如何在数学模式下绘制正多边形

Question

我有一个级数展开式，其中每个项都可以与一个正多边形相关联。我想在我正在写的一篇论文中指出这一点，方法是包括一个将图表视为数学表达式中的项的方程。下面显示了我脑海中的一个例子。

我很难弄清楚如何做到这一点，尤其是多边形的比例和居中。

如果有一个能够被 arxiv 编译器接受的解决方案就更好了。

Answer 1

欢迎光临！Ti钾Z 在库中内置了这些shapes.geometric。

\documentclass[fleqn]{article}
\usepackage{mathtools}
\usepackage{tikz}
\usetikzlibrary{shapes.geometric}
\newcommand{\Ngon}[2][]{\vcenter{\hbox{\begin{tikzpicture}
\node[regular polygon,regular polygon sides=#2,draw,minimum size=1cm,#1](#2-gon){};
\foreach \X in {1,...,#2}{\fill (#2-gon.corner \X) circle[radius=2pt];}
\end{tikzpicture}}}}
\begin{document}
\[
 \mathcal{Z}_X=\sum_{n=3}^\infty t_n
 =\Ngon{3}+\Ngon{4}+\Ngon{5}+\cdots
\]
or
\begin{align*}
 \mathcal{Z}_X&=\sum_{n=3}^\infty t_n\\
 &=\Ngon{3}+\Ngon{4}+\Ngon{5}+\cdots
\end{align*}
\end{document}

或者

\documentclass[fleqn]{article}
\usepackage{mathtools}
\usepackage{tikz}
\usetikzlibrary{shapes.geometric}
\newcommand{\Ngon}[2][]{\vcenter{\hbox{\begin{tikzpicture}
\node[regular polygon,regular polygon sides=#2,draw,minimum size=1cm,#1](#2-gon){};
\foreach \X in {1,...,#2}{\fill (#2-gon.corner \X) circle[radius=2pt];}
\end{tikzpicture}}}}
\begin{document}
\[
 \mathcal{Z}_X=\sum_{n=3}^\infty t_n
 =\foreach \Y in {3,4,5}
 {\unless\ifnum\Y=3 +\fi
 \Ngon{\Y}
 }+\cdots
\]
or
\begin{align*}
 \mathcal{Z}_X&=\sum_{n=3}^\infty t_n\\
 &=\foreach \Y in {3,4,5}
 {\unless\ifnum\Y=3 +\fi
 \Ngon{\Y}
 }+\cdots
\end{align*}
\end{document}

或者

\documentclass[fleqn]{article}
\usepackage{mathtools}
\usepackage{tikz}
\usetikzlibrary{shapes.geometric}
\newcommand{\Ngon}[2][]{\vcenter{\hbox{\begin{tikzpicture}
\node[regular polygon,regular polygon sides=#2,draw,minimum size=1cm,#1](#2-gon){};
\draw plot[only marks,mark=*,samples at={1,...,#2}] (#2-gon.corner \x);
\end{tikzpicture}}}}
\begin{document}
\[
 \mathcal{Z}_X=\sum_{n=3}^\infty t_n
 =\foreach \Y in {3,4,5}
 {\unless\ifnum\Y=3 +\fi
 \Ngon{\Y}
 }+\cdots
\]
or
\begin{align*}
 \mathcal{Z}_X&=\sum_{n=3}^\infty t_n\\
 &=\foreach \Y in {3,4,5}
 {\unless\ifnum\Y=3 +\fi
 \Ngon{\Y}
 }+\cdots
\end{align*}
\end{document}

虽然我没有明确测试过，但我很确定这在 arXiv 上可以工作。我已经在那里成功提交了几篇包含 TikZ 代码的论文。只有当你使用非常新的库时它才可能不起作用，但它shapes.geometric相当老旧。

Answer 2

我的这个答案可以很容易地适应您的情况，借助该curve2e包可以在环境中使用极坐标picture。当在一阶和高阶上标或下标中使用时，符号会自动按适当比例缩放，并遵循bold数学版本。

这是一个可编译的示例。有关详细信息和解释，请参阅引用的答案。

% My standard header for TeX.SX answers:
\documentclass[a4paper]{article} % To avoid confusion, let us explicitly 
                                 % declare the paper format.

