G'Day Folks,

FYI, I completed the 9-sided "dice" (it isn't a good dice though). :)

I found a website with general information about spheres, including lists on coords for "n points on sphere" (http://neilsloane.com/packings/). This gave me 9 (x,y,z) points. I then used Casio's online (x,y,z)->spherical calculator (https://keisan.casio.com/exec/system/1359533867) to convert the points to (lat,long). :)

The image below is the dice as viewed in open-scad.

The following code is for Open-SCAD; hopefully you'll be able to adapt it to other applications if you need to...

long = [0.0000, 3.5638, 41.3251, 93.5638, 145.8026, 183.5638, 221.3251, 273.5638, 325.8026];
lat  = [0.0000, 131.8103, 70.5288, 120.0000, 70.5288, 131.8103, 70.5288, 120.0000, 70.5288];

difference() {
  sphere(10, $fn=100);
  for (k=[0:8])
    rotate(long[k],[0,0,1]) rotate(lat[k],[1,0,0]) {
      translate([0,0,13.3]) cube([12,12,10],center=true);
      translate([-3.75,-4.5,7.805]) linear_extrude(1) text(str(k+1), size=9);
    }
}

To understand this code, I suggest looking at something like:

for (k=[0:4])
    rotate(long[k],[0,0,1]) rotate(lat[k],[1,0,0]) {
      translate([0,0,13.3]) cube([12,12,10],center=true);
      translate([-3.75,-4.5,7.805]) linear_extrude(1) text(str(k+1), size=9);
    }

This should give you 5 cubes (and extruded number text) rotated strangely around an empty middle. :)

cheers Steve

Yes, you are right! I was fooled by checking the genus of the first of the above example of dices as a central Voronoi cell:

genus[dice_] := Module[{faces, edges, points},
  faces = Length[MeshPrimitives[dice, 2]];
  edges = Length[MeshPrimitives[dice, 1]];
  points = Length[MeshPrimitives[dice, 0]];
  1 - (faces - edges + points)/2];

  genus[vc[{0., 0., 0.}]]
  (* Out:   -(19/2)  *)

This means that something else must be wrong here. In the second example (using mySpherePoints) we get 0 for the genus as the correct result.

Hello Hans,

Could you check with Spherepoints ...

using SpherePoints one gets for 11 points:

which is not really satisfying! I would very much like to see an improved version of SpherePoints in MM v12 !!! My little routine above gives in this case:

which seems to be more like what one expects.

But I am afraid your really interesting idea does not work in this case:

dice[p_] := dice[p] = ConvexHullMesh[mySpherePoints[p]];
faceNumber[dice_] := Length@MeshPrimitives[dice, 2];
faceNumber[dice[11]]
(* Out:  18   *)

I guess the reason for this is that the Euler formula holds only in case of triangulation, i.e. if the surface consists of triangles only.

Best regards, liebe Gruesse -- Henrik

Hi Peter,

... yes, a 10sided dice is doable:

dice10 = ConvexHullMesh[SpherePoints[7]];
polygs = MeshPrimitives[dice10, 2];
{dice10, Grid[{Length @@ #, Area[#]} & /@ polygs, Frame -> All]}

Regards -- Henrik

Addendum:

as a first criterion on the "quality" of such a dice might serve the identity of the area of its faces. If we try this with the above, we get:

polygs = MeshPrimitives[dice9, 2];
Grid[{Length @@ #, Area[#]} & /@ polygs, Frame -> All]

which means that the 4sided faces are probably less likely. This is somewhat disappointing.

Somewhere along the post on The Chaos Game by @Sander Huisman differences were found between SpherePoints and a different function (mySpherePoints) based on a simple idea of minimalization. Using this we get:

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]})]]

pts = Append[mySpherePoints[9], {0, 0, 0}];    
vc = VoronoiCells[pts];    
dice9 = vc[{0., 0., 0.}];    
polygs = MeshPrimitives[dice9, 2];    
Grid[{Length @@ #, Area[#]} & /@ polygs, Frame -> All]

which appears to be somewhat better.

Hi Henrik,

This has 14 faces though…

MeshPrimitives[ConvexHullMesh[SpherePoints[9]], 2] // Length

not the requested 9…