Although I don't have a complete solution,
https://openhome.cc/eGossip/OpenSCAD/TextSphere.html looks like it has most
of the missing pieces for you.

On Wed, Dec 29, 2021 at 8:21 AM Ari Diacou ari.diacou@gmail.com wrote:

The numbers start out with their normal vector pointing in the +z direction.  We need to rotate it so the normal is the avg(x) vector.  One way to do this is to tilt the letters away from the z-axis the right amount for the face by rotating around the y-axis by the z direction angle which is acos(z/norm) where norm is the length of the vector to the center of the face and z is the z-component of that vector.  Then rotate the result around the z-axis by the angle of the projection into the xy-plane from the x-axis.  Here, we have to compensate for the acos only returning results in the first 2 quadrants but looking at the y-component of the vector to the face.  Remembering that mathematical operations start at the inside and work outwards that gives the following for setting the texts.

for(i=[0:19])
translate(.5sizeavg(i)) color("yellow")
rotate([0,0,sign(avg(i)[1])acos(avg(i)[0]/sqrt((avg(i)[0])^2+(avg(i)[1])^2))])
rotate([0,acos(avg(i)[2]/norm(avg(i))),0])
text(str(i+1),size=sizeletter_ratio,valign="center",halign="center”);

This looks good except for 1 and 18.  1 looks like it’s tilted to match 2 and 18 looks like it’s tilted like 19.  So what have I forgotten?  These are both “special” because the y-component is zero and the x-component is negative.  The acos gives 0 <= theta <180.  I’ll fix this inelegantly with an if statement rather than thinking harder about rotations.

for(i=[0:19])
translate(.5sizeavg(i)) color("yellow")
if(avg(i)[1]==0 && avg(i)[0]<0){
rotate([0,-acos(avg(i)[2]/norm(avg(i))),0])
text(str(i+1),size=size*letter_ratio,valign="center",halign="center");
}
else{
rotate([0,0,sign(avg(i)[1])acos(avg(i)[0]/sqrt((avg(i)[0])^2+(avg(i)[1])^2))])
rotate([0,acos(avg(i)[2]/norm(avg(i))),0])
text(str(i+1),size=sizeletter_ratio,valign="center",halign="center");
}

Note that the text is 2D so won’t render.  Worse, if I extrude the text and union it with the polyhedron, the mesh isn’t closed and it still won’t render.  Maybe that has to be built of equilateral triangles moved into position the way the text is, but then we have to figure out how each triangle has to be rotated first.

Lenore

One Issue is, your icosahedron is inside out.
If you remove the  # and switch to combined view [F12], you see a lot
pink faces. These should be yellow.

Change faces to

faces=[
    [10,0,2],[0,8,2],[0,4,8],[0,6,4],[0,10,6],
    [10,2,7],[2,5,7],[2,8,5],[8,9,5],[8,4,9],
    [4,1,9],[6,1,4],[6,11,1],[6,10,11],[10,7,11],
    [3,7,5],[3,11,7],[3,1,11],[3,9,1],[3,5,9]];

and then your icosahedron is valid.

Am 29.12.21 um 17:20 schrieb Ari Diacou:

There is a simple solution using BOSL2:

include <BOSL2/std.scad>
include <BOSL2/polyhedra.scad>
difference(){
  regular_polyhedron("icosahedron", side=35);
  regular_polyhedron("icosahedron", side=35,draw=false)
    down(.3) linear_extrude(height=1)
      text(str($faceindex+1),halign="center",valign="center");
}

Am 29.12.21 um 17:20 schrieb Ari Diacou:

Thanks all.

There were 3 key parts. First like Lenore said, the arccosine has to be in
the y dimension, and has to be normalized by norm(avg(i)). The 2nd part was
that I had the wrong dimensions for the arctangent, which have to be
relative to y and x (not z and x). And the 3rd part is that both have to be
in the same rotation. So the rotation command is :
rotate([0,acos(avg(i)[2]/norm(avg(i))),atan2(avg(i)[1],avg(i)[0])])

JJVB's suggestion was correct, and I could remove the hull() that I had to
put in to make it manifold.

