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Dear Sander,

brilliant - as always! I already knew the 'chaos game', but I was not aware of the fact that it works with simple points - I would have tried something using a sub-triangular mapping ... Thank you very much for sharing!

I could not resist trying your method in 3D! Unfortunately this is much less spectacular ...

ClearAll["Global`*"]
sequence[n_, m_] := RandomChoice[Range[n], m]

fromSphericalCoordinates[{r_, \[Theta]_, \[Phi]_}] := 
  FromSphericalCoordinates[{r, \[Theta], \[Phi]}];
fromSphericalCoordinates[{1, 0., _}] = {0, 0, 1};

sphereDist[sph1_, sph2_] := 
 EuclideanDistance[fromSphericalCoordinates[Prepend[sph1, 1]], 
  fromSphericalCoordinates[Prepend[sph2, 1]]]

(* calculates n positions on a 3D-sphere with minimal 'energy' when \
repulsive  *)

sphereAewquiDist[n_Integer /; n > 2] := 
 Module[{vars, vpairs, energy, constr, sol},
  vars = (({Subscript[\[Theta], #1], Subscript[\[Phi], #1]} &) /@ 
     Range[n]); vpairs = Subsets[vars, {2}]; 
  energy = Total[1./Apply[sphereDist, vpairs, {1}]];
  constr = 
   Flatten[Apply[{0 <= #1 <= Pi, -Pi <= #2 <= Pi} &, vars, {1}]];
  sol = Last[NMinimize[{energy, constr}, constr[[All, 2]]]]; 
  Chop[fromSphericalCoordinates /@ (vars /. 
       sol /. {\[Theta]_, \[Phi]_} :> {1, \[Theta], \[Phi]})]]

gr = Table[
   sPts = sphereAewquiDist[n];
   chmLines = MeshPrimitives[ConvexHullMesh[sPts], 1];
   pts = FoldList[(#1 + sPts[[#2]])/2 &, RandomChoice[sPts], 
     sequence[n, 500000]];
   {n, Graphics3D[{Black, Thick, chmLines, Blue, PointSize[0.0001], 
      Point[pts]}, AspectRatio -> Automatic, ImageSize -> 600, 
     SphericalRegion -> False, Boxed -> False]},
   {n, 4, 6}];

Grid[gr, Frame -> All]

The result:

Best regards -- Henrik

@Henrik Schachner Since it is now public knowledge: https://reference.wolfram.com/language/ref/SpherePoints.html SpherePoints will be introduced in version 11.1 which is similar to your sphereAewquiDist function, with some hard-coded position for 'small' n. and for large 'n' a fast procedure.

Just as a minor addendum:

From the documentation:

SpherePoints[n] gives the positions of n uniformly distributed points on the surface of a unit sphere.

At least for certain small numbers of points I would not regard this function as "working". See the following examples using 10, 11 and 13 points (left: result of SpherePoints - right: more like expected result):

This is somewhat disappointing. My code for the above:

ClearAll["Global`*"]

fromSphericalCoordinates[{r_, \[Theta]_, \[Phi]_}] := 
  FromSphericalCoordinates[{r, \[Theta], \[Phi]}];
fromSphericalCoordinates[{1, 0., _}] = {0, 0, 1};
sphereDist[sph1_, sph2_] := 
 EuclideanDistance[fromSphericalCoordinates[Prepend[sph1, 1]], 
  fromSphericalCoordinates[Prepend[sph2, 1]]]

mySpherePoints[n_Integer /; n > 2] := 
 Module[{vars, vpairs, energy, constr, sol}, 
  vars = (({Subscript[\[Theta], #1], Subscript[\[Phi], #1]} &) /@ 
     Range[n]); vpairs = Subsets[vars, {2}];
  energy = Total[1./Apply[sphereDist, vpairs, {1}]];
  constr = 
   Flatten[Apply[{0 <= #1 <= Pi, -Pi <= #2 <= Pi} &, vars, {1}]];
  sol = Last[NMinimize[{energy, constr}, constr[[All, 2]]]];
  Chop[fromSphericalCoordinates /@ (vars /. 
       sol /. {\[Theta]_, \[Phi]_} :> {1, \[Theta], \[Phi]})]]

GraphicsGrid[(Show[
       Graphics3D[{Opacity[.3], Sphere[]}, SphericalRegion -> True], 
       ConvexHullMesh[#]] & /@ {SpherePoints[#], 
      mySpherePoints[#]}) & /@ {10, 11, 13}, Frame -> All, 
 ImageSize -> 600]

Regards -- Henrik

Very nice, in conception, code, and artistry. One comment: the 4-vertex (square) case that alone seems so boring is probably quite important. The fact that it does not give any "nice" fractal is, I suspect, relevant to why the Chaos Game Representation is so useful for making images from nucleotide sequences. Depending on level of pixelization these have a sort of n-gram "memory", hence (I think) implicitly encode some of the local sequence structure. This does not work for other vertex counts and that is perhaps related to why they give pictures that are nicer for other purposes (fractals, material design maybe, art definitely, ...)

On a different note, I like the phrase "three sided polygon". I'm thinking we could shorten that, perhaps to "trigon". If only the language had a word for this... (look, it was a great post, I gotta be allowed to make light of something).

Here some more hi-resolution examples: