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方形旋转和反射对称

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方形旋转和反射对称

刚开始学习群论，希望能画出正多边形的旋转和反射对称。从网站上获取了以下代码，无法将其拼凑起来得到我需要的东西，尤其是开放的弯曲箭头。researchgate.net 有等边三角形的对称性，但受版权保护。如果有人可以在 Tikz 中将它们制作出来供大家使用，那就太好了。如果没有制作所有多边形的代码，只需一个正方形的代码就好了。

% !TeX spellcheck = en_GB
\documentclass{article}
\usepackage{amssymb}
\usepackage{tikz}
\usetikzlibrary{shapes.geometric, calc}
\usetikzlibrary{shapes.arrows, fadings}

\begin{document}
    \begin{tikzpicture}[scale=3 ]
    \def\rps{4} % regular polygon sides
    \node (a) 
    [draw,  blue!0!black,rotate=0,minimum size=3cm,regular polygon, fill=blue!25,  regular polygon sides=\rps ] at (0, 0) {}; 
    
    
    \foreach \x in {1,2,...,\rps}{
        \draw [black,dashed, shorten <=-0.75cm,shorten >=-0.75cm](a.center) -- (a.side \x);
        \draw [black,dashed, shorten <=-0.75cm,shorten >=-0.75cm](a.center) -- (a.corner \x);}
    
    \end{tikzpicture}
    
    
    
\begin{tikzpicture}[every node/.append style={ draw=white, left color=white, double arrow}]
\node [
right color=blue,
double arrow,
minimum height=1.5cm
] {};

\end{tikzpicture}   
    
    
\end{document}

方形旋转反射对称性

答案1

使用arrows.meta庫。 在此处输入图片描述

\documentclass[tikz, border=1cm]{standalone}
\usetikzlibrary{arrows.meta}
\makeatletter
\tikzset{
  a/.style={
    double=green!10,
    double distance=2pt,
    draw=teal!50,
  },
  foo /.tip={Stealth[inset=0pt, length=4pt, width=6pt, fill=green!10]},
  pics/arc arrow/.style args={#1:#2:#3}{
    code={
      \draw[a, -foo] (0, 0) arc (#1:#2:#3) coordinate (arc@temp);
      \path (arc@temp) ++(270+#2:4pt) ++(#2:3pt) coordinate (-a);
    }
  },
  pics/straight arrow/.style={
    code={
      \begin{scope}[rotate=#1]
        \draw[a, foo-foo] (0, -0.5) -- (0, 0.5);
        \fill[green!10] (-1pt, 0.5cm-4.1pt) rectangle (1pt, 0.5cm-3.9pt+\pgflinewidth);
        \fill[green!10] (1pt, -0.5cm+4.1pt) rectangle (-1pt, -0.5cm+3.9pt-\pgflinewidth);
        \coordinate (-a) at (3pt, -0.5cm + 4pt);
      \end{scope}
    }
  },
  mydash/.style={dash pattern=on 2mm off 0.5mm}
}
\makeatother

\begin{document}
\begin{tikzpicture}
  \path[draw=blue!80, fill=purple!30] (-2, -2) rectangle (2, 2);
  \foreach \i in {0, 45, ..., 315} {
    \draw[mydash] (0, 0) -- (\i:4);
  }
  \foreach \r/\t/\p [count=\i] in {90/r/above right, 135/s/above left, 180/t/left, 225/u/below left} {
    \pic (arrow\i) at (\r:3.4cm) {straight arrow=\r};
    \node[\p, inner sep=1pt] at (arrow\i-a) {$\t$};
  }
  \foreach \x/\r/\t/\p [count=\i] in {0.4/360/c/right, 0.8/270/b/below left, 1.2/180/a/above left} {
    \pic (arc\i) at (0, \x) {arc arrow=90:\r:\x cm};
    \node[\p, inner sep=1pt] at (arc\i-a) {$\t$};
  }
\end{tikzpicture}
\end{document}

答案2

这是一个\poly显示正多边形任意边数的旋转的命令。

bending library标记最大角度旋转的内箭头（仅适用于五边形）存在一个小问题，我还没有解决，因为和之间似乎不兼容double arrows。

更新：箭头从 12 点开始

截屏

\documentclass{article}
\usepackage{tikz}
%\usetikzlibrary {bending}
\usetikzlibrary{math,calc}
\usetikzlibrary{shapes.geometric,shapes.arrows,arrows.meta}
\usepackage{alphalph}

\newcommand\poly[1]{
    \begin{tikzpicture}[auto=right,>={Triangle[open,fill=white,length=4pt,angle=60:1pt 2,inset=0pt]}]
     % regular polygon sides
    \def\rps{#1}
    % rayon du cercle circonscrit
    \def\rayonext{2.5cm}
    % couleur de fond du polygone
    \def\couleur{blue!25}
    % nombre de rotations
    \pgfmathsetmacro\nbrotation{int(\rps-1)}
    % rayon du cercle inscrit
    \pgfmathsetmacro\rayonint{multiply(\rayonext,cos(180/\rps))}
   
