D16 Contest

a triangle with 20mm and 4.61mm sides has 13deg angle.

Yep. But here, the distance from face to center is 10mm only (20mm is from face to face).

there we go that works out now.

So far it looks like your arrangement may very well be the ideal.

after ten billion computations I am 99.99999% sure yours is the best possible solution Magic. Will have to get one of these dies.

mctrivia ... you my good man are awesome. That is a great line to use for the KickStarter when we do it.

Magic ... that is enough reassurance for me. Go ahead and put numbers on your numbers on the faces D16 and we'll run with that design!

Tom @ Impact!

McTrivia, I do appreciate the compliment, in particular coming from a great dice designer like you. Thanks.

Tom, I'll work on the numbering of this D16 as soon as possible, but I keep the contest open until September 30th, in case someone else would like to participate.

To all, I should not be available to answer any post from 20th to 30th of September (unless I find an Internet access where I will be), but do no hesitate to post any solution you think could compete: I will have a look at it carefully at my return.

Thank you all!

Magic & McTrivia, great work! It is always fascinating if a design "by construction" also turns out as optimum. Genius at work

There is little to add from my side, as an old Dell PC can not compete with McTrivia's impressive server farm. Nevertheless, I can confirm that Magic's solution is indeed at least a local optimum, and I have not found anything better. However, if Magic's solution is indeed optimum, then I claim that there are infinitely many optimum solutions!

I tried to visualize the constellation of the points and the corresponding polyhedron with the applet on my home page:http://www.aleakybos.ch/sph_codes.htm : choose PackAnti from the drop down menu (packing optimzation criterion, antipodal = parallel faces) and 16 points.



Point 0 is the North pole (A1 is Magic's description), red lines correspond to the minimum distance between any two points. It can be seen that the points on the second layer (4,5,6,7 or B1-B4) all have minimum distance to the North pole and also to one of their two neighbors on the same layer (4-5 and 6-7). Layer 3, the equatorial plane, is interesting. Of the 6 points, there are two pairs with minimum distance to each other (e.g. 3-10) and to two points in the upper layer (3-5 and 10-7). However, there are also two points (1 and 9 or D1 and D2 in Magic's picture) that are so called rattlers: the circles on these points do not touch any other circle! This means that point 1 (and its antipode 9) can be moved inside a small area (about 1.4% of the spheres' radius) as long as the distance to its neighbours is not below the minimum distance, hence there are indeed infinitely many designs, Although I have not found a design that beats Magic's, I found many that are equally good

However, it makes sense to allocate points 1 (and 9) such that the design is symmetric, so kudos to Magic once more!

No worry, Tom, no need to produce infinitely many designs, one will do :laughing:

Given those symmetries, it is actually possible to express the coordinates analytically. Assuming that Theta is the elevation of the first plane (same notation as Magic) and Beta is the smaller azimuth angle between two points on the first plane (4-5 or B1-B2), it can be shown that
cos(Beta)=(cos(Theta)-cos^2(Theta))/(1-cos^2(Theta))
and
cos(Theta)=sin(Theta)*sin((Theta+Beta)/2)
Solving these two equations numerically yields
Theta=49.62703013°
Beta=66.85149262°

This relates to Magic's design as follows:
Second and 4th layer: Phi=+/- Beta/2 and +/- (180°-Beta/2)
Third layer, points C: Phi=+/- (90°-Theta/2) and +/- (90°+Theta/2)

So much for now.

Alea Kybos

aleakybos,

Great post ... very interesting reading even for a non-mathematican.

I took a stab at numbering the D16 and man even spending 3 hours on it ... I have a feeling Magic will show me a better way. So I have a lot of respect for the dice designers of Shapeways.

I think we the help we are getting the new dice that Impact! will be producing will be an awesome addition to the gaming community!

McTrivia and AleaKybos, thank you for your contributions. There are still 2 days left for the contest, so do not hesitate to post any idea.

AleaKybos, your calculations are really interesting, thanks again. I know you are a fan of Truncated Sphere Dice, it is always a pleasure to dicuss with you about this topic . I've seen your new PackAnti option. It is really awesome. But for 22, there is no clear symmetric solution... I was wondering if you could add a "CoveringAnti" (or add an "anti" checkbox so that we could use any optimisation algorithm with the parallel faces constraint). I am curious to see the result for other even faces dice...

