sep=0pt] (y0) at (1,0) {\tvdots};
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\node[inner sep=0pt] (y7) at (1,7) {\tvdots};
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\end{tikzpicture}\friezesep\hskip1pt
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%
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\begin{tikzpicture}[baseline=(desc.base)]%2\star\infty
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\node (desc) at (0.35,9.5) {${2}{\star}{\infty}$};
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\foreach \n in {1,...,4} {
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\pic at (0,0.05+2*\n) {tendril={1.2}{1.2}{-20}{black!20}};
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\pic at (0,-.65+2*\n) {tendril={-1.2}{1.2}{160}{black!20}};
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}
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\foreach \n in {1,...,3} {
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\pic at (.7,1.05+2*\n) {tendril={-1.2}{1.2}{-20}{black!20}};
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\pic at (.7,.35+2*\n) {tendril={1.2}{1.2}{160}{black!20}};
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}
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\node[inner sep=0pt] (d4) at (.7,8.35) {\tvdots};
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\node[inner sep=0pt] (c0) at (.7,1.05) {\tvdots};
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\end{tikzpicture}\hskip1pt\friezesep
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%
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\begin{tikzpicture}[baseline=(desc.base)]%\star\infty\infty
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\node (desc) at (0,9) {${\star}{\infty}{\infty}$};
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\foreach \n in {1,...,4} {
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\pic at (0,.35+1.7*\n) {tendril={1.5}{1.5}{-20}{black!20}};
426+
\pic at (0,-.5+1.7*\n) {tendril={-1.5}{1.5}{160}{black!20}};
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\node (a\n) at (0,.35+1.7*\n) {};
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\node (b\n) at (0,-.5+1.7*\n) {};
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}
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\node[inner sep=0pt] (b5) at (0,8) {\tvdots};
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\node[inner sep=0pt] (a0) at (0,.35) {\tvdots};
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\end{tikzpicture}\friezesep
433+
%
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\begin{tikzpicture}[baseline=(desc.base)]%\infty\star
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\node (desc) at (0.5,8.5) {${\infty}{\star}$};
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\foreach \n in {1,...,6} {
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\pic at (0,\n) {tendril={1.5}{1.5}{70}{black!20}};
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\pic at (1,\n) {tendril={-1.5}{1.5}{70}{black!20}};
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}
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\node[inner sep=0pt] (x0) at (0,0) {\tvdots};
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\node[inner sep=0pt] (x7) at (0,7) {\tvdots};
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\node[inner sep=0pt] (y0) at (1,0) {\tvdots};
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\node[inner sep=0pt] (y7) at (1,7) {\tvdots};
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\end{tikzpicture}\friezesep
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\begin{tikzpicture}[baseline=(desc.base)]%*22\infty
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\node (desc) at (0.5,9) {${\star}{2}{2}{\infty}$};
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\foreach \n in {1,...,4} {
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\pic at (0,.35+1.7*\n) {tendril={1.5}{1.5}{-20}{black!20}};
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\pic at (0,-.5+1.7*\n) {tendril={-1.5}{1.5}{160}{black!20}};
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\pic at (1,.35+1.7*\n) {tendril={1.5}{-1.5}{160}{black!20}};
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\pic at (1,-.5+1.7*\n) {tendril={1.5}{1.5}{160}{black!20}};
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}
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\node[inner sep=0pt] (b5) at (0,8) {\tvdots};
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\node[inner sep=0pt] (a0) at (0,.35) {\tvdots};
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\node[inner sep=0pt] (d5) at (1,8) {\tvdots};
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\node[inner sep=0pt] (c0) at (1,.35) {\tvdots};
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\end{tikzpicture}
458+
\end{sidecaption}
459+
\end{figure}
435460

436-
\end{center}
461+
See~\cref{fig:friezes,fig:friezes-gen}
462+
463+
\begin{figure}
464+
\newcommand*{\friezesep}{\hskip2pt\rule[-9cm]{\pgflinewidth}{9.2cm}\hskip2pt}%
465+
\begin{sidecaption}%
466+
{The seven frieze patterns up to isometry,
467+
with their orbifold symbols and superimposed generators.}[fig:friezes-gen]
468+
\begin{tikzpicture}[baseline=(desc.base)]%\infty\infty
469+
\node (desc) at (0,8.5) {${\infty}{\infty}$};
470+
\foreach \n in {1,...,6} {
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\pic at (0,\n) {tendril={1.5}{1.5}{70}{black!10}};
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\node[dot] (a\n) at (0,\n) {};
473+
}
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\node[inner sep=0pt] (a7) at (0,7) {\tvdots};
