After working with Orangery on this D32 (not numbered yet)

I decided to go on the idea of making dice from truncated spheres.

As you know, a round D6 can be defined as the intersection of a cube and a sphere:

Using the same method, I also designed a rounded D12 (intersection of a dodecahedron and a sphere).

As I originally wanted to make a D10, I put back two opposite portions of sphere to the D12 to get this round D10 with a 5-fold symmetry:

Then, instead of using the original portions of sphere, I replaced them by cones (tangent to the sphere) to get this pointed D10:

Another D10 with a 4-fold symmetry can be obtained by using a different underlying polygon with 10 faces (2 squares and 8 non-regular pentagons):

Note the strange way I had to numbered it: this is due to the fact that a face is not always opposed to another face but to an edge.

Finally, playing with truncated spheres and cones, I found this strange shape that is mid-way between a sphere and a cube.

Don't know exactly what to do with it (not a die anyway because it can stop on a "edge", even though it has no edge ).

Currently I am working on a 3rd D10 with a 3-fold symmetry (not finished yet) and also on a D9... So, more news will come soon!

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It looks like your method of intersecting spheres with dual polyhedra yields the same results as the one that I tried, with slicing identical circles off a sphere at the appropriate points (but yours is probably simpler).]]>

For the numbering, I thought doing it as the D10 4-fold (with numbers in the edges) but another method would be to put the numbers near the bottom, on the adjacent faces, as in a standard D4.]]>

Here is my last family of truncated sphere D10. After 5-fold and 4-fold symmetry, these are the 3-fold D10s.

There are two models:

- the rounded D10 with 3-fold symmetry

- the pointed D10 with 3-fold symmetry

Both are base on a D14 non-regular cuboctahedron. This truncated sphere D14 numbered twice from 1 to 7 is also available.

To obtain the D10s, I add 4 portions of sphere (rounded version) or 4 tangent cones (pointed version) to the D14 to "cancel" 4 out of the 14 faces. So the symmetry of those D10s is the same as the symmetry of a tetrahedron (regular D4).

Check this picture to see them or click the links above.

Next steps will be having the D14 with twice the days of the week, and finishing numbering the D32.]]>

http://www.shapeways.com/model/189987/d9___nine_sided_die.ht ml]]>

Look forward to seeing the D9.

Unfortunately I have still yet to decorate the D32's that you made for me as I have been bogged down with other things.

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I like all your D10's except the ones with cones (I'm funny like that ). My favourite is the one that is made up from 2 squares and 8 non-regular pentagons as it has the optimal surface area to land on and as far as I know, is unique.

I think It would be nice (as you have come this far) to finish off the set with a D4, D8 and D20 as an alternative to the usual D&D dice and give them a posh name.

The D9, D14 and D32 could be available as add-ons. I hope to buy most of these dice this year once some spare cash comes available.]]>

So many things to design, so little time...

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Because the numbers are written exactly on the vertices, it is much easier to read the result than on traditional D4 (tetrahedron). And it should also roll better.

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By the way have you tried 'Uni Posca' pens. They are great for colouring the dice. I tried them out on the D32's you made (pictures to follow). I painted the numbers in by hand (still have three to do). The colours look good (they still need to be varnished) but I want to try and improve on the numbering as I like things to look consistent (may try rubber stamp blocks).

Numbering of a normal D32 from 0-31 might be used for days of the month. If the die lands on a zero the player could choose any day of his/her choice (or throw again). Just a thought.

I started on my web site... http://the-orangery.weebly.com/]]>

You mean a set including the still missing D8 and D20?

Well, I am in the process of beginning the D8, but it will take some time. In the meantime, if you wish a special set of existing dice, just PM me.

The numbering of the D32 is a nightmare for me: each time I beging to look into it, I see new problem. I have to find the exact angles between the faces and, say, the horizontal plane if I want to do it proprerly (well, basically I need some courage ).

Numbering from 1 to 31 with an extra symbol (like a star) to represent either 0 or 32 or "replay" can be an idea.

I didn't try Uni Posca pens, but I heard about them in this forum. I have to see if they are available in France.

For now, I let Shapeways dye the dice and for inking the numbers, I use "Hybrid Gel Grip Mettalic", some gel ink rollers with 0.8mm balls from the Japaneese Pentel. The white in particular works very great. Then I finish them with some coats of acrylic varnish.

