Re: Truncated Spheres [message #28433 is a reply to message #28407 ] Tue, 31 May 2011 18:10 UTC 


Happy to see the colored dice.
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 (0123, 1235?)...
So many things to design, so little time...



Re: Truncated Spheres [message #28475 is a reply to message #28433 ] Wed, 01 June 2011 15:30 UTC 


The football dice are numbered in such a way that they should produce realistic scores. There is also a chart you have to refer to determine which dice are used for a certain match. I can PM you with the details if you like. I made up a computer version of it some years ago which I'll dig out and send you although it was based on 14 sided dice and not 32.
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.





Re: Truncated Spheres [message #31456 is a reply to message #30551 ] Sun, 24 July 2011 17:19 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] So many things to design, so little time...



Re: Truncated Spheres [message #31464 is a reply to message #21786 ] Mon, 25 July 2011 06:54 UTC 


Looks great!
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.



Re: Truncated Spheres [message #31487 is a reply to message #31464 ] Mon, 25 July 2011 17:53 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 2116=5.
And you can check that 24+6+29+32+3+5=99.
So many things to design, so little time...



Re: Truncated Spheres [message #32229 is a reply to message #31487 ] Sun, 07 August 2011 16:59 UTC 


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 1315 time(s))
[Updated on: Sun, 07 August 2011 17:04 UTC] So many things to design, so little time...



Re: Truncated Spheres [message #32326 is a reply to message #32229 ] Tue, 09 August 2011 07:27 UTC 


Impressive... do these shapes have names?




Re: Truncated Spheres [message #33025 is a reply to message #32444 ] Sun, 21 August 2011 19:26 UTC 


I was looking for a nice design for a D7. Generally the Pentagonal Prism is used for this number of faces. Unfortunately, I don't like the Truncated Sphere that results from this polyhedron. So I was looking for something different.
For a long time, I though no other interesting 7sided 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!
So many things to design, so little time...



Re: Truncated Spheres [message #33068 is a reply to message #33025 ] Mon, 22 August 2011 13:15 UTC 


Do you think it is a new discovery? What did the shape look like before dissection? I'll be rating it!




Re: Truncated Spheres [message #33326 is a reply to message #33073 ] Thu, 25 August 2011 20:44 UTC 


Well I'm not sure how you found this shape but I'm glad you did! It has to be new.




Re: Truncated Spheres [message #36562 is a reply to message #36561 ] Thu, 13 October 2011 06:02 UTC 


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: righthanded and lefthanded.
It's good to have choice...
[Updated on: Thu, 13 October 2011 20:44 UTC] So many things to design, so little time...



Re: Truncated Spheres [message #36568 is a reply to message #36562 ] Thu, 13 October 2011 08:08 UTC 


Left and right handed dice  that's a new one on me... I think you should show them on mathpuzzle.com.



Re: Truncated Spheres [message #36608 is a reply to message #36568 ] Thu, 13 October 2011 20:07 UTC 


Ahahah! Lefthanded and right handed players can find the D24 that suit them properly, now...
I will check if I can propose something on mathpuzzle.com (the numbering of the D50, for instance).
So many things to design, so little time...



Re: Truncated Spheres [message #36747 is a reply to message #36608 ] Sun, 16 October 2011 20:03 UTC 


A new D7 with a 3fold 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 2fold symmetry: 1+4+2).
I will post a picture to show the difference.

Attachment: D73f_all.jpg
(Size: 13.12KB, Downloaded 1027 time(s))
So many things to design, so little time...



Re: Truncated Spheres [message #36750 is a reply to message #36747 ] Sun, 16 October 2011 20:36 UTC 


Here is the pictures that shows both underlying polyhedra:
The first one (in red) this the one with a 2fold 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...
[Updated on: Sun, 30 October 2011 14:16 UTC] So many things to design, so little time...



Re: Truncated Spheres [message #36763 is a reply to message #36750 ] Mon, 17 October 2011 08:04 UTC 


These polyhedra are interesting  how do you find them  trial and error? Anyway, the dice look great as usual.