475+
\node[inner sep=0pt] (a0) at (0,0) {\tvdots};
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\foreach \p/\n in {0/1,1/2,2/3,3/4,4/5,5/6,6/7} {
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\draw[gena] (a\p) -- (a\n);
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}
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\end{tikzpicture}\friezesep
480+
%
481+
\begin{tikzpicture}[baseline=(desc.base)]%\infty\times
482+
\node (desc) at (0.5,9) {${\infty}{\times}$};
483+
\foreach \n in {1,...,4} {
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\pic at (0,-.5+1.7*\n) {tendril={1.5}{1.5}{70}{black!10}};
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\pic at (1,.35+1.7*\n) {tendril={-1.5}{1.5}{70}{black!10}};
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\node[dot] (a\n) at (0,-.5+1.7*\n) {};
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\node[dot] (b\n) at (1,.35+1.7*\n) {};
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}
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\node[inner sep=0pt] (a5) at (0,8) {\tvdots};
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\node[inner sep=0pt] (b0) at (1,.35) {\tvdots};
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\foreach \p/\n in {0/1,1/2,2/3,3/4,4/5} {
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\draw[gena] (b\p) -- (a\n);
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}
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\foreach \n in {1,2,3,4} {
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\draw[gena] (a\n) -- (b\n);
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}
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\end{tikzpicture}\friezesep
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%
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\begin{tikzpicture}[baseline=(desc.base)]%22\infty
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\node (desc) at (0.5,8.5) {${2}{2}{\infty}$};
501+
\foreach \n in {1,...,6} {
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\pic at (0,\n) {tendril={1.5}{1.5}{70}{black!10}};
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\pic at (1,\n) {tendril={1.5}{1.5}{250}{black!10}};
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\node[dot] (x\n) at (0,\n) {};
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\node[dot] (y\n) at (1,\n) {};
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}
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\node[inner sep=0pt] (x0) at (0,0) {\tvdots};
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\node[inner sep=0pt] (x7) at (0,7) {\tvdots};
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\node[inner sep=0pt] (y0) at (1,0) {\tvdots};
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\node[inner sep=0pt] (y7) at (1,7) {\tvdots};
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\begin{scope}[gena]
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\foreach \p/\n in {0/1, 1/2, 2/3, 3/4, 4/5, 5/6, 6/7} {
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\draw (x\p)--(x\n);
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\draw (y\n)--(y\p);
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}
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\end{scope}
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\begin{scope}[genb]
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\foreach \n in {1,2,...,6} {
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\draw (x\n) to[bend right] (y\n);
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\draw (y\n) to[bend right] (x\n);
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}
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\end{scope}
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\end{tikzpicture}\friezesep\hskip1pt
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%
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\begin{tikzpicture}[baseline=(desc.base)]%2\star\infty
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\node (desc) at (0.35,9.5) {${2}{\star}{\infty}$};
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\foreach \n in {1,...,4} {
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\pic at (0,0.05+2*\n) {tendril={1.2}{1.2}{-20}{black!10}};
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\pic at (0,-.65+2*\n) {tendril={-1.2}{1.2}{160}{black!10}};
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\node[dot] (a\n) at (0,.05+2*\n) {};
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\node[dot] (b\n) at (0,-.65+2*\n) {};
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}
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\foreach \n in {1,...,3} {
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\pic at (.7,1.05+2*\n) {tendril={-1.2}{1.2}{-20}{black!10}};
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\pic at (.7,.35+2*\n) {tendril={1.2}{1.2}{160}{black!10}};
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\node[dot] (c\n) at (.7,1.05+2*\n) {};
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\node[dot] (d\n) at (.7,.35+2*\n) {};
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}
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\node[inner sep=0pt] (d4) at (.7,8.35) {\tvdots};
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\node[inner sep=0pt] (c0) at (.7,1.05) {\tvdots};
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\foreach \n in {1,2,3,4} {
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\draw[dblgena] (a\n) -- (b\n);
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}
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\foreach \n in {1,2,3} {
