How to draw 3 equal circles inside a circle in tikz or other way?

Question

I want to draw equal circles inside a circle using Tikz-pgf or other way as in the below image (please forget the numbers inside the small circles):

To draw a circle, we can use:

\begin{tikzpicture}
\draw[color=red] (0,0) circle [radius=1];
\end{tikzpicture}

But how to create an equal circle inside this unit circle, as in the first picture ?

One way is to find the center of each small circle and its radius. But that is very technical.

Is their any drawing tool that can easily do it? For example, there are drawing tools in Wolfram, but they need a subscription.

Remark: To give some idea about the first image above, it is a number system called the p-adic number system, and the image shows the distribution of 3-adic numbers. It is an interesting number system in mathematics.

Answer 1

Here you go!

\documentclass[tikz,border=3.14mm]{standalone}
\begin{document}
    \newcommand{\RA}{2}
    \pgfmathsetmacro{\RB}{\RA*(1+2/sqrt(3)}
    \begin{tikzpicture}
        \draw (0,0) circle[radius=\RB cm];
        \foreach \ang in {90,210,330}
            \draw (\ang:\RB-\RA) circle[radius=\RA cm];
    \end{tikzpicture}   
\end{document}

Of course, if you want to draw the outer circle first and define the inner circles from it, you may change a bit your code:

\documentclass[tikz,border=3.14mm]{standalone}
\begin{document}
    \newcommand{\RA}{2}
    \pgfmathsetmacro{\RB}{\RA/(1+2/sqrt(3)}
    \begin{tikzpicture}
        \draw (0,0) circle[radius=\RA cm];
        \foreach \ang in {90,210,330}
            \draw (\ang:\RA-\RB) circle[radius=\RB cm];
    \end{tikzpicture}   
\end{document}

BONUS: SECOND ITERATION

\documentclass[tikz,border=3.14mm]{standalone}
\usetikzlibrary{calc}
\begin{document}
    \newcommand{\RA}{2}
    \pgfmathsetmacro{\RB}{\RA/(1+2/sqrt(3)}
    \begin{tikzpicture}
        \draw (0,0) circle[radius=\RA cm];
        \foreach \ang [count = \i from 1] in {90,210,330}
            \draw (\ang:\RA-\RB) coordinate (center-\i) circle[radius=\RB cm];
        \pgfmathsetmacro{\RC}{\RB/(1+2/sqrt(3)}
        \foreach \i in {1,2,3}
            \foreach \ang in {90,210,330}
                \draw ($(center-\i)+(\ang:\RB-\RC)$) circle[radius=\RC cm];
    \end{tikzpicture}   
\end{document}

It was made very roughly if you just need two iterations. If you need more, it will be needed to make the code more legible and useful.

Answer 2

The mathematical problem was solved by SebGlav already in the comments and they also helped you a big deal with their answer.

I want to provide a recursive solution using circular nodes and labels (i.e. more nodes).

It uses the ext.pgfkeys-plus library of my tikz-ext package for the /utils/TeX/ifnum={<test>}{<true>}{<false>} key to end the recursion when level zero has been reached. (And if you can't use that the definition of it is in the code below.)

The idea behind this is to anchor the labels (i.e. the circles inside another circle) at the angles 90, 210 and 330 and at their parent node. That way, we only need to evalute the radii and PGF/TikZ figures out the trigonometry of it.

---

The second picture moves the recursion check inside cic/label so we can use level 0 to do something totally different. Furthermore another argument is introduced (#3 in cic/@ which replaced cic which is now only a “starter”) which gets carried down and appears as #2 in cic/node'.

It just contains a list of directions in the form of u (90°), l (210°) and r (330°).

Since this is just a ternary number, we could also replace this with some calculations using the level #1 and the digits 0, 1 and 2 to evaluate the values on-the-fly but here I've opted to hardcode these.

Code

\documentclass[tikz]{standalone}
%\usepgfkeyslibrary{ext.pgfkeys-plus}
\makeatletter
\pgfkeys{/utils/TeX/ifnum/.code n args={3}{%
    \ifnum#1\relax\expandafter\pgfutil@firstoftwo\else\expandafter\pgfutil@secondoftwo\fi
    {\pgfkeysalso{#2}}{\pgfkeysalso{#3}}}}
\makeatother
\begin{document}
\tikz[
  cic/node/.style={
    shape=circle, inner sep=+0pt, outer sep=+0pt, minimum size={#1}},
  cic/.style n args={2}{
    cic/node={#2}, cic/level #1/.try,
    /utils/TeX/ifnum={#1=0}{}{
      /utils/exec=%
        \pgfmathsetlengthmacro\rad{#2/(1+2/sqrt(3))}%
        \pgfmathtruncatemacro\level{#1-1},
      cic/label/.expanded={\level}{\rad}{ 90},
      cic/label/.expanded={\level}{\rad}{210},
      cic/label/.expanded={\level}{\rad}{330}}},
  cic/label/.style n args={3}{
    label={[anchor=#3, label position=#3, cic={#1}{#2}]}},
  cic/level setter/.style args={#1=#2}{cic/level #1/.style={fill=#2}},
  cic/level setter/.list={3=purple, 2=red!50, 1=green!50!black, 0=blue!50},
] \node[cic={3}{2cm}]{};
\tikz[
  cic/node/.style={
    shape=circle, inner sep=+0pt, outer sep=+0pt, minimum size={#1}},
  cic/node'/.style 2 args={% this node contents is just a debug fall back
    cic/node={#1}, path only, node contents={#2}, cic/node #2/.try},
  cic/.style 2 args={cic/@={#1}{#2}{cic-}},
  cic/@/.style n args={3}{
    cic/node={#2}, cic/level #1/.try,
    /utils/exec=\pgfmathsetlengthmacro\rad{#2/(1+2/sqrt(3))},
    cic/label/.expanded={\inteval{#1-1}}{\rad}{ 90}{#3u},
    cic/label/.expanded={\inteval{#1-1}}{\rad}{210}{#3l},
    cic/label/.expanded={\inteval{#1-1}}{\rad}{330}{#3r}},
  cic/label/.style n args={4}{
    /utils/TeX/ifnum={#1=0}{
      label={[anchor=#3, label position=#3, cic/node'={#2}{#4}]},
    }{
      label={[anchor=#3, label position=#3, cic/@={#1}{#2}{#4}]}}},
  cic/level setter/.style args={#1=#2}{cic/level #1/.style={fill=#2}},
  cic/level setter/.list={2=purple, 1=red!50},
  cic/node setter/.style args={#1=#2}{
    cic/node cic-#1/.style={node contents={#2}}},
  cic/node setter/.list={uu=0, ru=1, lu=2, ur=3, rr=4, lr=5, ul=6, rl=7, ll=8}
] \footnotesize\node[cic={2}{2cm}]{};
\end{document}

Output