The final code is going up on thingiverse at:
https://www.thingiverse.com/thing:5179144

Thanks so much all.

size=25;
letter_ratio=.3;
depth_ratio=.03;
sides=[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,"\u263A"];
golden_ratio=(1+sqrt(5))/2+0;
g=golden_ratio;
//See:
https://en.wikipedia.org/wiki/Regular_icosahedron#Cartesian_coordinates
rect1=[[0,+1,+g],[0,+1,-g],[0,-1,+g],[0,-1,-g]];
rect2=[[+1,+g,0],[+1,-g,0],[-1,+g,0],[-1,-g,0]];
rect3=[[+g,0,+1],[+g,0,-1],[-g,0,+1],[-g,0,-1]];
points=concat(rect1, rect2, rect3); //echo(points);
faces=[
[10,0,2],[0,8,2],[0,4,8],[0,6,4],[0,10,6],
[10,2,7],[2,5,7],[2,8,5],[8,9,5],[8,4,9],
[4,1,9],[6,1,4],[6,11,1],[6,10,11],[10,7,11],
[3,7,5],[3,11,7],[3,1,11],[3,9,1],[3,5,9]]1;
difference(){
polyhedron(.5sizepoints,faces);
for(i=[0:19])
translate(.5size*avg(i)) color("yellow")

rotate([0,acos(avg(i)[2]/norm(avg(i))),atan2(avg(i)[1],avg(i)[0])])
translate([0,0,-depth_ratiosize+.05])
linear_extrude(depth_ratiosize)

text(str(sides[i]),size=size*letter_ratio,valign="center",halign="center");
}
function avg(x)
//calculates the vector of the center of the face x, based on the average
of the vertices
=points[faces[x][0]]/3
+points[faces[x][1]]/3
+points[faces[x][2]]/3;

On Thu, Dec 30, 2021 at 2:31 AM jjvb-openscad@bassklampfe.de wrote:

On 30.12.2021 08:26, jjvb-openscad@bassklampfe.de wrote:

There is also another problem in the original code. The code sequence
below places text objects in the face centers:

for(i=[0:19])
translate(.5sizeavg(i)) color("yellow")
//rotate([1acos(avg(i)[2]/size),0,0])
//rotate([0,0,90-atan2(avg(i)[2],avg(i)[0])])
text(str(i+1),size=sizeletter_ratio,valign="center",halign="center");

But: text is a 2D geometric object (ref
https://en.wikibooks.org/wiki/OpenSCAD_User_Manual/Text ), so it only
works in preview (a design mistake to draw 2d objects as 3d if you ask
me). If you try to render with F6, the text objects disappear because
they do not have a 3d representation. This can be fixed using extrusion,
i.e.

for(i=[0:19])
translate(.5sizeavg(i)) color("yellow")
//rotate([1acos(avg(i)[2]/size),0,0])
//rotate([0,0,90-atan2(avg(i)[2],avg(i)[0])])
linear_extrude(height=1.0)
text(str(i+1),size=sizeletter_ratio,valign="center",halign="center");

As for the original problem, you have to compute the homogeneous
transformation matrix of the text letters. The calculation of the face
centers represents only the 4th column in the 4x4 homogeneous
transformation matrix, the orientations are specified in columns 1,2 and 3.

However, for each face there is an infinite number of coplanar
orientations of a letter in a face, the specification is incomplete. You
have to add some other requirement to define the transformation. I.e.
where is the "top" of each letter supposed to be pointing?

Carsten Arnholm

1. in your sides array, consider re-ordering the face labels. Vi Hart has
   a video https://www.youtube.com/watch?v=-p7C5FrgAzU&t=960s on how the
   faces of a d8 should be labeled to model the d6 where opposite faces have
   the same sum.

2. On a Mac, I changed the sides array to:

sides=["𓄿","𓅔","𓅢","𓃌","𓃍","𓃎","𓆉", "𓃰", "𓃠", "𓃟", "𓁶", "𓂖",
"𓂀", "𓁿", "𓁰", "𓁀", "𓁥", "𓁔", "𓁁", "𓀨"];

and text line to:

   text(str(sides[i]), size=size*letter_ratio, font="Noto Sans Egyptian

Hieroglyphs", valign="center",halign="center");

to produce:

[image: Screen Shot 2021-12-30 at 9.02.06 AM.png]

On Thu, Dec 30, 2021 at 5:34 AM Ari Diacou ari.diacou@gmail.com wrote:

3. after rewatching that Vi Hart video, at least either 6 or 9 should be
   replaced by something, so they won't be confusable on the actual dice.

On Thu, Dec 30, 2021 at 9:03 AM David Phillip Oster <
davidphilliposter@gmail.com> wrote: