Just for reference, here is the logic behind the numbering of the D18 Sphere:


The faces can be viewed as following 3 large circles (here in yellow, pink and blue):

The sum of the 8 numbers lying on any large circle is 76 (that is 8 times the average value, which is 9.5). There is nothing extraordinary here, this is only due to the fact that numbers on opposite faces sum to 19 (twice the average).
The most tricky property is that the sum of the 6 numbers surrounding any of the triangular rounded areas sum to 57 (six times 9.5).
There are 8 triangular areas that are in white in the scheme (including the white zone surrounding the 3 large circles). Each traingluar are is surrounded by 6 numbers: 3 on the edges (yellow, pink and blue) and 3 on the vertices (orange, purple and green). You can check: if you make the sum of any of such 6 numbers you'll find 57.

I guess there are several ways of numbering the D18 while complying with these properties.
But perhaps you can find another numbering with even more proprerties. Actually I tried also to have a nice symmetric numbering if you take the rest of the division by 3 of those numbers.

Let me know if you find something better! (but nothing to win this time )

• Attachment: D18numbering.jpg
  (Size: 29.45KB, Downloaded 950 time(s))

[Updated on: Sun, 10 July 2011 14:17 UTC]

Finally, the D32 has been numbered. It took a long time (it was designed in November 2010).


I decided this time to ignore the usual rule "numbers on opposite faces must sum to 33" and prefered the rule "difference between numbers on opposite faces must be 16". This allowed me to add interesting properties. In this case, you have faces surrounded by 6 faces and faces surrounded by 5 faces: if you take the number on a face surrounded by 5 faces and add to it the numbers on the 5 neighbours you will always got 99. This guarantees a certain fairness in the repartition of the numbers (basically, there is not an area of the die with more "value" than another).
As for the D33, there is a Regular Edition (diameter is 30mm, thickness is 1.5mm) and a Frosted Edition (diameter is 20 mm and thickness is 1.0 mm). Both version contain the support material.

[Updated on: Mon, 25 July 2011 06:45 UTC]

Thanks Orangery.
For your reference, here is the layout of the D32 numbering:

There is just half of the die since the numbers on opposite faces can be deduced easily (the difference between numbers on opposite faces is 16).
For instance, around the number 24 that is on a pentagonal face, you have 5 numbers on hexagonal faces: 6, 29, 32 and 3 (that are visible) and the number on the opposite face of the hexagonal face having the number 21, that is 21-16=5.
And you can check that 24+6+29+32+3+5=99.

• Attachment: D32numbering.jpg
  (Size: 53.49KB, Downloaded 893 time(s))

Still work in progress: the D22 Sphere.

It is constructed by intersecting a sphere with an unusual polyhedron. This polyhedron is based on a tetrahedron (and as the same symmetries as it): there are 3 faces for each vertex, one face by edge as well as the 4 original faces (3x4 + 6 + 4 = 22).

I have still to number it (as well as the D50)...

• Attachment: D22.jpg
  (Size: 13.38KB, Downloaded 918 time(s))

[Updated on: Sun, 07 August 2011 17:04 UTC]

Well, the shape before being intersected with a sphere look like a rhombus for the upper face, 4 identical quadrangles as lateral faces and two triangles for the lower faces (1+4+2 = 7).


I don't know if it's new, but I know I have never seen it before...
In my opinion, it is less interesting that the corresponding truncated sphere, though...
And I think i should be able to create a new D7 with a 3-fold symmetry (based on the sum 1+3+3 instead of 1+4+2).
Stay tuned!

[Updated on: Mon, 22 August 2011 15:36 UTC]

Here is a new addition to the Truncated Sphere dice family:
The D13 Sphere.

It has a 4-fold symmetry and is based on a 1+4+4+4 repartition of the faces (there is 1 single face surrounded by 4 faces at the bottom and 2 additional layers of 4 faces).
I had search along time how to make a D13 and there was no obvious solution: having this single face at one end allows new combinations, so expect more dice with odd number of faces to come...

• Attachment: D13_top.jpg
  (Size: 13.11KB, Downloaded 774 time(s))

The Alternative D24 is another D24 Sphere Dice, based on the Pentagonal Icositetrahedra.

This shape is more efficient than the classical D24 Sphere: for a same sphere radius, the faces are bigger. But the numbers are not located on faces (unless you read them on the bottom face, that is under the die).
The shape is chiral, so it comes in two flavours: right-handed and left-handed.
It's good to have choice...

• Attachment: D24Snub_R_Top.jpg
  (Size: 17.85KB, Downloaded 760 time(s))

[Updated on: Thu, 13 October 2011 20:44 UTC]

A new D7 with a 3-fold symmetry:

There is one face surrounded by a layer of 3 faces and completed by an additional layer of 3 other faces (1+3+3).

It is different from the one already mentionned (that has a 2-fold symmetry: 1+4+2).
I will post a picture to show the difference.

