Creating figure to represent ordinal numbers using tikzpicture

Question

I am working on overleaf and I want to make a diagram to represent the ordinal numbers (a mathematical term) in the following way...

I am using tikzpicture to make this diagram. My attempt is as follows (not completed!).

\begin{tikzpicture}
  \draw[gray] (-4,-4) grid (4,4);
  \draw (0,0) node[] {$0$};
  \draw (-0.3,-0.2) node[] {$1$};
  \draw (-0.5,-0.4) node[] {$2$};
  \draw (-0.4,-0.7) node[] {$3$};
  \draw[-> , dotted]  (-0.3,-0.8) .. controls (0.2,-1.2) and (1,-0.3) .. (0.3,0.4) ;
  \draw (0,0.5) node[] {$\omega$};
  \draw (-0.8,0.3) node[] {$\omega+1$};
  \draw (-1.3,-0.3) node[] {$\omega+2$};
  \draw (-1.1,-0.8) node[] {$\omega+3$};
  \draw[-> , dotted]  (-1,-1) .. controls (-0.9,-1.3) and (-0.6,-1.5) .. (-0.35,-1.5) ;
  \draw (0,-1.5) node[] {$2 \cdot \omega$};
  \draw[-> , dotted]  (.55,-1.5) .. controls (1.2,-1.3) and (1.3,-0.8) .. (1.3,-0.5) ;
  \draw (1.3,-0.3) node[] {$3 \cdot \omega$};
  \draw[-> , dotted]  (1.3,0) .. controls (1.3,0.5) and (1,1.3) .. (0.3,1.3) ;
  \draw (0,1.3) node[] {$\omega^{\omega}$};
\end{tikzpicture}

whose output is as follows

But the problem is, it is very time-consuming and is not looking that much beautiful. Also, the entries like $\omega + 1, \omega + 2$, etc are feeling so big in this picture. Can we reduce the size? Can anyone tell me any other way to create such diagrams? any link to check or any source someone provide? any sort of help will be very helpful.

Note: I am using the grid on purpose to draw the entries accordingly. By beautiful, I did not mean about the grid, I can comment that line.

Thanks in advance.

Answer 1

I found it an interesting challenge, since it is about placing terms in something similar to a spiral, searching I found this very detailed code from Guilherme Zanotelli to draw parametric spirals, and then I adapted it to obtain a similar result, I have used some nesting to place the nodes.

RESULT:

MWE:

\documentclass[tikz,border=0.5cm]{standalone}
%this code is from Guilherme Zanotelli  in a "short" explanation in https://tex.stackexchange.com/a/333824/154390 [start]
\newcommand\bonusspiral{} % just for safety
% \bonusspiraldraw options(start angle:end angle)(start radius:final radius)[revolutions]
\def\bonusspiral#1(#3:#4)(#5:#6)[#7]{
    \pgfmathsetmacro{\domain}{#4+#7360}
    \pgfmathsetmacro{\growth}{180(#6-#5)/(pi(\domain-#3))}
    \draw [#1,
    shift={(#2)},
    domain=#3pi/180:\domainpi/180,
    variable=\t,
    smooth,
    samples=int(\domain/5)] plot ({\t r}: {#5+\growth\t-\growth#3pi/180});
}
%this code is from Guilherme Zanotelli  in a "short" explanation in https://tex.stackexchange.com/a/333824/154390 [end]
%Modification to put a node in specific point of the function
%\NodeOnSpiraldraw options(start angle:end angle)(start radius:final radius)[revolutions]{node_position_in_degrees}{node content}
\def\NodeOnSpiral#1(#3:#4)(#5:#6)[#7]#8#9{
    \pgfmathsetmacro{\dom}{#4+#7360}
    \pgfmathsetmacro{\grow}{180(#6-#5)/(pi(\dom-#3))}
    \path [
        shift={(#2)},
        domain=#3pi/180:\dompi/180,
        variable=\j,
        smooth,
        samples=int(\dom/5),
        samples at=#8pi/180
    ]
        plot ({\j r}: {#5+\grow\j-\grow#3*pi/180})
        node[
            #1,
            rectangle,
            fill=white,
            inner sep=2pt,
            scale=0.8
        ]{#9};
}
\begin{document}
    \begin{tikzpicture}[>=latex]
    %Start drawing the thing.
    % First spiral is used as workspace a spiral that starst from 90 at radious 1 and ends at 90 at radious 6.
    \bonusspiralblack!10,dotted, thick(90:90)(1:6)[5]
% Draw a line from (90:1) to (90:6)
\draw[thick] (90:1) -- (90:4);
\draw[thick,dashed] (90:4) -- (90:6);

% Draw the first spiral arrow in one revolution from 90:1 to 85:2. no to 90:2 to not put the arrow interrupting the node
\bonusspiral[red!50!black,dashed, thick,->](0,0)(90:85)(1:2)[1]
% Put the nodes in their corresponding angle position in the spiral.
\foreach  \degpos/\nodetext in {
    90/0,
    135/1,
    180/2,
    225/3%
}{\NodeOnSpiral[green!50!black](0,0)(90:90)(1:3)[2]{\degpos}{$\nodetext$}}

% Draw the second spiral arrow in one revolution from 90:2 to 85:3. 
\bonusspiral[blue!50!black,dashed, thick,->](0,0)(90:86)(2:3)[1]
% same thing in the corresponding spiral and revolution angle.
\foreach  \degpos/\nodetext in {
    450/w,
    480/w+1,
    510/w+2,
    630/2w,
    660/2w+1,
    690/2w+2,
    720/3w,
    760/4w%
}{\NodeOnSpiral[red!50!black](0,0)(90:90)(1:3)[2]{\degpos}{$\nodetext$}}

% Draw the third spiral arrow in one revolution from 90:3 to 85:4. 
\bonusspiral[blue!50!black,dashed, thick,->](0,0)(90:87)(3:4)[1]
% same thing in the corresponding spiral and revolution angle.
\foreach  \degpos/\nodetext in {
    810/w^2,
    840/w^2+1,
    870/w^2+2,
    945/w^2+w,
    990/w^2+dw,
    1035/w^2\cdot2,
    1080/w^2\cdot3%
}{\NodeOnSpiral[red!50!black](0,0)(90:90)(1:3)[2]{\degpos}{$\nodetext$}}

% Draw the last spirals arrow in two revolutions from 95:4 to 88:6. 
\bonusspiral[black, thick](0,0)(93:87)(4:5)[1]
\bonusspiral[black, thick,->](0,0)(93:88)(5:6)[1]
% same thing in the corresponding spiral and revolution angle.
\foreach  \degpos/\nodetext in {
    1170/w^3,
    1890/w^w%
}{\NodeOnSpiral[red!50!black](0,0)(90:90)(1:3)[2]{\degpos}{$\nodetext$}}


\end{tikzpicture} 

\end{document}