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# Truncated Spheres

Truncated Spheres [message #21786] Tue, 21 December 2010 21:46 UTC
Hi all,

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!

[Updated on: Tue, 21 December 2010 22:23 UTC]

Re: Truncated Spheres [message #21787 is a reply to message #21786 ] Tue, 21 December 2010 22:08 UTC
Nice. I had made a d9 using the same technique. I was going to un-"hide" it, but looking at it I realize that I just numbered the faces even though it doesn't land with a face up. Woops. Well, something else to fix. I'd like to make a bunch of other odd-sided dice ("odd" both in the sense of an unusual number for sides of a die and also some that are odd and not even), and do some randomness tests on them, see if they're close to fair.

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).
Re: Truncated Spheres [message #21788 is a reply to message #21787 ] Tue, 21 December 2010 22:32 UTC
Yes, I suspect that the 2 methods leads to the same result and our D9 are more or less the same (yours is actually "rounder" that mine, I think).
Feel free to show it in this thread if you wish, once it is finished.

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.
Re: Truncated Spheres [message #22067 is a reply to message #21788 ] Mon, 03 January 2011 13:52 UTC
Hi all,

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.

Re: Truncated Spheres [message #22071 is a reply to message #22067 ] Mon, 03 January 2011 13:59 UTC
I renumbered my d9, with numbers on the "corners", and un-hid it. Haven't bought a copy yet, so caveat emptor.

http://www.shapeways.com/model/189987/d9___nine_sided_die.ht ml
Re: Truncated Spheres [message #22228 is a reply to message #21786 ] Thu, 06 January 2011 15:06 UTC
Magic, the dice look great. I didn't think the D10 with four fold symmetry would work. It looks fair too! I have not seen anything like it before.

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.

[Updated on: Thu, 06 January 2011 15:18 UTC]

Re: Truncated Spheres [message #22248 is a reply to message #22228 ] Thu, 06 January 2011 21:32 UTC
Thanks Orangery. On my side, I have still to number the D32 and the D9.
I ordered all the other truncated sphere dice today. I will test their fairness as soon as possible.
Re: Truncated Spheres [message #22278 is a reply to message #21786 ] Fri, 07 January 2011 10:15 UTC
Personally I think the number on a die should be read face up whether on a face or an edge. I don't particularly like the 'normal' D4 where you have to read along the bottom.

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.
Re: Truncated Spheres [message #22294 is a reply to message #22278 ] Fri, 07 January 2011 18:02 UTC
Yes you are right, I can do a D4 (numbered on the round part), a D8 and a D20 to complete the set.
And perhaps also a D12 based on a rhombic dodecahedron.

So many things to design, so little time...
Re: Truncated Spheres [message #22588 is a reply to message #22294 ] Thu, 13 January 2011 22:13 UTC
Re: Truncated Spheres [message #24073 is a reply to message #22588 ] Sun, 20 February 2011 07:52 UTC
Another member of the Truncated Sphere family (asked by Orangery ): the D4 Sphere.

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.

• Attachment: D4Sphere.jpg

[Updated on: Sun, 20 February 2011 08:02 UTC]

So many things to design, so little time...
Re: Truncated Spheres [message #24074 is a reply to message #24073 ] Sun, 20 February 2011 10:01 UTC
This looks a fun die... nice work. Do you think you will get round to making up a 'set'?

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/
Re: Truncated Spheres [message #24078 is a reply to message #24074 ] Sun, 20 February 2011 11:17 UTC
Hi Orangery,

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.

[Updated on: Sun, 20 February 2011 11:18 UTC]

So many things to design, so little time...
Re: Truncated Spheres [message #24081 is a reply to message #24078 ] Sun, 20 February 2011 17:11 UTC

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!

• Attachment: D8Sphere.jpg

So many things to design, so little time...
Re: Truncated Spheres [message #24168 is a reply to message #24081 ] Wed, 23 February 2011 20:26 UTC
I wasn't sure what you meant about the numbers failing as I don't think you have mentioned having trouble with any of the other die. I'm okay with 2D stuff, not 3D,

Re: Truncated Spheres [message #24170 is a reply to message #24168 ] Wed, 23 February 2011 21:10 UTC
Carving numbers on dice, as any boolean operation (union, intersection, substration...) is a very... "random" operation. Sometimes it works, sometimes it fails...
It is (and has always been) a pain to do...

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...

[Updated on: Wed, 23 February 2011 21:11 UTC]

So many things to design, so little time...
Re: Truncated Spheres [message #24961 is a reply to message #21786 ] Fri, 18 March 2011 11:09 UTC
I have just received the three different versions of the D10's. They are brilliant! Every dice collector should have these. Now I just need to paint them without messing up.
Re: Truncated Spheres [message #24980 is a reply to message #24961 ] Fri, 18 March 2011 19:26 UTC
Thanks Orangery: I am very happy that you like them (and I can only agree with you: any dice collector should have them ).
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 33333th die.

• Attachment: D33Kevin.jpg

So many things to design, so little time...
Re: Truncated Spheres [message #24981 is a reply to message #21786 ] Fri, 18 March 2011 19:29 UTC
wouldn't that be 33333rd? Keep up the good work. These are definitely different.

I learned a long time ago the wisest thing I can do is be on my own side, be an advocate for myself and others like me. -Maya Angelou
Re: Truncated Spheres [message #24983 is a reply to message #24981 ] Fri, 18 March 2011 20:24 UTC
I should have pronounced it mentally to avoid my mistake...
Thank you for the kind words, Mike.
The D9 and D11 are already available. Other will probably come later.

So many things to design, so little time...
Re: Truncated Spheres [message #24986 is a reply to message #24983 ] Fri, 18 March 2011 21:19 UTC
Wow, that D33 is the weirdest die I have even seen. I thought my eyes were going funny when I first saw it. I also looked at the D9 and D11... they are also mad things. You seem to be on a roll .