%    % definition de l'angle de départ des flèches des rotations
    \pgfmathsetmacro\startarrow{90}

    \node (a) 
    [draw,blue!0!black,rotate=0,minimum size=2*\rayonext,regular polygon,fill=\couleur,regular polygon sides=\rps] at (0, 0) {}; 

    \foreach \x in {1,...,\rps}{
        \draw [black,dashed, shorten >=-1.5cm,shorten <=0cm](a.center) -- (a.side \x);
        \draw [black,dashed, shorten >=-0.75cm,shorten <=0cm](a.center) -- (a.corner \x);
        }
    
\foreach \i in {1,...,\rps}{
        \draw[<->,double distance=1.4pt] ({\startarrow+(\i-1)*180/\rps}:\rayonext+5mm) ++([rotate=\startarrow+90+(\i-1)*180/\rps]-4mm,0) -- ++([rotate=\startarrow+90+(\i-1)*180/\rps]8mm,0)node[pos=.02,auto=right,outer sep=2pt,inner sep=1pt,]{\alphalph{\i+17}};
    }

\foreach \i [evaluate=\i as \ratio using \i/\rps] in {1,...,\nbrotation}{
         \draw [->,double distance=1.4pt]               
          ($(a.center)!\rayonint pt!(a.90)!\ratio!(a.center)$)arc[start angle=\startarrow,delta angle=360*\ratio,radius={\rayonint pt*(1-\ratio)}]
         node[pos=.98,auto=right,outer sep=2pt,inner sep=0pt,node font=small,font=\bf,fill=\couleur,text=white]{\alphalph{\i}};
    }
\end{tikzpicture}
}
    
 \begin{document}   
\poly{3}
\qquad
\poly{4}
\bigskip

\bigskip

\poly{5}\qquad
\poly{6}
\end{document}

旧答案：箭头从线段平分线开始

截屏

\documentclass{article}
\usepackage{tikz}
%\usetikzlibrary {bending}
\usetikzlibrary{math,calc}
\usetikzlibrary{shapes.geometric,shapes.arrows,arrows.meta}
\usepackage{alphalph}

\newcommand\poly[1]{
    \begin{tikzpicture}[auto=right,>={Triangle[open,fill=white,length=4pt,angle=60:1pt 2,inset=0pt]}]
     % regular polygon sides
    \def\rps{#1}
    % rayon du cercle circonscrit
    \def\rayonext{2.5cm}
    % couleur de fond du polygone
    \def\couleur{blue!25}
    % nombre de rotations
    \pgfmathsetmacro\nbrotation{int(\rps-1)}
    % rayon du cercle inscrit
    \pgfmathsetmacro\rayonint{multiply(\rayonext,cos(180/\rps))}
   
    % definition de l'angle de départ des flèches des rotations
    \pgfmathparse{iseven(\rps)} \pgfmathresult
    \ifnum \pgfmathresult=1
    \pgfmathsetmacro\startarrow{90}
    \else
    \pgfmathsetmacro\startarrow{90+180/\rps}
    \fi

    \node (a) 
    [draw,blue!0!black,rotate=0,minimum size=2*\rayonext,regular polygon,fill=\couleur,regular polygon sides=\rps] at (0, 0) {}; 

    \foreach \x in {1,...,\rps}{
        \draw [black,dashed, shorten >=-1.5cm,shorten <=0cm](a.center) -- (a.side \x);
        \draw [black,dashed, shorten >=-0.75cm,shorten <=0cm](a.center) -- (a.corner \x);
        }
    
\foreach \i in {1,...,\rps}{
        \draw[<->,double distance=1.4pt] ({\startarrow+(\i-1)*180/\rps}:\rayonext+5mm) ++([rotate=\startarrow+90+(\i-1)*180/\rps]-4mm,0) -- ++([rotate=\startarrow+90+(\i-1)*180/\rps]8mm,0)node[pos=.02,auto=right,outer sep=2pt,inner sep=1pt,]{\alphalph{\i+17}};
    }

\foreach \i [evaluate=\i as \ratio using \i/\rps] in {1,...,\nbrotation}{
         \draw [->,double distance=1.4pt]               
         ($(a.side 1)!\ratio!(a.center)$)arc[start angle=\startarrow,delta angle=360*\ratio,radius={\rayonint pt*(1-\ratio)}]
         node[pos=.98,auto=right,outer sep=2pt,inner sep=0pt,node font=small,font=\bf,fill=\couleur,text=white]{\alphalph{\i}};
    }
\end{tikzpicture}
}
    
 \begin{document}   
\poly{3}\qquad
\poly{4}
\bigskip

\bigskip

\poly{5}\qquad
\poly{6}
\end{document}

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