Good luck with your website!]]>

It was quite easy to do the first time, but there were not enough polygons in the rounded part. Adding more polygons made the process of "extruding" (should I say intruding?) the numbers fail quite often...

But finally, I did it! ]]>

Anyway, another great dice (over 50 downloads). Will you be getting these made (D4, D8)?

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I will probably order the D4 and the D8 but not immediately: as an anti-addiction policy, I do not place a new order as long as the current one has not been delivered. And my latest order has not arrived yet and does not include dice, sorry...

PS: the downloads are just people reading the message, not peple ordering the die, unfortunately... ]]>

After dying or painting them, I recommend to ink the numbers with some gel ink pen. Gel is important so that the ink stay in the indentation instead of diffusing in the neighbourhood... And then some varnish...

Let us know the outcome.

I take advantage of this post to show you a die I did for Kevin Cook: this is a D33 to celebrate his 33333

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Thank you for the kind words, Mike.

The D9 and D11 are already available. Other will probably come later. ]]>

I painted the D10's but I got in a mess with them. I didn't have a gel pen. Will try a different technique.

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Thanks again.

If you like my designs (in particular the ones you received), do not hesitate to rate them!

Good luck with your painting!]]>

This is the intersection of an icosahedron with a sphere. As you can see, due to the fact that a vertex is surrounded by 5 triangles, the rounded area is larger than in other Truncated Sphere dice. So the faces are smaller and the numbers too.

The distance from one face to the opposite one is 2 cm. This die is hollow, with a thickness of 1.5 mm (suitable for Alumide for instance).]]>

As a consequence the Truncated Sphere Dice Set is now available !

This set is composed of 7 regular Truncated Sphere Dice:

- D4

- D6

- D8

- D10

- D%

- D12

- D20

Ideal for dice collectors and RPG players!

I will probably create later another set for unusual Truncated Sphere dice (D9, D11, D17 etc.)]]>

I think there is only the Rhombic Dodecahedron left. I'm assuming it will have different properties than the regular shape?

The two D10's you made (not the one with the 14 sides) roll really well and for about the same length of time (on average) even though they are constructed quite differently.

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The Rhombic Dodecahedron is definitvely different from the regular Dodecahdron and will come with the Rhombic Triacontahdron (D30).

By the way, the Rhombic Dodecahedron is from the same family as the D9 and the future D15 (because 9=3+3+3, 12=4+4+4 and 15=5+5+5)

I plan also to make a different D8.

Concerning the D10s, you are speaking of the D10 with 4-fold symmetry (polyhedron with squares and pentagons) and the D10 with 5-fold symmetry (derived from a regular dodecahedron), right?

Is there a problem with the 3rd one (D10 with 3-fold symmetry, derived from the D14) or is is just that you did not test it?]]>

Going off on a tangent, I thought I had read somewhere that the D11 was constructed using the '1+n+n+n+1 formula (1+3+3+3+1). Same with the D17 (1+5+5+5+1). Does this mean that a D5 might be possible (1+1+1+1+1) and also the 'different' D8 you mention (1+2+2+2+1)?

As for the D10 (fourteen flat sides), there is no problem with it. I didn't mention it simply because it rolls for longer than the other two (as expected). My favourite D10 is the one made up of squares and pentagons. ]]>

This website looks like it has some data you might be able to use (although not much in the way of pictures):

http://www.buddenbooks.com/jb/pack/sphere/intro.htm]]>

For instance, 1+3+3+3+1 implies that the face of the poles are triangles (because n=3). If you put n=2 then the poles degenerate into simple edges (a polygon with two vertices is a "double-edge") and what you actually get is 2+2+2 which is a simple cube (D6)...

Same thing for 1+n+n+1 (n=4 gives your favorite D10 )

For n=3 you get the regular octahedron (D8) and for n=2 you can consider that the poles degenerate into perpendicular edges and you obtain the regular tetrahedron (D4).

The new D8 will actually be a 4+4, like the regular octahedron, but with a twist of 45° on the four lower faces.

I also have some ideas for a D3...

