Sphere packing in a sphere diagrams

Question

Inspired by How to create nice-looking nuclei in TikZ?, I am trying to draw arrangements of spheres, in particular the sphere packing in a sphere problem.

From Wikipedia - "Sphere packing in a sphere is a three-dimensional packing problem with the objective of packing a given number of equal spheres inside a unit sphere. It is the three-dimensional equivalent of the circle packing in a circle problem in two dimensions."

For small numbers, the results are trivial:

But here are some larger examples which are more complicated:

These images were taken from the Wikipedia article linked above.

My question is how could one go about creating such diagrams using TikZ? Using the tikz-3dplot package springs to mind. But one tricky aspect is you have to draw the balls in the right order, so they overlap correctly to give the desired 3D view.

My first attempt at drawing the trivial cases:

MWE

\documentclass[margin=0.5cm]{standalone}
\usepackage{tikz}
\usepackage{tikz-3dplot}
\begin{document}
\tdplotsetmaincoords{30}{120}
\begin{tikzpicture}[tdplot_main_coords]

\shade [ball color=red] (0,0,0) circle (1cm);

\begin{scope}[xshift=4cm]
\draw [gray] (0,0,0) circle (1cm);
\shade [ball color=red] (0:0.5) circle (0.5cm);
\shade [ball color=blue] (180:0.5) circle (0.5cm);
\end{scope}

\begin{scope}[xshift=2cm,yshift=-3cm]
\draw [gray] (0,0,0) circle (1cm);
\shade [ball color=red] (240:0.536) circle (0.4641cm);
\shade [ball color=blue] (0:0.536) circle (0.4641cm);
\shade [ball color=green] (120:0.536) circle (0.4641cm);
\end{scope}

\end{tikzpicture}
\end{document}

I'm primarily interested in a TikZ solution, but solutions using other packages (PSTricks/Asymptote) are welcome. I know Asymptote in particular is probably better suited to such diagrams.

A related question is How to draw a series of simple circle packing illustrations, possibly with Tikz?, which deals with circle packing in a circle.

A good question for me and it is very interested to draw a kernel of an atom. — Sebastiano, Sep 01 '18 at 06:14
Very nice question indeed! But I do think it depends a lot on whether or not you know the correct drawing order on beforehand. If you know the Cartesian coordinates of the inner spheres then the z buffer of pgfplots can order the drawing for you I think. — Max, Sep 01 '18 at 07:36

score 10 · Accepted Answer · 2018-09-01T15:13:32.970

The theory behind this is actually not very difficult. One way (out of two ways) to make the spheres maximally packed is to put them on the root lattice of A_3=SU(4). The simple roots of A_3 can be chosen to be

\alpha_1=(1,0,0)
\alpha_2=(-1/2,1/\sqrt{2},-1/2)
\alpha_3=(0,0,1)

A lattice point has then the coordinates \sum_i n_i\alpha_i where the n_i\in\mathbbm{Z}. UPDATE: I gave up trying to do that by TeX only and asked Mathematica to compute the projections of the center coordinates of the spheres on the normal of the visible plane. Objects that should be hidden have a more negative projection than objects that could cover them. This yields a lengthy "master list" that can be used to draw the spheres in the right (?) order. In principle, this could be done with pgfplotstable also, but it would be considerably more effort for pgfplotstable dummies like me. The downside of the current answer is that the list has to be recreated for each new set of view angles.