Re: Truncated Spheres [message #36799 is a reply to message #36763 ] Mon, 17 October 2011 20:37 UTC 


The first one was realy designed on paper before doing the calculations and trying it on the modeling software. I was trying to put the 7 circles on a sphere in an optimal way.
The second one is more classical. You can compare it to the D11 (5 layers: 1+3+3+3+1) but with less layers and with the layers not being symmetrical (there is no ending 1): 1+3+3.
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.
So many things to design, so little time...




Re: Truncated Spheres [message #37736 is a reply to message #37619 ] Tue, 01 November 2011 17:10 UTC 


I am currently working on two D21.
The red one has a 2fold symmetry, the yellow one a 4fold 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
 2fold: 2+4+2+4+2+2+4+1
 4fold: 4+4+4+4+4+1
[Updated on: Tue, 01 November 2011 19:01 UTC] So many things to design, so little time...




Re: Truncated Spheres [message #40441 is a reply to message #40439 ] Sun, 18 December 2011 13:36 UTC 


This looks good! I'm trying to work out whether it is a fair die or not.
I notice that you don't have a five sided die... impossible?
Season's greetings.



Re: Truncated Spheres [message #40445 is a reply to message #40441 ] Sun, 18 December 2011 15:03 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 doublepyramid 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 5sided 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] So many things to design, so little time...




Re: Truncated Spheres [message #45902 is a reply to message #40988 ] Sat, 24 March 2012 08:26 UTC 


I have been following this blog on truncated spheres with great interest!
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/distributingpointsspher 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!



Re: Truncated Spheres [message #45907 is a reply to message #45902 ] Sat, 24 March 2012 10:28 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] So many things to design, so little time...




Re: Truncated Spheres [message #45945 is a reply to message #45927 ] Sun, 25 March 2012 08:28 UTC 


Thank you for your interest!
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 3fold symmetry with packing, and a 5prism with all the others.
For 8 points packing and energy yield a 4trapezohedron (albeit with different edge lengths!), covering and volume yield a 2+4+2 shape with 2fold symmetry, again with different edge lengths.
For 9 points the results are:
packing: 1+4+4 shape with 2fold symmetry
others: 3+3+3, 3fold symmetry (different but pretty close edge lengths)
10 points:
packing: 2+6+2 with 2fold symmetry
others: 1+4+4+1 with 4fold 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)!



Re: Truncated Spheres [message #45951 is a reply to message #45945 ] Sun, 25 March 2012 10:55 UTC 


I did exactly the same investigation. Volume is not working for me (or it is very slow?).
Very often Energy lead to the same result as Covering, sometimes as Packing, but with a better accuracy (more symmetries, less different lengths).
 I agree with your results for 4, 5, 6, 7 and 12.
 For 8 as I said the Covering result is very interesting. But I would call it 2222 (not 242).
 For 9 if you turn your 144 shape (which I would rather call 1424) in another direction, you will find the 333 again (with different sizes).
 10 leads to a 2422 for Packing (very interesting indeed) and a 1441 for the others.
 11 is hard to understand. Packing leads to a dodecahedron with a missing face (I did a D10 as a dodecahedron with 2 opposite faces missing). And with Energy two pentagons of the original dodecahdron have fused into one elongated hexagon. Covering is unclear...
 13 leads to 1444 (the one I did) for Packing and to something like 122422 (not sure of the order) the the two others. I'd like to make this one...
 14 is 142241 for Packing (new one) and 1661 for the others.
 15 is very interesting. It always leads to a 33333 shape but only Coverage has a 3fold symmetry. The others are enantiomers.
I still have to look at the others, but i can say that this exploration is very exciting.
[Updated on: Sun, 25 March 2012 14:22 UTC] So many things to design, so little time...





Re: Truncated Spheres [message #51825 is a reply to message #51818 ] Wed, 25 July 2012 11:56 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 renumbered D32!
[Updated on: Wed, 25 July 2012 13:55 UTC] So many things to design, so little time...




Re: Truncated Spheres [message #51880 is a reply to message #51833 ] Thu, 26 July 2012 05:30 UTC 


Hey, nice trick! It looks great!
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
So many things to design, so little time...


