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如何绘制三个相交的黄金矩形来创建二十面体的顶点？

tag-icon • tag-icon tikz-pgf 3d tikz-3d
如何绘制三个相交的黄金矩形来创建二十面体的顶点？

由 3 个黄金矩形组成的二十面体

如何在 Tikz 中创建如图所示的三个相交的黄金矩形？

答案1

钛钾Z 没有 3D 引擎，所以我们需要按正确的顺序绘制元素。对于飞机的情况，已经开发了一些工具，我在这里使用。你需要3dtools 库来编译这个。

\documentclass[tikz,border=3mm]{standalone}
\usepackage{tikz-3dplot}
\usetikzlibrary{3dtools}
\begin{document}
% define a command for the projections (to become a part of the library)
\makeatletter
\newcommand\GetProj[2]{\begingroup
  \pgfutil@tempcnta=0%
  \edef\pgfutil@tmpa{0}%
  \pgfutil@for\my@item:={#1}\do{%
    \pgfmathsetmacro\mycoord{TD("\my@item")}%
    \advance\pgfutil@tempcnta by1\relax
    \pgfmathsetmacro{\pgfutil@tmpa}{\pgfutil@tmpa-screendepth(\mycoord)}%
  }%
  \pgfmathsetmacro{#2}{\pgfutil@tmpa/\pgfutil@tempcnta}%
  \pgfmathsmuggle#2\endgroup}
\makeatother  
% counter for 3d ordering  
\newcounter{tdorder}
% set a sufficient number of layers (for more complex scenarios you need more)
\foreach \X in {1,...,40}
{\pgfdeclarelayer{layer\X}
\ifnum\X=1
\xdef\LstLayers{layer\X}
\else
\xdef\LstLayers{\LstLayers,layer\X}
\fi}
\pgfsetlayers\LstLayers
\tikzset{closed polygon/.style={insert path={foreach \Coord [count=\nCoord] in {#1}
 {\ifnum\nCoord=1
  \Coord
 \else
  -- \Coord
 \fi} -- cycle}},polygon/.style={insert path={foreach \Coord [count=\nCoord] in {#1}
 {\ifnum\nCoord=1
  \Coord
 \else
  -- \Coord
 \fi}}},
 vertex poly/.style={insert path={foreach \Coord [count=\nCoord] in {#1}
 {\ifnum\nCoord=1
  \Coord node[vertex,vname=\Coord]{}
 \else
  -- \Coord node[vertex,vname=\Coord]{}
 \fi}}},vertex/.style={circle,draw,fill=white,inner sep=2pt},
 vname/.code={\def\vnameadd(##1){\edef\myvname{v-##1}}%
 \expandafter\vnameadd#1\relax
 \tikzset{alias=\myvname}%
 }}
% main code
\foreach \X in {5,15,...,355}
{\tdplotsetmaincoords{90+30*sin(\X)}{\X}
\begin{tikzpicture}[tdplot_main_coords,scale=pi,
    font=\sffamily,fill opacity=1,
    declare function={phi=(1+sqrt(5))/2;},
    line join=round,line cap=round]
  \path[tdplot_screen_coords,use as bounding box] (-2,-2) rectangle (2,2);
  % define the vertices (there are certainly superior naming conventions)
  \path (0,0,0) coordinate (O)
     (phi/2,1/2,0)  coordinate (xypp) (phi/2,0,0)  coordinate (xyp0)
     (phi/2,-1/2,0)  coordinate (xypm) (0,1/2,0)  coordinate (xy0p)
     (-phi/2,-1/2,0)  coordinate (xymm) (-phi/2,0,0)  coordinate (xym0)
     (-phi/2,1/2,0)  coordinate (xymp) (0,-1/2,0)  coordinate (xy0m)
     (0,phi/2,1/2)  coordinate (yzpp) (0,phi/2,0) coordinate (yzp0) 
     (0,phi/2,-1/2)  coordinate (yzpm) (0,0,1/2) coordinate (yz0p) 
     (0,-phi/2,-1/2)  coordinate (yzmm) (0,-phi/2,0) coordinate (yzm0) 
     (0,-phi/2,1/2)  coordinate (yzmp) (0,0,-1/2) coordinate (yz0m) 
     (1/2,0,phi/2)  coordinate (xzpp) (0,0,phi/2)  coordinate (xzp0) 
     (-1/2,0,phi/2)  coordinate (xzpm) (1/2,0,0)  coordinate (xz0p) 
     (-1/2,0,-phi/2)  coordinate (xzmm) (0,0,-phi/2)  coordinate (xzm0) 
     (1/2,0,-phi/2)  coordinate (xzmp) (-1/2,0,0)  coordinate (xz0m) ;