@Henryseg Yes, I had a look at them and at the repulsion force polyhedra of Martin Trump, but basically I found it difficult to take advantage of this information (maximum angle for the cones or coordinates of the center of the points). So, sometime I use them to fin a good approximate position for the face but then I try to solve equations to maximize the radius of the circles.]]>

It is not based on the regular octahedron but on an trapezohedron ("antidiamant" in French) that are dual of the antiprisms.

It is a co-creator, so you can choose your numbering.

With numbers on edges, the number will be on top of the die, when it lies on a horizontal plane. With the numbers on the bottom faces, you will have to look under the die to find out the result. Choose no numbers if you prefer contemplating the shape or if you would like to take care of the numbering by yourself!]]>

@Dizingof: thanks and welcome back!]]>

3 rows of 5 faces (5 at each tropic, 5 at the equator).

The numbering follow the usual rule (as for the D9 for instance): the sum of each row is constant (it's 40 in this case), the sum of two numbers of the tropic symmetic to the equator plan, and the sum of two numbers of the equator circle symmetric to the "middle number" (8 here) is constant (it's16 in this case). Note that there must be an exception (all the numbers cannot go by pair since there is an odd number of face ) and that is 8: 8 cannot have a symmetric since the sum of two symmetric numbers is 16 (and only 8+8 = 16).]]>

As the number of faces is even, each number is exactly on a face (sometimes, these dice look very usual ).

Instead, the numbering is very interesting in this case.

Opposite faces numbers sum to 19, obviously. As a consequence any group of 8 numbers of a large diameter sum to 76. A more unexpected consequence is that any group of 5 numbers follows this strange rule: if you sum the double of the central number to the 4 other surrounding numbers, you obtain 57.

Example in the rendering image:

...4..

9 18 5

...3..

2x18 + 4 + 9 + 5 + 3 = 57

But the most interesting property in this particular numbering is that the sum of the six numbers surrounding a spherical zone is always 57.

Example here: 18 + 5 + 7 + 11 +13 + 3 = 57

I had to use a computer to find the good combination (there are a lot of them, by the way).

I could number this die from 1 to 9 twice (I found a way of doing it where the 8 numbers of the large diameters always sum to 40) or from 1 to 6 three time or even from 1 to 3 six times. Let me know if you want some particular numbering.]]>

Going back to a previous post, I now understand how a cube is made up of six trapezohedrons... but what happens if you stick two (squashed) tetrahedrons together?

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I am not forgetting the Rhombic Dodecahdron and the Triacontahedron, but I am currently working on a D24 and some other secret stuff...

Stay tuned! ]]>

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I used 'Uni Posca' pens for the base colours and the numbering (obviously hand written!).

]]>

Good idea coloring the flat and the rounded part in different colors for the black and white D10.

For the numbers, I use a gel ink pen to ink them. The gel stay in the numbers without expanding out of the indentation. The problem is finding a pen thin enough to reach the inside of the numbers...

For the D32, I love the vivid colors of Uni Posca. But I am curious to know which kind of numbering you are using (0-1-2-3, 1-2-3-5?)...]]>

This guy is making similar dice to yours... http://www.shapeways.com/shops/propmodule?sg49347%5Bpage%5D= 27#sg49347. The circular faces on his have less surface area but are nice all the same.]]>

You can PM me the charts for my own footballistic culture

And yes, the Truncated Sphere style is use by several dice designers at Shapeways (actually, we often discuss together) and elsewhere. You can see most of them here: http://www.dicecollector.com/THE_DICE_THEME_SPHERICAL.html

Cheers,

Magic.

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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 )

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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.

]]>

A lot of thought has gone into numbering all of your dice. It is nice when the maths works out just as you want it.

Thank you again for helping out.]]>

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.

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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)...

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And so is this new D16:

This one has already been numbered and thus is available for sale.]]>

For a long time, I though no other interesting 7-sided polyhedron did exist, but after a lot of research with pen and paper and some calculations on Excel, I found out a polyhedron that leads to this Truncated Sphere D7:

I am more than happy with it. I think it has unexpected "partial" symmetries. It is difficult to explain, but as soon as I get a printed version, I will try to show you.

So, it you are bored with the pentagonal prism, this D7 is for you!

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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!

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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...

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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...

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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.]]>

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...

]]>

I did not try it before because the equations are difficult to write and most of all difficult to solve. So I had to do a program to find approximations.

But the efforts are rewarding once you get a nice shape...