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.

Re: Truncated Spheres [message #25023 is a reply to message #24986 ] Sun, 20 March 2011 16:26 UTC

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

So many things to design, so little time...
Re: Truncated Spheres [message #25890 is a reply to message #25023 ] Mon, 11 April 2011 05:57 UTC
The long awaited Truncated Sphere D20 is now available!

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).

• Attachment: D20.jpg

So many things to design, so little time...
Re: Truncated Spheres [message #25892 is a reply to message #25890 ] Mon, 11 April 2011 06:12 UTC
I also designed a D10 for percentage and a new version if the D6 (circle faces nearly in contact and wall thickness of 1.5 mm only).
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.)

So many things to design, so little time...
Re: Truncated Spheres [message #25895 is a reply to message #25890 ] Mon, 11 April 2011 06:53 UTC
I wish these dice could turn out exactly as the 3D CAD drawings (not a mark on them), .

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.
Re: Truncated Spheres [message #25941 is a reply to message #25895 ] Mon, 11 April 2011 21:15 UTC
I have seen outstanding results in metal after sanding and polishing, but my post-production skills stop at inking and varnishing plastic...

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?

[Updated on: Mon, 11 April 2011 21:19 UTC]

So many things to design, so little time...
Re: Truncated Spheres [message #25957 is a reply to message #25941 ] Tue, 12 April 2011 07:29 UTC
Re D9 family of dice, does this mean there is a D3 (1+1+1)?

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.
Re: Truncated Spheres [message #25970 is a reply to message #25895 ] Tue, 12 April 2011 15:07 UTC
Have you looked at circle packings on the sphere? That is, finding the arrangement of n identical non-overlapping circles on the sphere so that the circles have maximum radius.

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
Re: Truncated Spheres [message #26010 is a reply to message #25970 ] Wed, 13 April 2011 10:35 UTC
@Orangery No, unfortunately, n cannot be 1 or 2. Or more precisely when it is 2 the polyhedron is degenerated.

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 45Â° 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.

[Updated on: Wed, 13 April 2011 10:36 UTC]

So many things to design, so little time...
Re: Truncated Spheres [message #26143 is a reply to message #26010 ] Sun, 17 April 2011 08:38 UTC
As announced in the previous message, I am glad to present the Alternative D8 Sphere.
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!

So many things to design, so little time...
Re: Truncated Spheres [message #26144 is a reply to message #26143 ] Sun, 17 April 2011 13:30 UTC
I like the look of this one. Could this configuration work for a D6 (instead of a cube?).
Re: Truncated Spheres [message #26145 is a reply to message #21786 ] Sun, 17 April 2011 13:42 UTC
Very clever

http://www.3Dizingof.com
Re: Truncated Spheres [message #26147 is a reply to message #26145 ] Sun, 17 April 2011 16:49 UTC
@Orangery: I knew you would like it: it is like the D10 with 4-fold symmetry but without the top and bottom faces (and the angles are different). Unfortunately for the D6, the trapezohedron that maximize the diameter of the circles is precisely... the cube! So, this method gives nothing new. For n=5, I think you would get the D10 with 5-fold symmetry and for n=6 the rounded part would probably become too large (larger than a face).

@Dizingof: thanks and welcome back!

So many things to design, so little time...
Re: Truncated Spheres [message #26148 is a reply to message #21786 ] Sun, 17 April 2011 16:55 UTC
Thanks Magic, good to be back

http://www.3Dizingof.com
Re: Truncated Spheres [message #26802 is a reply to message #26148 ] Sun, 01 May 2011 06:13 UTC
I am glad to introduce the D15 Sphere:

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).

• Attachment: D15.jpg

So many things to design, so little time...
Re: Truncated Spheres [message #26803 is a reply to message #26802 ] Sun, 01 May 2011 06:35 UTC
This a a die I designed a long time ago and that I forgot numbering.
So, after a long time on the drawing table, here is the D18 Sphere:

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.

• Attachment: D18.jpg

[Updated on: Sun, 01 May 2011 08:49 UTC]

So many things to design, so little time...
Re: Truncated Spheres [message #26804 is a reply to message #26803 ] Sun, 01 May 2011 08:44 UTC
I wasn't expecting a D18. I like the way it could be also set up to be a D9, D6 and D3... very interesting!

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?

Re: Truncated Spheres [message #26805 is a reply to message #26804 ] Sun, 01 May 2011 09:10 UTC
Ah, yes, i understand what you mean. If you stick together two pyramids you obtain a "dipyramid" like this one from Friz. If you squash it and intersect it with a sphere you will indeed obtain a new D6. I had no plan to design this one though, because it is less optimal than the cube: for a given sphere radius, the circles would be smaller. So it is not a solution of the repulsion force polyhedra. But not all my Truncated Dice have as an underlying polyhedron the dual of a repulsion force polyhedron, so perhaps one day, you will see it coming...

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!

So many things to design, so little time...
Re: Truncated Spheres [message #28404 is a reply to message #26805 ] Tue, 31 May 2011 08:27 UTC
I have sent along a couple of pictures of the D10's that I bought from you (3 sorts). The black and white one turned out pretty good but I hope to redo the other two (I didn't have a fine enough pen for the brown die).

• Attachment: Dice 10.JPG

Re: Truncated Spheres [message #28407 is a reply to message #28404 ] Tue, 31 May 2011 08:40 UTC
As for the D32 football dice, well I must admit I got in a bit of a muddle with the numbering. Some are not even finished. I tried using rubber number stamps for one of them but the white ink did not cover very well and it was difficult positioning numbers .

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

• Attachment: Dice 32.JPG