\usepackage[T1]{fontenc}         % Not always necessary, but recommended.
% End of standard header.  What follows pertains to the problem at hand.

\usepackage{halloweenmath} % also loads "amsmath" and "pict2e"
\usepackage{curve2e} % for polar coordinate in "picture" environments


%== \BEGIN { WIZARDRY } ========================================
\makeatletter

\@ifdefinable\SuCmathpictvertex{} % check that this name can be used
\@ifdefinable\@SuC@reserved@dimen{\newdimen\@SuC@reserved@dimen}

% The code that implements the following environment could also be 
% in-lined inside the definition of "\@SuC@general@pict"; however, 
% providing an additional layer might prove helpful in the future.
\newenvironment*{@SuC@math@picture}[8]{%
  % #1 := height of the symbol, in picture coordinate units
  % #2 := depth of the symbol, in picture coordinate units
  % #3 := width of the symbol, in picture coordinate units
  % #4 := x-coordinate of leftmost point of the symbol
  % #5 := font selector (e.g., "\textfont")
  % #6 := diameter of the vertices
  % #7 := width of sidebearings, in picture coordinate units
  % #8 := thickness of the edges
  \def\SuCmathpictvertex{\circle*{#6}}%
  \setlength\unitlength{\fontdimen 22 #5\tw@}%
  \setlength\@SuC@reserved@dimen{#7\unitlength}%
  \kern\@SuC@reserved@dimen
  \@HwM@d@pict@strut{#2}%
  \picture(#3,#1)(#4,-1)%
    \roundcap
    \roundjoin
    \linethickness{#8\@HwM@thickness@units@for #5}%
}{%
  \endpicture
  \kern\@SuC@reserved@dimen
}
\newcommand*\@SuC@general@pict[9]{%
  % #1 := code that draws the tree
  % #2 := height of the symbol, in picture coordinate units
  % #3 := depth of the symbol, in picture coordinate units
  % #4 := width of the symbol, in picture coordinate units
  % #5 := x-coordinate of leftmost point of the symbol
  % #6 := font selector (e.g., "\textfont")
  % #7 := diameter of the vertices
  % #8 := width of sidebearings, in picture coordinate units
  % #9 := thickness of the edges
  \begin{@SuC@math@picture}%
            {#2}{#3}% height / depth
            {#4}{#5}% width / min-x
            #6% font selector (e.g., "\textfont")
            {#7}% diameter of the vertices
            {#8}% width of sidebearings
            {#9}% thickness of the edges
    #1%
  \end{@SuC@math@picture}%
}
\newcommand*\@SuC@math@version@shunt[7]{%
  % #1 := code that draws the tree
  % #2 := height of the symbol, in picture coordinate units
  % #3 := depth of the symbol, in picture coordinate units
  % #4 := width of the symbol, in picture coordinate units
  % #5 := x-coordinate of leftmost point of the symbol
  % #6 := style selector (e.g., "\displaystyle"), here unused
  % #7 := font selector (e.g., "\textfont")
  \@HwM@choose@thicknesses{\@SuC@general@pict {#1}{#2}{#3}{#4}{#5}#7}%
      % Two lists of parameters follow, each in the form
      %
      %   {{<diam.>}{<sidebearing>}{<line thickn.>}}
      %
      %  where:
      %    <diam.> is the diameter of the vertices, in picture
      %        coordinate units ("\unitlength");
      %    <sidebearing> is the width of an extra space that goes
      %        at both sides of the symbol, in the same units as above;
      %    <line thickn.> is the thickness of the edges, in units of
      %        the default-rule-thickness "\fontdimen" (note that "{}"
      %        can be used to mean "{1}").
      % The second list is used when the "bold" math version is in
      % force, the first pair in all other cases (usually, this means
      % that the "normal" math version is in force).
      {{.8}{.4}{}}% for "normal" (actually, non-"bold") math version
      {{1}{.5}{1.5}}% for "bold" math version
}