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\draw[dblgena] (c\n) -- (d\n);
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}
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\begin{scope}[genb]
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\foreach \n in {1,2,3,4} {
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\draw (a\n) to[bend right] (d\n);
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\draw (d\n) to[bend right] (a\n);
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}
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\foreach \b/\c in {1/0,2/1,3/2,4/3} {
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\draw (b\b) to[bend right] (c\c);
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\draw (c\c) to[bend right] (b\b);
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}
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\end{scope}
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\end{tikzpicture}\hskip1pt\friezesep
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%
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\begin{tikzpicture}[baseline=(desc.base)]%\star\infty\infty
560+
\node (desc) at (0,9) {${\star}{\infty}{\infty}$};
561+
\foreach \n in {1,...,4} {
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\pic at (0,.35+1.7*\n) {tendril={1.5}{1.5}{-20}{black!10}};
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\pic at (0,-.5+1.7*\n) {tendril={-1.5}{1.5}{160}{black!10}};
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\node[dot] (a\n) at (0,.35+1.7*\n) {};
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\node[dot] (b\n) at (0,-.5+1.7*\n) {};
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}
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\node[inner sep=0pt] (b5) at (0,8) {\tvdots};
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\node[inner sep=0pt] (a0) at (0,.35) {\tvdots};
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\foreach \n in {1,2,3,4} {
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\draw[dblgena] (a\n) -- (b\n);
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}
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\foreach \p/\n in {0/1,1/2,2/3,3/4,4/5} {
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\draw[dblgenb] (a\p) -- (b\n);
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}
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\end{tikzpicture}\friezesep
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%
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\begin{tikzpicture}[baseline=(desc.base)]%\infty\star
578+
\node (desc) at (0.5,8.5) {${\infty}{\star}$};
579+
\foreach \n in {1,...,6} {
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\pic at (0,\n) {tendril={1.5}{1.5}{70}{black!10}};
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\pic at (1,\n) {tendril={-1.5}{1.5}{70}{black!10}};
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\node[dot] (a\n) at (0,\n) {};
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\node[dot] (b\n) at (1,\n) {};
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}
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\node[inner sep=0pt] (a0) at (0,0) {\tvdots};
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\node[inner sep=0pt] (a7) at (0,7) {\tvdots};
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\node[inner sep=0pt] (b0) at (1,0) {\tvdots};
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\node[inner sep=0pt] (b7) at (1,7) {\tvdots};
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\foreach \p/\n in {0/1,1/2,2/3,3/4,4/5,5/6,6/7} {
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\draw[gena] (a\p) -- (a\n);
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\draw[gena] (b\p) -- (b\n);
592+
}
593+
\foreach \n in {1,...,6} {
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\draw[dblgenc] (a\n) -- (b\n);
595+
}
596+
\end{tikzpicture}\friezesep
597+
\begin{tikzpicture}[baseline=(desc.base)]%*22\infty
598+
\node (desc) at (0.5,9) {${\star}{2}{2}{\infty}$};
599+
\foreach \n in {1,...,4} {
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\pic at (0,.35+1.7*\n) {tendril={1.5}{1.5}{-20}{black!10}};
601+
\pic at (0,-.5+1.7*\n) {tendril={-1.5}{1.5}{160}{black!10}};
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\pic at (1,.35+1.7*\n) {tendril={1.5}{-1.5}{160}{black!10}};
603+
\pic at (1,-.5+1.7*\n) {tendril={1.5}{1.5}{160}{black!10}};
604+
\node[dot] (a\n) at (0,.35+1.7*\n) {};
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\node[dot] (b\n) at (0,-.5+1.7*\n) {};
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\node[dot] (c\n) at (1,.35+1.7*\n) {};
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\node[dot] (d\n) at (1,-.5+1.7*\n) {};
608+
}
609+
\node[inner sep=0pt] (b5) at (0,8) {\tvdots};
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\node[inner sep=0pt] (a0) at (0,.35) {\tvdots};
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\node[inner sep=0pt] (d5) at (1,8) {\tvdots};
612+
\node[inner sep=0pt] (c0) at (1,.35) {\tvdots};
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\foreach \p/\n in {0/1,1/2,2/3,3/4,4/5} {
614+
\draw[dblgenb] (a\p) -- (b\n);
615+
\draw[dblgenb] (c\p) -- (d\n);
616+
}
617+
\foreach \n in {1,2,3,4} {
618+
\draw[dblgenc] (b\n) -- (d\n);
619+
\draw[dblgenc] (a\n) -- (c\n);
620+
\draw[dblgena] (a\n) -- (b\n);
621+
\draw[dblgena] (c\n) -- (d\n);
622+
}
623+
\end{tikzpicture}
624+
\end{sidecaption}
625+
\end{figure}
437626

438627
\section{Incidence geometries and the Levi graph}
439628

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