• Attachment: D7-3f_all.jpg
  (Size: 13.12KB, Downloaded 741 time(s))

Here is the pictures that shows both underlying polyhedra:


The first one (in red) this the one with a 2-fold symmetry (1 rhombus + 4 quads + 2 isosceles triangles) and the second (in yellow) the one with a 3 fols symmetry (1 equilateral triangle + 3 irregular pentagons + 3 quads)

I hope that it is now easier to understand the difference...

• Attachment: D7_compare.jpg
  (Size: 25.12KB, Downloaded 772 time(s))

[Updated on: Sun, 30 October 2011 14:16 UTC]

I used the same program I wrote for making the D13 to make the D19 Sphere:

Once again, the shape is inspired by the repulsive force polyhedra but is slightly different in order to maximize the diameter of the faces.
As for the D13 there is a face in one end that is "alone". The repartition of the faces by layers is 1+4+2+4+2+2+4.
There is only a 2-fold symmetry, but the underlying polyhedron is still attractive (even though its repulsive origin... ).

• Attachment: D19_poly.jpg
  (Size: 13.12KB, Downloaded 709 time(s))

I am currently working on two D21.
The red one has a 2-fold symmetry, the yellow one a 4-fold symmetry.

You can see the underlying polyhedra (wireframe and solid).
The interesting point is that, although different, these two arrangements of 21 circles around the sphere seem as efficient (the diameter of the circles is the same for a given sphere). This is quite unsual...
The repartition of the faces in layers of faces with the same angle is
- 2-fold: 2+4+2+4+2+2+4+1
- 4-fold: 4+4+4+4+4+1

• Attachment: D21-compare.jpg
  (Size: 27.55KB, Downloaded 676 time(s))

[Updated on: Tue, 01 November 2011 19:01 UTC]

Hi Orangery,

Yes, I think it is fair. It made of two pyramids with an equilateral triangular base stuck together. In this case, the height of the pyramid is caluclated to maximize the size of the circular faces obtained after the intersection with the sphere, but whatever the height, the double-pyramid is fair, so we can suppose that the truncated sphere is fair either. Of course this is true as long as the die does not stop on the rounded part...

For the 5-sided die, it is possible but it will not be a really surprising die: a triangular prism (once again with the appropriate height) intersecting a sphere: I looked for other shapes, without success.
I will make it very soon, I will finish the ones that are still unfinished (the two D21, the D22, and the D50) and I will probably make a pause with the dice in general and the truncated spheres in particular...

Cheers,

Magic

[EDIT] typos

[Updated on: Sat, 31 December 2011 15:24 UTC]

Hi Alea,

I am very pleased to meet you.

Actually, I know your website and I wanted to contact you because you mention that some shapes (like the Pentagonal Icositetrahedron) are not usable as die because basically the opposite of one face is not one face: this did not prevent me from positioning numbers where needed on the truncated spheres including out of the faces

I know the program from Bob (I did not contact him instead I was in touch with Martin Trump). Basicaly Repulsive Force polyhedra are great for inspiration, but I had to recalculate all the values probably because as you mentionned they use repulsion forces and I need packing. Generally the shapes are very close, only the values of the angles and sometime the shape of the faces are different (like irregular hexagons instead of triangles).

That's why I am very happy you wrote this new program.
On my side, the biggest difference I noted between the two methods was for the D13.
With Repulsion Force Polyedra, the D13 looks like a dodecahdron with an additional face replacing an edge (and repulsing all the other faces). This additional face is a rectangle, and the polyhedron has a 2 fold symmetry (identical when rotating it at 180°).
When I tried on my side to maximize the size of the face, this additional face turned square and the whole polyhedron had a 4 fold symmetry (identical when rotating it at 90°).

That's not what I obtain with your program... Perhaps I made a mistake in the design of my d13, when forcing the constraints.
What do you think?

Regards,

Magic
PS: is Alea your real first name? Alea sounds like aléa to me, the root of "aléatoire" meaning "random"...

[Updated on: Sat, 24 March 2012 17:11 UTC]

[Updated on: Sun, 25 March 2012 14:22 UTC]

Hey, I only thought of you the other day. It all seemed to kick off with those D32's I forced you to make for me . I had one re-numbered, (picture to follow) although I am now thinking of downsizing to something like a D14 (curiously).

Although I am unable to donate at this time, I will try to make people aware of the KickStarter Project as I have recently joined the Board Game Designers Forum. It seems that most of the members in the group use dice of some kind.

Best of luck with it.

Dave.

[Updated on: Wed, 25 July 2012 07:40 UTC]

Hi Dave,

If you can make people being aware of this project on game forums, I would really appreciate, thanks.
And you are right, before the D32, the only truncated sphere I had done was the D6, which is not really original
So, thank you also for giving me this tricky problem to solve!

Vincent aka Magic
Ps: waiting to see your re-numbered D32!

[Updated on: Wed, 25 July 2012 13:55 UTC]

Here is a picture of one of the higher scoring dice (D32). A friend of mine with a nice handwriting style was responsible for the numbers. I used small white self adhesive circles (yet to fall off). It is a vast improvement on the ones that I numbered.



By the way what is the circle diameter on the D14?

• Attachment: 008.JPG
  (Size: 117.18KB, Downloaded 203 time(s))