\documentclass[tikz,border=3.14mm]{standalone}
\usepackage{tikz-3dplot}
\usetikzlibrary{3d}
\tikzset{declare function={posx(\x,\y,\z)=\x-\y/2;
posy(\x,\y,\z)=\y/sqrt(2);
posz(\x,\y,\z)=-\y/2+\z;
}}
\newsavebox\Proton
\newsavebox\Neutron
\sbox\Proton{\tikz{\shade[ball color=red] circle({1/sqrt(2)});}}
\sbox\Neutron{\tikz{\shade[ball color=blue] circle({1/sqrt(2)});}}
\begin{document}
% this list has been generated by Mathematica for the present projection
\xdef\MasterList{{{0, 0, 0}}, {{0, 0, 1}, 
  {-1, -1, 0}, {0, -1, 0}, 
  {0, 1, 1}, {-1, 0, 0}, {1, 1, 1}, 
  {0, 0, 0}, {-1, -1, -1}, 
  {1, 0, 0}, {0, -1, -1}, 
  {0, 1, 0}, {1, 1, 0}, 
  {0, 0, -1}}, {{-1, 0, 1}, 
  {0, 0, 1}, {-1, -1, 0}, 
  {1, 0, 1}, {0, -1, 0}, 
  {-1, -2, -1}, {0, 1, 1}, 
  {-1, 0, 0}, {1, 1, 1}, {0, 0, 0}, 
  {-1, -1, -1}, {1, 0, 0}, 
  {0, -1, -1}, {1, 2, 1}, 
  {0, 1, 0}, {-1, 0, -1}, 
  {1, 1, 0}, {0, 0, -1}, 
  {1, 0, -1}}, {{-1, -1, 1}, 
  {0, -1, 1}, {-1, -2, 0}, 
  {0, 1, 2}, {-1, 0, 1}, 
  {-2, -1, 0}, {1, 1, 2}, 
  {0, 0, 1}, {-1, -1, 0}, 
  {-2, -2, -1}, {-1, 1, 1}, 
  {1, 0, 1}, {0, -1, 0}, 
  {-1, -2, -1}, {1, 2, 2}, 
  {0, 1, 1}, {-1, 0, 0}, 
  {-2, -1, -1}, {1, -1, 0}, 
  {0, -2, -1}, {1, 1, 1}, 
  {0, 0, 0}, {-1, -1, -1}, 
  {0, 2, 1}, {-1, 1, 0}, {2, 1, 1}, 
  {1, 0, 0}, {0, -1, -1}, 
  {-1, -2, -2}, {1, 2, 1}, 
  {0, 1, 0}, {-1, 0, -1}, 
  {1, -1, -1}, {2, 2, 1}, 
  {1, 1, 0}, {0, 0, -1}, 
  {-1, -1, -2}, {2, 1, 0}, 
  {1, 0, -1}, {0, -1, -2}, 
  {1, 2, 0}, {0, 1, -1}, 
  {1, 1, -1}}, {{0, 0, 2}, 
  {-1, -1, 1}, {-2, -2, 0}, 
  {0, -1, 1}, {-1, -2, 0}, 
  {0, 1, 2}, {-1, 0, 1}, 
  {-2, -1, 0}, {0, -2, 0}, 
  {1, 1, 2}, {0, 0, 1}, 
  {-1, -1, 0}, {-2, -2, -1}, 
  {0, 2, 2}, {-1, 1, 1}, 
  {-2, 0, 0}, {1, 0, 1}, 
  {0, -1, 0}, {-1, -2, -1}, 
  {1, 2, 2}, {0, 1, 1}, {-1, 0, 0}, 
  {-2, -1, -1}, {1, -1, 0}, 
  {0, -2, -1}, {2, 2, 2}, 
  {1, 1, 1}, {0, 0, 0}, 
  {-1, -1, -1}, {-2, -2, -2}, 
  {0, 2, 1}, {-1, 1, 0}, {2, 1, 1}, 
  {1, 0, 0}, {0, -1, -1}, 
  {-1, -2, -2}, {1, 2, 1}, 
  {0, 1, 0}, {-1, 0, -1}, 
  {2, 0, 0}, {1, -1, -1}, 
  {0, -2, -2}, {2, 2, 1}, 
  {1, 1, 0}, {0, 0, -1}, 
  {-1, -1, -2}, {0, 2, 0}, 
  {2, 1, 0}, {1, 0, -1}, 
  {0, -1, -2}, {1, 2, 0}, 
  {0, 1, -1}, {2, 2, 0}, 
  {1, 1, -1}, {0, 0, -2}}, 
 {{-1, 0, 2}, {-2, -1, 1}, 
  {0, 0, 2}, {-1, -1, 1}, 
  {-2, -2, 0}, {-1, 1, 2}, 
  {-2, 0, 1}, {1, 0, 2}, 
  {0, -1, 1}, {-1, -2, 0}, 
  {-2, -3, -1}, {0, 1, 2}, 
  {-1, 0, 1}, {-2, -1, 0}, 
  {1, -1, 1}, {0, -2, 0}, 
  {-1, -3, -1}, {1, 1, 2}, 
  {0, 0, 1}, {-1, -1, 0}, 
  {-2, -2, -1}, {0, 2, 2}, 
  {-1, 1, 1}, {2, 1, 2}, 
  {-2, 0, 0}, {1, 0, 1}, 
  {0, -1, 0}, {-1, -2, -1}, 
  {-2, -3, -2}, {1, 2, 2}, 
  {0, 1, 1}, {-1, 0, 0}, {2, 0, 1}, 
  {-2, -1, -1}, {1, -1, 0}, 
  {0, -2, -1}, {-1, -3, -2}, 
  {2, 2, 2}, {1, 1, 1}, {0, 0, 0}, 
  {-1, -1, -1}, {-2, -2, -2}, 
  {1, 3, 2}, {0, 2, 1}, {-1, 1, 0}, 
  {2, 1, 1}, {-2, 0, -1}, 
  {1, 0, 0}, {0, -1, -1}, 
  {-1, -2, -2}, {2, 3, 2}, 
  {1, 2, 1}, {0, 1, 0}, 
  {-1, 0, -1}, {2, 0, 0}, 
  {-2, -1, -2}, {1, -1, -1}, 
  {0, -2, -2}, {2, 2, 1}, 
  {1, 1, 0}, {0, 0, -1}, 
  {-1, -1, -2}, {1, 3, 1}, 
  {0, 2, 0}, {-1, 1, -1}, 
  {2, 1, 0}, {1, 0, -1}, 
  {0, -1, -2}, {2, 3, 1}, 
  {1, 2, 0}, {0, 1, -1}, 
  {-1, 0, -2}, {2, 0, -1}, 
  {1, -1, -2}, {2, 2, 0}, 
  {1, 1, -1}, {0, 0, -2}, 
  {2, 1, -1}, {1, 0, -2}}, 
 {{-1, 0, 2}, {-2, -1, 1}, 
  {0, 0, 2}, {-1, -1, 1}, 
  {-2, -2, 0}, {-1, 1, 2}, 
  {-2, 0, 1}, {1, 0, 2}, 
  {0, -1, 1}, {-1, -2, 0}, 
  {-2, -3, -1}, {0, 1, 2}, 
  {-1, 0, 1}, {-2, -1, 0}, 
  {1, -1, 1}, {0, -2, 0}, 
  {-1, -3, -1}, {1, 1, 2}, 
  {0, 0, 1}, {-1, -1, 0}, 
  {-2, -2, -1}, {0, 2, 2}, 
  {-1, 1, 1}, {2, 1, 2}, 
  {-2, 0, 0}, {1, 0, 1}, 
  {0, -1, 0}, {-1, -2, -1}, 
  {-2, -3, -2}, {1, 2, 2}, 
  {0, 1, 1}, {-1, 0, 0}, {2, 0, 1}, 
  {-2, -1, -1}, {1, -1, 0}, 
  {0, -2, -1}, {-1, -3, -2}, 
  {2, 2, 2}, {1, 1, 1}, {0, 0, 0}, 
  {-1, -1, -1}, {-2, -2, -2}, 
  {1, 3, 2}, {0, 2, 1}, {-1, 1, 0}, 
  {2, 1, 1}, {-2, 0, -1}, 
  {1, 0, 0}, {0, -1, -1}, 
  {-1, -2, -2}, {2, 3, 2}, 
  {1, 2, 1}, {0, 1, 0}, 
  {-1, 0, -1}, {2, 0, 0}, 
  {-2, -1, -2}, {1, -1, -1}, 
  {0, -2, -2}, {2, 2, 1}, 
  {1, 1, 0}, {0, 0, -1}, 
  {-1, -1, -2}, {1, 3, 1}, 
  {0, 2, 0}, {-1, 1, -1}, 
  {2, 1, 0}, {1, 0, -1}, 
  {0, -1, -2}, {2, 3, 1}, 
  {1, 2, 0}, {0, 1, -1}, 
  {-1, 0, -2}, {2, 0, -1}, 
  {1, -1, -2}, {2, 2, 0}, 
  {1, 1, -1}, {0, 0, -2}, 
  {2, 1, -1}, {1, 0, -2}}, 
