Drawing text in vertices using polygon regular tikz

Question

I'm trying plot the regular polygon induced by the equation x^(11)+1=0. I've drawn the polygon and the circunference with this code:

\documentclass[a4paper]{article}

\usepackage[english]{babel}
\usepackage[utf8x]{inputenc}
\usepackage{amsmath}
\usepackage{graphicx}
\usepackage{tikz}
\usetikzlibrary{shapes.geometric}


\begin{document}

\begin{tikzpicture}[scale=3]
\draw [very thick,<->] (-1.5,0)--(1.5,0);
\draw [very thick,<->] (0,-1.5)--(0,1.5);
\draw[thick,red!90!black] (0,0) circle (1cm);
\node [draw, thick, blue!90!black,rotate=90,minimum size=6cm,regular polygon, regular polygon sides=11] at (0,0) {};    

\end{tikzpicture}
\end{document}

Now, I want to add the text w_i to the i-th vertex. How can I do it?

Answer 1

Updated

I've decided to update my answer by adding a counter to the foreach statement. This way it is much more automatized and the code is shorter.

The foreach adds the nodes at each corner of the polygon, which can be referred to using corner # (you can find this in the Tikz Manual, page 229), but it starts the subscript number from 0 at a certain corner and adds 1 to the previous value.

Anyway, here it is:

\documentclass[border=1cm]{standalone}
\usepackage{amsmath}
\usepackage{graphicx}
\usepackage{tikz}
\usetikzlibrary{shapes.geometric, calc}

\begin{document}
\begin{tikzpicture}[scale=3]
\draw [very thick,<->] (-1.5,0)--(1.5,0);
\draw [very thick,<->] (0,-1.5)--(0,1.5);
\draw[thick,red!90!black] (0,0) circle (1cm);

\node (pol) [draw, thick, blue!90!black,rotate=90,minimum size=6cm,regular polygon, regular polygon sides=11] at (0,0) {}; 

\foreach \n [count=\nu from 0, remember=\n as \lastn, evaluate={\nu+\lastn}] in {7,8,...,10,11,1,2,...,5,6} 
\node[anchor=\n*(360/11)]at(pol.corner \n){$\omega_{\nu}$};

\end{tikzpicture}
\end{document}

Answer 2

The correct solution (from mathematics point of view) with PSTricks. I used zero-based index here (because the OP wants it).

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

\makeatletter

\def\Atom#1{%
\begin{pspicture}(-3,-3)(3,3)
    \psaxes[labels=none,ticks=none,linecolor=lightgray!50](0,0)(-3,-3)(3,3)
    \pscircle[dimen=m,linecolor=lightgray]{2}
    \degrees[#1]
    \curvepnodes[plotpoints=\numexpr#1+1]{0}{#1}{2 t .5 add \pst@angleunit PtoC}{P}  
    \psnline[linecolor=blue](0,\Pnodecount){P}
    \multido{\i@=0+1}{\Pnodecount}{\qdisk(P\i@){2pt}\uput{2.2}[(P\i@)](0,0){$\omega_{#1}^{\i@}$}}
\end{pspicture}}

\makeatother

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

A bonus puzzle

Why did I add .5 to t in 2 t .5 add \pst@angleunit PtoC? :-)

Latest edit

Another method that seems to be more complicated but it is still fun!

\documentclass[pstricks,border=12pt,12pt]{standalone}
\usepackage{pst-eucl,pst-plot}

\psset{CurveType=polygon}

\makeatletter
\def\Atom#1{%
\begin{pspicture}[showgrid=false](-3,-3)(3,3)
  \psaxes[labels=none,ticks=none,linecolor=lightgray!50](0,0)(-3,-3)(3,3)
    \pscircle[linecolor=red]{2}
    \degrees[#1]
    \def\points{}\def\names{}\def\angles{}
    \multido{\i@=0+1,\n@=.5+1.0}{#1}
    {
        \xdef\points{\points(!2 \n@\space \pst@angleunit PtoC){A\i@}}
        \xdef\names{\names \omega_{\i@},}
        \xdef\angles{\angles \n@,}
    }
    \edef\args{[PointName={\names},PosAngle={\angles}]\points}
    \expandafter\pstGeonode\args
\end{pspicture}}
\makeatother

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

Answer 3

Only for more clearly I've rearrenged the order of vertices in the correct mathematical order:

\documentclass[border=1cm]{standalone}
\usepackage{amsmath}
\usepackage{graphicx}
\usepackage{tikz}
\usetikzlibrary{shapes.geometric, calc}


\begin{document}
\begin{tikzpicture}[scale=3]
\draw [very thick,<->] (-1.5,0)--(1.5,0);
\draw [very thick,<->] (0,-1.5)--(0,1.5);
\draw[thick,red!90!black] (0,0) circle (1cm);

\node (pol) [draw, thick, blue!90!black,rotate=90,minimum size=6cm,regular polygon, regular polygon sides=11] at (0,0) {}; 

\foreach \anchor/\label/\placement in
    {corner 1/$w_5$/above left, 
     corner 2/$w_6$/left, 
     corner 3/$w_7$/left, 
     corner 4/$w_8$/below,
     corner 5/$w_9$/below,   
     corner 6/$w_{10}$/right,
     corner 7/$w_0$/right,
     corner 8/$w_1$/above,
     corner 9/$w_2$/above,
     corner 10/$w_3$/above,
     corner 11/$w_4$/above}
\draw[shift=(pol.\anchor)] plot coordinates{(0,0)}
node[font=\scriptsize,\placement] {\label};

\end{tikzpicture}
\end{document}