Cheers,

Magic

Magic, thank you for your feedback. Actually, the applet on my home does not compute dice, it just displays known configurations. So, if you would like to add one of the dice to the applet, just send me the coordinates and I will upload the new design to visualize it.

McTrivia: Some of your truncated spherical dice in my collection seem to have parallel faces, the D22, D28, D30, D40, D60, and D144 (although there I am not completely sure, the faces are so small). I assume that you calculated them the same way as the D16 in this contest, with your huge server farm. Would you mind sharing your D22 design with me, the applet might reveal some symmetries.

Of the optimum configurations in the sense of maximum separation (or "packing"), only 2 have parallel faces: the D6 (the good old cube), D12 (pentagonal dodecahedron). The optimum D8 is not the octahedron, but the 4-antiprism, which has no parallel faces.

So, there are more spherical dice with parallel faces to be designed, e.g. the D8 (is this the octahedron?), D14, D18 and D20 (is this the dodecahedron?)

all my newer spherical dice with even faces of parallel faces. I computed them in the same fashion though I did not take the time to guarantee optimality just close. Unfortunately he computer I designed those dice on died a while back and took the point clouds with it. I have a much better backup system in place now to guarantee that never happens again but it is to late to recover those dice.

AleaKybos, i will send you the coordinate of the D16 by PM. But is think there are the same as your configuration.
I'd like also to see if we get for the D14 and D18 is the same configuration as the ones I already designed.
For the D20, I must say that there is a lot of round areas for the icosahedron, because there are always 5 faces meeting at each vertex. So perhaps we can do better. And the good think is that, by trying to design the D16, I found a lot of ways to make a D20 (because very often I obtained 16 faces and 4 by round areas that could contain themselves a face after some deformation)...

McTrivia, if you re-write your program, it will be interesting to see if we can find symmetries in a dice that begins with random positions.

Thanks to both of you.

To all, the contest is finished! I hope you will soon see the D16 numbered in my Shapeways shop, and later mass-produced by Tom from Impact! Minatures !
I'll keep you updated!

While I appreciate the creativity required to build the perfect die, with the D16 I'm not sure you know what you have. This is way bigger than a a D16. You have also invented the 2D4 die.

After you've got the die just the way you want it, build one with numbers 2,3,3,4,4,4,5,5,5,5,6,6,6,7,7,8 for me please.

I'm sure Magic could design one of those for Shapeways ... and we can put it in the long term cue to try and build.

Tom

So many things to design, so little time... (did I already say that?)
Toscrawford, you are right: each time we design a dice with a number of faces that is a square number, we can simulate two dice with this one. I will do this numbering for you if you wish (that's the beauty of 3D printing, you can personalized objects as you wish). But, if you cannot wait, McTrivia already did this kind of numbering for its own D16 called the Dual D4

Thanks for pointing out the dual d4. Its cool that all this is possible. In the case of the dual d4, I'm not really a fan of how the numbers are placed. For the time being I think I'd rather wait to see how yours comes out. I'm definitely buying something, but I have the luxury of time before making my decision.

As you wish. Il will make this double-D4 numbering as soon as possible (with opposite summing to 10 and numbers as well balanced as possible).

If you like the idea of combining two dice in one, you will perhaps also like this two models:
- the Double Coin D4
- the Double D6
Unlike the D16, they have moving parts though.

I keep you updated.

Magic, wow, the Double Coin D4 and the Double D6 are cool designs, keep going!

I am still working on the D14 and thd D18. For the D14 I haven't found anything better than your "cuboctahedron" yet (phi=54.74° = arccos(1/sqrt(3)).

For the D18, however, I found a design that is better than your Rhombi-cuboctahedron. It is very irregular, and I am sure that by a combination of manual and computational optimization it can still be improved. I will keep you posted.

Thanks AleaKybos. Can't wait to see this D18.

Toscrawford, the D16 numbered as two D4 is now available:

Enjoy!

That's awesome. Thanks!