 {{-1, -2, 1}, {-1, 0, 2}, 
  {-2, -1, 1}, {0, 0, 2}, 
  {-1, -1, 1}, {-2, -2, 0}, 
  {-1, 1, 2}, {-2, 0, 1}, 
  {1, 0, 2}, {0, -1, 1}, 
  {-1, -2, 0}, {-2, -3, -1}, 
  {1, 2, 3}, {0, 1, 2}, {-1, 0, 1}, 
  {-2, -1, 0}, {1, -1, 1}, 
  {-3, -2, -1}, {0, -2, 0}, 
  {-1, -3, -1}, {1, 1, 2}, 
  {0, 0, 1}, {-1, -1, 0}, 
  {-2, -2, -1}, {0, 2, 2}, 
  {-1, 1, 1}, {2, 1, 2}, 
  {-2, 0, 0}, {1, 0, 1}, 
  {0, -1, 0}, {-1, -2, -1}, 
  {-2, -3, -2}, {1, 2, 2}, 
  {0, 1, 1}, {-1, 0, 0}, {2, 0, 1}, 
  {-2, -1, -1}, {1, -1, 0}, 
  {0, -2, -1}, {-1, -3, -2}, 
  {-1, 2, 1}, {2, 2, 2}, {1, 1, 1}, 
  {0, 0, 0}, {-1, -1, -1}, 
  {-2, -2, -2}, {1, -2, -1}, 
  {1, 3, 2}, {0, 2, 1}, {-1, 1, 0}, 
  {2, 1, 1}, {-2, 0, -1}, 
  {1, 0, 0}, {0, -1, -1}, 
  {-1, -2, -2}, {2, 3, 2}, 
  {1, 2, 1}, {0, 1, 0}, 
  {-1, 0, -1}, {2, 0, 0}, 
  {-2, -1, -2}, {1, -1, -1}, 
  {0, -2, -2}, {2, 2, 1}, 
  {1, 1, 0}, {0, 0, -1}, 
  {-1, -1, -2}, {1, 3, 1}, 
  {0, 2, 0}, {3, 2, 1}, 
  {-1, 1, -1}, {2, 1, 0}, 
  {1, 0, -1}, {0, -1, -2}, 
  {-1, -2, -3}, {2, 3, 1}, 
  {1, 2, 0}, {0, 1, -1}, 
  {-1, 0, -2}, {2, 0, -1}, 
  {1, -1, -2}, {2, 2, 0}, 
  {1, 1, -1}, {0, 0, -2}, 
  {2, 1, -1}, {1, 0, -2}, 
  {1, 2, -1}}}
\xdef\LstCol{"red","blue"}
\tdplotsetmaincoords{-90+109.471}{-90+70}
\foreach \Lst in \MasterList
{\typeout{\Lst}
\begin{tikzpicture}
\path[use as bounding box] (-3.5,-3.5) rectangle (3.5,3.5);
\draw (0,0) circle ({1}); % /sqrt(2)
%\node at (1,1) {\Y,\X};
\begin{scope}[tdplot_main_coords]
 \draw[-latex] (0,0,0) coordinate (O) -- (1,0,0) node[right]{$\alpha_1$};
 \draw[-latex] (O) -- (-1/2,{1/sqrt(2)},-1/2) node[right]{$\alpha_2$};
 \draw[-latex] (O) -- (0,0,1) node[right]{$\alpha_3$};
 \draw[red,-latex] (O) -- (1/2,{1/sqrt(2)},1/2) node[right]{$-\theta$};
 \foreach \Z in \Lst
  {\typeout{\Z}
  \pgfmathsetmacro{\myx}{{\Z}[0]}
  \pgfmathsetmacro{\myy}{{\Z}[1]}
  \pgfmathsetmacro{\myz}{{\Z}[2]}
  \pgfmathtruncatemacro{\mycol}{int(2*rnd)}
  \ifnum\mycol=1
  \node at ({posx(\myx,\myy,\myz)},
  {posy(\myx,\myy,\myz)},{posz(\myx,\myy,\myz)}) {\usebox\Neutron};
  \else
  \node at ({posx(\myx,\myy,\myz)},
  {posy(\myx,\myy,\myz)},{posz(\myx,\myy,\myz)}) {\usebox\Proton};
  \fi}
\end{scope}
\end{tikzpicture}}
\end{document}

