Golden spiral in TiKZ: how do I get the right shape and background?

Question

I want to use tikz to draw a golden spiral, growing clockwise, ideally (but not necessarily) superimposed over a set of golden rectangles (you all know the image I'm talking about). A little bit of searching led me to this code, which yields a counterclockwise spiral:

\documentclass[tikz]{standalone}
\usetikzlibrary{decorations.markings,calc}
\tikzset{nctopath/.style={
     to path=(\tikztostart) ..controls ($(\tikztostart)!1cm*#1!-90:(\tikztotarget)$) and 
        ($(\tikztotarget)!($(\tikztostart)!1cm*#1!-50:(\tikztotarget)$)!70:    (\tikztostart)$).. 
    (\tikztotarget)
    },
}
\begin{document}    
\begin{tikzpicture}
\def\totx{1}
\coordinate (n-1-1) at (0,0) {};
\foreach \x[count=\xi from 2, evaluate=\x as \temptotx using int(\x+\totx)] in {1,...,9}{
\draw[decoration={
    markings,mark=between positions 0 and 1 step 0.249 with {
            \coordinate (n-\x-\pgfkeysvalueof{/pgf/decoration/mark info/sequence number})     ;
        }
   },postaction=decorate
] 
(n-\x-1) arc (\x*90+180:(\x+1)*90+180:{(\temptotx)*3mm}) 
coordinate (n-\xi-1) 
\pgfextra{\xdef\totx{\temptotx}};
}

\end{tikzpicture}
\end{document}

The thing is that I'm still new to tikz and thus not sure what code here does what. My questions are:

1. How do I reverse the growth direction?

2. How do I set phi as the growth factor (which I don't think it is currently, just from looking at the resulting spiral)?

3. How cumbersome would it be to add in the golden rectangles (i.e., would I have to draw each one individually, or is there some way to define a pattern)?

Answer 1

Basically, the same approach as JLDiaz but using a recursive macro and letting tikz scopes and transformations do all the work.

\documentclass[tikz, border=0.125cm]{standalone}
\begin{document}

\def\spiral#1{%
  \pgfmathparse{int(#1)}%
  \ifnum\pgfmathresult>0
    \draw [help lines] (0,0) rectangle ++(1,1);
    \begin{scope}[shift={(1,1)}, rotate=90, scale=1/1.6180339887]
      \spiral{#1-1}
    \end{scope}
    \draw [red] (0,0) arc (270:360:1);
  \fi
}

\tikz[scale=2]{\spiral{12}}

\end{document}

And perhaps...

\documentclass[border=0.125cm, tikz]{standalone}

\begin{document}

\def\spiral#1#2#3{%
  \pgfmathparse{int(#1)}%
  \ifnum\pgfmathresult>0
    \draw [help lines] (0,0) rectangle ++(1,1);
    \begin{scope}[shift={(1,1)}, rotate=#2, scale=1/#3]
      \spiral{#1-1}{#2}{#3}%
    \end{scope}
    \pgfmathparse{int(\a)}%
    \ifnum\pgfmathresult=0
      \draw [red] (0,0) -- (1,1);
    \else%
      \draw [red] (0,0) arc (270+45-#2/2:360-45+#2/2:{1/sqrt(2) / sin(#2/2)});
    \fi
  \fi
}

\foreach \a in {90,85,...,-90,-85,-80,...,85}{
 \tikz[scale=2]{\spiral{12}{\a}{1.6180339887}\useasboundingbox (-0.5,-0.5) rectangle (3.5,3.5);}
}

\end{document}

Answer 2

Code

Improved a little the code to make easier the explanations, and to put the "golden ratio rectangles" in a background layer. Explanations below the picture.

Updated after egreg's comments.

\documentclass{article}
\usepackage{tikz}
\begin{document}
\thispagestyle{empty}
\usetikzlibrary{calc}

\pgfdeclarelayer{background}
\pgfsetlayers{background,main}

\newcommand\GoldenRatio{1.6180339887}
\newcommand\Side{10}
\newcommand\Sqrtwo{1.4142135624}

\begin{tikzpicture}[line cap=round]
\coordinate (origin) at (0,0);
\foreach \angle in {0,90,...,1000} {
  \coordinate (corner) at ($(origin)+(45+\angle:\Side*\Sqrtwo)$);
  \begin{pgfonlayer}{background}
      \draw[help lines] (origin) rectangle (corner);
  \end{pgfonlayer}
  \draw[very thick,red] (origin) arc(\angle-90:\angle:\Side);
  \coordinate (origin) at (corner);
  \pgfmathsetmacro{\Side}{\Side/\GoldenRatio}
  \xdef\Side{\Side}
}
\end{tikzpicture}
\end{document}

Result

Explanations

The picture is built recursively, from a fundamental building block, which is a square containing a red arc.