By the way both models can be seen as a tranformation of a cube:

- the first as a cube where one of the square face is cut into two in diagonal

- the second as a cube with one vertex cut to get an extra triangular face

Of course, this is just an initial construction: the vertices have to be moved to maximize the radii of the circles than can hold in the faces.]]>

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... ).

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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

]]>

Instead of being based on a cube, it is based on a double pyramid.

As a consequence, the numbers are not positionned on the faces but rather on the edges.

If you remove the faces, this model leads the the D6 Shell.

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I notice that you don't have a five sided die... impossible?

Season's greetings.]]>

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]]>

Happy New Year!

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I like the idea of making dice out of truncated spheres. I would like to have dice that are „fair" or at least „optimum" in some sense.

In the strict sense, only isohedral dice are fair (platonic, catalan, dipyramids, trapeozohedra and some strange cousins, http://www.aleakybos.ch/sha.htm ). If a sphere is intersected with an isohedral polyhedron, I would call it fair, neglecting the rare case when such a die would stop on a curved part of the surface.

"Optimum" truncated spherical dice can be constructed by distributing the n circles on the sphere in some optimum way. There are plenty of choices, this is a good overview: http://www.maths.unsw.edu.au/about/distributing-points-spher e

Some of Magic's designs are based repulsion force polyhedra http://members.ozemail.com.au/~llan/mpol.html . In my view this corresponds to the optimization criterion "minimal potential energy", best known solutions are here: http://www2.research.att.com/~njas/electrons/dim3/

Magic writes that he optimized some of his designs by maximizing the diameter of the circles. I think that this is the "packing" criterion, best known solutions are here: http://www2.research.att.com/~njas/packings/dim3/

I got in touch with Bob who wrote the original visualization applet for repulsion force polyhedra and got his permission to modify it. I included various optimization criteria, the results of my endeavor are visible here: http://www.aleakybos.ch/sph_codes.htm

At this time the computation of the dual for some larger odd numbers is still buggy, but I will work on it

Comments are welcome. Play & enjoy!

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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"...

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I like a lot the shape obtained with 8 point minimizing covering. I should be able to make a new die from this one.

And the D10 obtained by packing is different from any of the ones I have.

It seems that there are a lot of new shapes to explore...

Thank you for that!

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I like the way you put numbers on dice whose faces are not parallel, like the D24 you mentioned. Could you use the same approach for numbering the isohedral D24 Pentagonal Icositetrahedron and the D60 Pentagonal Hexecontahedron (I mean the regular shapes, not the truncated spheres), That would be unique.

Comparing the various optimization criteria with my applet, I observed that the optimum solutions are identical for 4,5,6 points.

For 7 points you get a 1+3+3 shape with 3-fold symmetry with packing, and a 5-prism with all the others.

For 8 points packing and energy yield a 4-trapezohedron (albeit with different edge lengths!), covering and volume yield a 2+4+2 shape with 2-fold symmetry, again with different edge lengths.

For 9 points the results are:

packing: 1+4+4 shape with 2-fold symmetry

others: 3+3+3, 3-fold symmetry (different but pretty close edge lengths)

10 points:

packing: 2+6+2 with 2-fold symmetry

others: 1+4+4+1 with 4-fold symmetry (different but pretty close edge lengths)

For 11 and 13 points things start getting interesting

For 12 points you get the dodecahedron for all 4 criteria.

For 24 points packing yields the Pentagonal Icositetrahedron (dual to the sub cube)!

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Very often

- I agree with your results for 4, 5, 6, 7 and 12.

- For 8 as I said the

- For 9 if you turn your 1-4-4 shape (which I would rather call 1-4-2-4) in another direction, you will find the 3-3-3 again (with different sizes).

- 10 leads to a

- 11 is hard to understand.

- 13 leads to 1-4-4-4 (the one I did) for

- 14 is

- 15 is very interesting. It always leads to a 3-3-3-3-3 shape but only

I still have to look at the others, but i can say that this exploration is very exciting.

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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.]]>

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!

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

Sometimes I use varnish glue to coat the dice. I wonder if it could make the stickers more durable (unsure how the ink would behave though)...

For the D14 (which is 2cm high) the circles are circa 1cm, a little bit more than your D32 (which is 3cm high), so the stickers should fit ]]>