OLD ANSWER: The only problem I have (I think) is to adjust the order in which the spheres are drawn (or, equivalently, to dial the right view angles for the simple-minded order I chose).

\documentclass[tikz,border=3.14mm]{standalone}
\usepackage{tikz-3dplot}
\usetikzlibrary{3d}%{n1 - n2/2, n2/Sqrt[2], -n2/2 + n3}
\tikzset{declare function={posx(\x,\y,\z)=\x-\y/2;
posy(\x,\y,\z)=\y/sqrt(2);
posz(\x,\y,\z)=-\y/2+\z;
}}
\newsavebox\Proton
\newsavebox\Neutron
\sbox\Proton{\tikz{\shade[ball color=red] circle({1/sqrt(2)});}}
\sbox\Neutron{\tikz{\shade[ball color=blue] circle({1/sqrt(2)});}}
\begin{document}
\xdef\LstCol{"red","blue"}
\foreach \Lev in {0,...,4}
{\tdplotsetmaincoords{109.471}{0}
\begin{tikzpicture}
\path[use as bounding box] (-3.5,-3.5) rectangle (3.5,3.5);
\draw (0,0) circle ({1}); % /sqrt(2)
%\node at (1,1) {\Y,\X};
\begin{scope}[tdplot_main_coords]
 \draw[-latex] (0,0,0) coordinate (O) -- (1,0,0) node[right]{$\alpha_1$};
 \draw[-latex] (O) -- (-1/2,{1/sqrt(2)},-1/2) node[right]{$\alpha_2$};
 \draw[-latex] (O) -- (0,0,1) node[right]{$\alpha_3$};
 \draw[red,-latex] (O) -- (1/2,{1/sqrt(2)},1/2) node[right]{$-\theta$};
 % level 0
 \ifnum\Lev>0
  \pgfmathtruncatemacro{\mycol}{int(2*rnd)}
  \ifnum\mycol=1
  \node at (0,0,0) {\usebox\Neutron};
  \else
  \node at (0,0,0) {\usebox\Proton};
  \fi
 \fi
 % level 1
 \ifnum\Lev>1
  \foreach \Z in {{-1, -1, -1}, {-1, -1, 0}, 
  {-1, 0, 0}, {0, -1, -1}, 
  {0, -1, 0}, {0, 0, -1}, {0, 0, 1}, 
  {0, 1, 0}, {0, 1, 1}, {1, 0, 0}, 
  {1, 1, 0}, {1, 1, 1}}
  {\pgfmathsetmacro{\myx}{{\Z}[0]}
  \pgfmathsetmacro{\myy}{{\Z}[1]}
  \pgfmathsetmacro{\myz}{{\Z}[2]}
  \pgfmathtruncatemacro{\mycol}{int(2*rnd)}
  \ifnum\mycol=1
  \node at ({posx(\myx,\myy,\myz)},
  {posy(\myx,\myy,\myz)},{posz(\myx,\myy,\myz)}) {\usebox\Neutron};
  \else
  \node at ({posx(\myx,\myy,\myz)},
  {posy(\myx,\myy,\myz)},{posz(\myx,\myy,\myz)}) {\usebox\Proton};
  \fi}
 \fi
 % level 2
 \ifnum\Lev>2
 \foreach \Z in {{-1, -2, -1}, {-1, 0, -1}, {-1, 0, 1}, {1, 0, -1}, {1, 0, 1}, {1, 2, 
  1}}
  {\pgfmathsetmacro{\myx}{{\Z}[0]}
 \pgfmathsetmacro{\myy}{{\Z}[1]}
 \pgfmathsetmacro{\myz}{{\Z}[2]}
 \pgfmathtruncatemacro{\mycol}{int(2*rnd)}
 \ifnum\mycol=1
 \node at ({posx(\myx,\myy,\myz)},
 {posy(\myx,\myy,\myz)},{posz(\myx,\myy,\myz)}) {\usebox\Neutron};
 \else
 \node at ({posx(\myx,\myy,\myz)},
 {posy(\myx,\myy,\myz)},{posz(\myx,\myy,\myz)}) {\usebox\Proton};
 \fi}
 \fi
 % level 3
 \ifnum\Lev>3 
  \foreach \Z in {{-2, -2, -1}, {-2, -1, -1}, {-2, -1, 0}, {-1, -2, -2}, {-1, -2, 
                0}, {-1, -1, -2}, {-1, -1, 1}, {-1, 1, 0}, {-1, 1, 
   1}, {0, -2, -1}, {0, -1, -2}, {0, -1, 1}, {0, 1, -1}, {0, 1, 2}, {0,
    2, 1}, {1, -1, -1}, {1, -1, 0}, {1, 1, -1}, {1, 1, 2}, {1, 2, 
   0}, {1, 2, 2}, {2, 1, 0}, {2, 1, 1}, {2, 2, 1}}  
   {\pgfmathsetmacro{\myx}{{\Z}[0]}
  \pgfmathsetmacro{\myy}{{\Z}[1]}
  \pgfmathsetmacro{\myz}{{\Z}[2]}
  \pgfmathtruncatemacro{\mycol}{int(2*rnd)}
  \ifnum\mycol=1
  \node at ({posx(\myx,\myy,\myz)},
  {posy(\myx,\myy,\myz)},{posz(\myx,\myy,\myz)}) {\usebox\Neutron};
  \else
  \node at ({posx(\myx,\myy,\myz)},
  {posy(\myx,\myy,\myz)},{posz(\myx,\myy,\myz)}) {\usebox\Proton};
  \fi}
 \fi
\end{scope}
\end{tikzpicture}}%}}
\end{document}

If I were to skip level 2, I think it would be fine. Perhaps that's the way to go because these spheres are partly covered by those of level 1.

Max · Answer 2 · 2020-09-15T09:17:25.830

This is only a start and I used some 'unconventional' methods so maybe it's not really helpful, but I will just throw it out here. Lets start with the result so maybe you keep on reading :)

As mentioned in my comment, I assumed that the 3D Cartesian coordinates of the inner spheres are known, and I use the z buffer=sort of the pgfplots package to determine the drawing order.

I defined a style sphere packing axis that sets the needed axis options, and re-defines the view key to set the x, y, and z vectors to unit length.

\makeatletter
\pgfplotsset{
    sphere packing axis/.style={
        hide axis,
        clip=false,
        z buffer=sort,
        % Redefine view={<azimuth>}{<elevation>} key
        view/.code 2 args={%
            % Set elevation and azimuth angles
            \pgfmathsetmacro\view@az{##1}
            \pgfmathsetmacro\view@el{##2}
            % Calculate projections of rotation matrix
            \pgfmathsetmacro\xvec@x{cos(\view@az)}
            \pgfmathsetmacro\xvec@y{-sin(\view@az)*sin(\view@el)}
            \pgfmathsetmacro\yvec@x{sin(\view@az)}
            \pgfmathsetmacro\yvec@y{cos(\view@az)*sin(\view@el)}
            \pgfmathsetmacro\zvec@x{0}
            \pgfmathsetmacro\zvec@y{cos(\view@el)}
            % Set base vectors
            \pgfkeysalso{
                x={(\xvec@x cm,\xvec@y cm)},
                y={(\yvec@x cm,\yvec@y cm)},
                z={(\zvec@x cm,\zvec@y cm)},
            }
        },
    }
}
\makeatother

I use a outer- and inner radius key to set the dimensions of the spheres. This is not really needed but it is convenient I think.

\tikzset{
    outer sphere radius/.store in=\spherepackingouterradius,
    inner sphere radius/.store in=\spherepackinginnerradius,
}

I defined a new plot mark, that is simply a shaded circle. Still have to figure out the coloring.