The square is defined from two points. The first one is called origin, and initially is located at (0,0), but it will move in successive iterations of the loop. The second one is called corner, and its coordinates are dependent of those of origin in the following way:

  \coordinate (corner) at ($(origin)+(45+\angle:\Side*\Sqrtwo)$);

The expression +(45+\angle:\Side*\Sqrtwo) uses polar coordinates, relative to origin. The angle is 45 plus the "current angle" (initially zero), and the distance is the diagonal of the square whose side has \Side units of length.

Once origin and corner are known, the building block is drawn as (omitted in this case the background layer for clarity):

  \draw[help lines] (origin) rectangle (corner);
  \draw[very thick,red] (origin) arc(\angle-90:\angle:\Side);

In the first iteration, since \angle=0, \Side=10, this produces the following building block, whose side has 10 units of length:

Now, the neat trick is to update all the variables which define the building block, so that the next building block has:

• Its origin at the previous block corner
• Its \angle increased in 90 degrees
• Its \Side reduced in the golden ratio.

This is the role of the next lines:

  \coordinate (origin) at (corner);
  \pgfmathsetmacro{\Side}{\Side/\GoldenRatio}
  \xdef\Side{\Side}

The following figure shows the two first iterations, and thus the two first building blocks. I decided to draw also in blue the vector which places the "corner" relative to the origin, to better understand the procedure:

And so on, at each step the building block is reduced by the golden ratio, rotated 90 degrees, and shifted to the latest "corner".

Answer 3

Just for fun with PSTricks. No recursively-invoked macro is used here. Instead, the next node is determined by the previous one in a loop.

Linear complexity

\documentclass[pstricks,border=12pt]{standalone}
\usepackage{pst-node,multido}
\psset{dimen=monkey}

\pstVerb
{
    /R   5 def
    /P   {2 5 sqrt 1 add div} def
    /p   {1 P div} def
    /OmP {1 P sub} def
    /C   {2 sqrt mul 45 PtoC} def
}

\def\Atom#1{%
\begin{pspicture}(-4,-1)(6,6)
    \pnode(!p 1 sub R mul neg 0 ){A-1}
    \multido{\ic=0+1,\ip=-1+1}{#1}
    {%
        \nodexn{(A\ip)+(!\ic\space dup 1 sub P exch exp R mul OmP mul exch 90 mul PtoC)}{A\ic}
        \rput(A\ic){\rput{!90 \ic\space mul}(0,0){\psframe(!R P \ic\space exp mul C)}}
    }
    \pscustom[linecolor=red]{\multido{\ic=0+1}{#1}{\psarc(A\ic){!R P \ic\space exp mul}{!\ic\space 90 mul}{!\ic\space 90 mul 90 add}}}
\end{pspicture}}

\begin{document}
\multido{\i=0+1}{12}{\Atom{\i}}
\end{document}

Quadratic complexity

It is not recommended as creating the i-th node needs i iterations.

\documentclass[pstricks,border=12pt]{standalone}
\usepackage{pst-node,pst-plot}
\psset{dimen=monkey}

\pstVerb
{
    /R   5 def
    /P   {2 5 sqrt 1 add div} def
    /p   {1 P div} def
    /OmP {1 P sub} def
    /C   {2 sqrt mul 45 PtoC} def
}

\def\Atom#1{%
\begin{pspicture}(-4,-1)(6,6)
    \curvepnodes[plotpoints=#1]{0}{\the\numexpr#1-1\relax}
    {
        p 1 sub R mul neg 0 
        0 1 t
        {
          dup 1 sub P exch exp R mul OmP mul exch 90 mul PtoC
            3 -1 roll add 3 1 roll add exch 
        } for 
    }{A}
    \multido{\ic=0+1}{#1}{\rput(A\ic){\rput{!90 \ic\space mul}(0,0){\psframe(!R P \ic\space exp mul C)}}}
    \pscustom[linecolor=red]{\multido{\ic=0+1}{#1}{\psarc(A\ic){!R P \ic\space exp mul}{!\ic\space 90 mul}{!\ic\space 90 mul 90 add}}}
\end{pspicture}}

\begin{document}
\multido{\i=2+1}{12}{\Atom{\i}}
\end{document}

Closed form

In progress...