  % define the plane data as a list of <drawing options>/<shape>/<vertices> 
  \def\PlaneData{%
  {draw=none,fill=red}/closed polygon/{(xypp),(xyp0),(O),(xy0p)},%
  {draw=none,fill=red}/closed polygon/{(xypm),(xyp0),(O),(xy0m)},%
  {draw=none,fill=red}/closed polygon/{(xymp),(xym0),(O),(xy0p)},%
  {draw=none,fill=red}/closed polygon/{(xymm),(xym0),(O),(xy0m)},%
  {draw=none,fill=blue}/closed polygon/{(yzpp),(yzp0),(O),(yz0p)},%
  {draw=none,fill=blue}/closed polygon/{(yzpm),(yzp0),(O),(yz0m)},%
  {draw=none,fill=blue}/closed polygon/{(yzmp),(yzm0),(O),(yz0p)},%
  {draw=none,fill=blue}/closed polygon/{(yzmm),(yzm0),(O),(yz0m)},%
  {draw=none,fill=green!70!black}/closed polygon/{(xzpp),(xzp0),(O),(xz0p)},%
  {draw=none,fill=green!70!black}/closed polygon/{(xzpm),(xzp0),(O),(xz0m)},%
  {draw=none,fill=green!70!black}/closed polygon/{(xzmp),(xzm0),(O),(xz0p)},%
  {draw=none,fill=green!70!black}/closed polygon/{(xzmm),(xzm0),(O),(xz0m)},%
  {draw,thick}/vertex poly/{(xypp),(xzpp)},%
  {draw,thick}/vertex poly/{(xypm),(xzpp)},%
  {draw,thick}/vertex poly/{(xypp),(yzpp)},%
  {draw,thick}/vertex poly/{(xymp),(yzpp)},%
  {draw,thick}/vertex poly/{(xypm),(yzmp)},%
  {draw,thick}/vertex poly/{(xymm),(yzmp)},%
  {draw,thick}/vertex poly/{(xymp),(xzpm)},%
  {draw,thick}/vertex poly/{(yzmp),(xzpm)},%
  {draw,thick}/vertex poly/{(yzpp),(xzpm)},%
  {draw,thick}/vertex poly/{(yzmp),(xzpp)},%
  {draw,thick}/vertex poly/{(yzpp),(xzpp)},%
  {draw,thick}/vertex poly/{(yzmm),(xzmm)},%
  {draw,thick}/vertex poly/{(yzpm),(xzmm)},%
  {draw,thick}/vertex poly/{(yzmm),(xzmp)},%
  {draw,thick}/vertex poly/{(yzpm),(xzmp)},%
  {draw,thick}/vertex poly/{(xymm),(xzmm)},%
  {draw,thick}/vertex poly/{(xzpp),(xzpm)},%
  {draw,thick}/vertex poly/{(xypp),(xzmp)},%
  {draw,thick}/vertex poly/{(xypm),(xzmp)},%
  {draw,thick}/vertex poly/{(xypp),(yzpm)},%
  {draw,thick}/vertex poly/{(xymp),(yzpm)},%
  {draw,thick}/vertex poly/{(xypm),(yzmm)},%
  {draw,thick}/vertex poly/{(xymm),(yzmm)},%
  {draw,thick}/vertex poly/{(xymm),(xzmm)},%
  {draw,thick}/vertex poly/{(xymp),(xzmm)}}
  % normal of screen (last row of the rotation matrix)
  \path[overlay] ({sin(\tdplotmaintheta)*sin(\tdplotmainphi)},
       {-1*sin(\tdplotmaintheta)*cos(\tdplotmainphi)},
       {cos(\tdplotmaintheta)}) coordinate (n); 
  % build up the list of projections       
  \foreach \Style/\Poly/\CoordLst [count=\nC] in \PlaneData
  {%
  \GetProj{\CoordLst}{\currproj}
  \ifnum\nC=1
    \xdef\LstProj{\currproj}
  \else
    \xdef\LstProj{\LstProj,\currproj}
  \fi}
  % draw the planes in appropriate layers
  \foreach \Style/\Poly/\CoordLst [count=\nC] in \PlaneData
  {%
  \GetProj{\CoordLst}{\currproj}
  \setcounter{tdorder}{1}
  \foreach \Proj in \LstProj
  {\pgfmathtruncatemacro{\itest}{ifthenelse(\Proj<\currproj,1,0)}
  \ifnum\itest=1
  \stepcounter{tdorder}
  \fi}
  \begin{pgfonlayer}{layer\number\value{tdorder}}
  \edef\temp{\noexpand\path[\Style,\Poly={\CoordLst}];}
  \temp
  \end{pgfonlayer}
  }
\end{tikzpicture}}
\end{document}

在此处输入图片描述

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