\pgfdeclareplotmark{sphere}{
    \fill[ball color=red,draw=none] (0,0) circle (\spherepackinginnerradius);
}

Finally I use the definitions from above to draw the inner spheres with known coordinates, and the outer sphere on top of it.

\begin{document}
    \begin{tikzpicture}[outer sphere radius=1cm,inner sphere radius=0.4142cm]
        \begin{axis}[sphere packing axis,view={25}{30}]
            \addplot3[mark=sphere,draw=none] coordinates{
                (0,0,0.5858)
                (0,0,-0.5858)
                ( 0.4142, 0.4142,0)
                ( 0.4142,-0.4142,0)
                (-0.4142,-0.4142,0)
                (-0.4142, 0.4142,0)
            };
            \shade[ball color=black,opacity=0.1] (axis cs:0,0,0) circle (\spherepackingouterradius);
        \end{axis}
    \end{tikzpicture}
\end{document}

Edit
I had to make up for not including a MWE by adding some more examples, for N=2,3,4:

Edit 2
I got the colors working, and apparently there was already a ball mark defined.

MWE:

\documentclass[margin=2mm]{standalone}
\usepackage{pgfplots}
\makeatletter
\pgfplotsset{
    sphere packing axis/.style={
        hide axis,
        clip=false,
        z buffer=sort,
        colormap={bluered}{
            rgb255(0cm)=(0,0,180); rgb255(1cm)=(0,255,255); rgb255(2cm)=(100,255,0);
            rgb255(3cm)=(255,255,0); rgb255(4cm)=(255,0,0); rgb255(5cm)=(128,0,0)},
        every axis plot/.style={
            mark=ball,
            scatter,
            point meta=explicit,
            mark size=\spherepackinginnerradius,
            scatter/use mapped color={ball color=mapped color},
            mark options={draw opacity=0},
        },
        % Redefine view={<azimuth>}{<elevation>} key
        view/.code 2 args={%
            % Set elevation and azimuth angles
            \pgfmathsetmacro\view@az{##1}
            \pgfmathsetmacro\view@el{##2}
            % Calculate projections of rotation matrix
            \pgfmathsetmacro\xvec@x{cos(\view@az)}
            \pgfmathsetmacro\xvec@y{-sin(\view@az)sin(\view@el)}
            \pgfmathsetmacro\yvec@x{sin(\view@az)}
            \pgfmathsetmacro\yvec@y{cos(\view@az)sin(\view@el)}
            \pgfmathsetmacro\zvec@x{0}
            \pgfmathsetmacro\zvec@y{cos(\view@el)}
            % Set base vectors
            \pgfkeysalso{
                x={(\xvec@x cm,\xvec@y cm)},
                y={(\yvec@x cm,\yvec@y cm)},
                z={(\zvec@x cm,\zvec@y cm)},
            }
        },
    }
}
\makeatother
\tikzset{
    outer sphere radius/.store in=\spherepackingouterradius,
    inner sphere radius/.store in=\spherepackinginnerradius,
}
\begin{document}
    \begin{tikzpicture}[outer sphere radius=1cm,inner sphere radius=0.5cm]
        \begin{axis}[sphere packing axis,view={25}{30}]
            \addplot3[] coordinates{
                (0,0, 0.5) [0]
                (0,0,-0.5) [1]
            };
            \shade[ball color=black,opacity=0.1] (axis cs:0,0,0) circle (\spherepackingouterradius);
        \end{axis}
    \end{tikzpicture}
    \begin{tikzpicture}[outer sphere radius=1cm,inner sphere radius=0.4641cm]
        \begin{axis}[sphere packing axis,view={25}{30}]
            \addplot3[] coordinates{
                ( 0,      0.5359,0) [0]
                (-0.4641,-0.2679,0) [1]
                ( 0.4641,-0.2679,0) [2]
            };
            \shade[ball color=black,opacity=0.1] (axis cs:0,0,0) circle (\spherepackingouterradius);
        \end{axis}
    \end{tikzpicture}
    \begin{tikzpicture}[outer sphere radius=1cm,inner sphere radius=0.4494cm]
        \begin{axis}[sphere packing axis,view={25}{30}]
            \addplot3[]coordinates{
                ( 0,      0,      0.5505) [0]
                (-0.4495,-0.2595,-0.1835) [1]
                ( 0.4495,-0.2595,-0.1835) [2] 
                ( 0,      0.5190,-0.1835) [3]
            };
            \shade[ball color=black,opacity=0.1] (axis cs:0,0,0) circle (\spherepackingouterradius);
        \end{axis}
    \end{tikzpicture}
    \begin{tikzpicture}[outer sphere radius=1cm,inner sphere radius=0.4142cm]
        \begin{axis}[sphere packing axis,view={25}{30}]
            \addplot3[] coordinates{
                ( 0,      0,     0.5858) [0]
                ( 0,      0,    -0.5858) [1] 
                ( 0.4142, 0.4142,0     ) [2]
                ( 0.4142,-0.4142,0     ) [3]
                (-0.4142,-0.4142,0     ) [4]
                (-0.4142, 0.4142,0     ) [5]
            };
            \shade[ball color=black,opacity=0.1] (axis cs:0,0,0) circle (\spherepackingouterradius);
        \end{axis}
    \end{tikzpicture}
\end{document}