% The user-level command:
\newcommand*\DeclareNewSuCMathPict[6]{%
  % #1 := control sequence being defined (e.g., "\SuCTreeOne")
  % #2 := height of the symbol, in picture coordinate units
  % #3 := depth of the symbol, in picture coordinate units
  % #4 := width of the symbol, in picture coordinate units
  % #5 := x-coordinate of leftmost point of the symbol
  % #6 := code that draws the tree
  \newcommand*{#1}{%
    \@HwM@general@ordinary@symbol
      {\@SuC@math@version@shunt {#6}{#2}{#3}{#4}{#5}}%
  }%
}

\makeatother
%== \END { WIZARDRY } ==========================================


% Now we use our new declaration:

\DeclareNewSuCMathPict{\trigon}
            {3}{1}  % height / depth
            {4}{-2} % width  / min-x
{% code that draws the picture
    \polygon(90:2)(210:2)(330:2)%
    \put (90:2){\SuCmathpictvertex}%
    \put(210:2){\SuCmathpictvertex}%
    \put(330:2){\SuCmathpictvertex}%
}

\DeclareNewSuCMathPict{\tetragon}
            {3}{1}  % height / depth
            {4}{-2} % width  / min-x
{% code that draws the picture
    \polygon(45:2)(135:2)(225:2)(315:2)%
    \put (45:2){\SuCmathpictvertex}%
    \put(135:2){\SuCmathpictvertex}%
    \put(225:2){\SuCmathpictvertex}%
    \put(315:2){\SuCmathpictvertex}%
}

\DeclareNewSuCMathPict{\pentagon}
            {3}{1}  % height / depth
            {4}{-2} % width  / min-x
{% code that draws the picture
    \polygon(90:2)(162:2)(234:2)(306:2)(378:2)%
    \put (90:2){\SuCmathpictvertex}%
    \put(162:2){\SuCmathpictvertex}%
    \put(234:2){\SuCmathpictvertex}%
    \put(306:2){\SuCmathpictvertex}%
    \put(378:2){\SuCmathpictvertex}%
}

\DeclareNewSuCMathPict{\hexagon}
            {3}{1}  % height / depth
            {4}{-2} % width  / min-x
{% code that draws the picture
    \polygon(0:2)(60:2)(120:2)(180:2)(240:2)(300:2)%
    \put  (0:2){\SuCmathpictvertex}%
    \put (60:2){\SuCmathpictvertex}%
    \put(120:2){\SuCmathpictvertex}%
    \put(180:2){\SuCmathpictvertex}%
    \put(240:2){\SuCmathpictvertex}%
    \put(300:2){\SuCmathpictvertex}%
}



\begin{document}

The polygons:
\( \trigon + \tetragon + \pentagon + \hexagon \).

\begingroup \bfseries \boldmath
Bold math:
\( \trigon + \tetragon + \pentagon + \hexagon \).
\endgroup

Note that both the vertices and the edges get thicker.

A formula with subscripts, etc.; compare the diameter of the poygons 
and the height-plus-depth of the parentheses.
\begin{gather*}
    f(\trigon) + g(\tetragon) + h(\hexagon) = 0 \\
    \sum_{i=\trigon}^{\hexagon} H_{i}(x)
        = H_{\trigon}(x) + H_{\tetragon}(x)
            + H_{\pentagon}(x) + H_{\hexagon}(x)
        = \varOmega(\trigon * \tetragon * \pentagon * \hexagon)
\end{gather*}
And here is an example with a double script:
\( \mathit{T}^{\mathit{S}_{\hexagon}} \).

\end{document}

这是输出：

请注意，在此示例中，为简单起见，所有四个多边形都被视为具有与其外接圆相同的宽度、高度和深度：这可能并不完全合适。特别是，您可能需要微调 的第四和第五个参数中的值\DeclareNewSuCMathPict。