Edit 3
As requested in the comments. It is possible to define the size of the spheres separately using table instead of coordinates.

MWE:

\documentclass[margin=2mm]{standalone}
\usepackage{pgfplots}
\makeatletter
\pgfplotsset{
    sphere packing axis/.style={
        hide axis,
        clip=false,
        z buffer=sort,
        colormap={bluered}{
            rgb255(0cm)=(0,0,180); rgb255(1cm)=(0,255,255); rgb255(2cm)=(100,255,0);
            rgb255(3cm)=(255,255,0); rgb255(4cm)=(255,0,0); rgb255(5cm)=(128,0,0)},
        every axis plot/.style={
            mark=ball,
            only marks,
            scatter,
            point meta=explicit,
            mark size=\spherepackinginnerradius,
            scatter/use mapped color={ball color=mapped color},
            mark options={draw opacity=0},
        },
        % Redefine view={<azimuth>}{<elevation>} key
        view/.code 2 args={%
            % Set elevation and azimuth angles
            \pgfmathsetmacro\view@az{##1}
            \pgfmathsetmacro\view@el{##2}
            % Calculate projections of rotation matrix
            \pgfmathsetmacro\xvec@x{cos(\view@az)}
            \pgfmathsetmacro\xvec@y{-sin(\view@az)sin(\view@el)}
            \pgfmathsetmacro\yvec@x{sin(\view@az)}
            \pgfmathsetmacro\yvec@y{cos(\view@az)sin(\view@el)}
            \pgfmathsetmacro\zvec@x{0}
            \pgfmathsetmacro\zvec@y{cos(\view@el)}
            % Set base vectors
            \pgfkeysalso{
                x={(\xvec@x cm,\xvec@y cm)},
                y={(\yvec@x cm,\yvec@y cm)},
                z={(\zvec@x cm,\zvec@y cm)},
            }
        },
    }
}
\makeatother
\tikzset{
    outer sphere radius/.store in=\spherepackingouterradius,
    inner sphere radius/.store in=\spherepackinginnerradius,
}
\begin{document}
    \begin{tikzpicture}[outer sphere radius=1cm,inner sphere radius=0.4641cm]
        \begin{axis}[sphere packing axis,view={25}{30}]
            \addplot3[
                point meta=\thisrow{color},
                visualization depends on={\thisrow{size}*\spherepackinginnerradius \as \mysize},
                scatter/@pre marker code/.append style={
                    /tikz/mark size=\mysize}
            ] table {
                x       y       z    color size
                0       0.5359  0    0     0.7
                -0.4641 -0.2679 0    1     0.7
                0.4641  -0.2679 0    2     0.7
                0       0       0    3     0.5
        };
        \shade[ball color=black,opacity=0.1] (axis cs:0,0,0) circle (\spherepackingouterradius);
    \end{axis}
\end{tikzpicture}

\end{document}

From your last MWE, is there any way that when I add a ball at (0,0,0) to make it appear smaller? — Felix, Sep 14 '20 at 17:10
@Felix You can change the size of the inner spheres with the key inner sphere radius and the size of the outer sphere with outer sphere radius. The position of the spheres is irrelevant for the size. — Max, Sep 15 '20 at 06:19
Thanks, I already found that out after a while. But by changing inner sphere radius all balls changed size. I only wanted to edit the size of one single ball without changing the size of the other balls. Haven't found out how to do that though. — Felix, Sep 15 '20 at 08:35
@Felix Sorry for misunderstanding, see the latest edit. It seems like an obscure hack but actually it is also used in the manual. — Max, Sep 15 '20 at 09:21

Sphere packing in a sphere diagrams

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