D16 Contest

Discussion in 'General Discussion' started by Magic, Sep 10, 2012.

  1. Magic
    Magic Well-Known Member
    Hello all,

    As you know, the Kickstater launched by Impact! Miniatures to mass produced some of my Truncated Sphere dice was very successful. But during this Kickstater Tom from Impact! challenged me to create a D16 with numbers on faces (my own version had numbers on edges).
    I found what I think is the best solution for this problem.
    But Tom and I are not fully satisfied with this design.

    So now, it's my turn to challenge you! If you can find a better design than the one described bellow, Tom will offer you a Shapeways prepaid gift coupon of $100.

    The die must satisfy the following requirements:
    - Must be a 16-sided die (this one was obvious :D )
    - Must be a Truncated Sphere: this means that you must first design a polyhedron with all the faces at the same distance from the center, and then intersect it with a sphere of the appropriate size - not to small so that most of the disc forming the faces of the die will touch and not too big so that those faces remain circular and do not overlap.
    - Must have numbers on faces: to achieve that, each face must have an opposite face that is parallel.
    - Must be better than my solution: the faces must not be too small - 48° or more of separation (I will explain what it is) are recommended and the rounded part must be evenly distributed. This excludes the dipyramid for instance.

    You can now participate to the contest, but I'd like to give you some extra information.

    Here is a full description of my own solution, so that you do not submit the same design.
    I describe my dice by the numbers of faces by layers and, for each layer, the angle Theta that is between the normal of the face with the vertical axis (or if you prefer: the angle between the face and the horizontal plane) and then, for each face of the layer, the angle Phi giving its orientation around the vertical axis (a kind of lattitude).

    My D16 has its 16 faces distributed on 5 layers: 1+4+6+4+1
    - First and last layer: one horizontal face. Basically, the die has two poles (faces A1 and A2). Theta=0 and 180°, Phi is meaningless in this case.
    - Second and 4th layer: 4 faces of type B. For the 1st layer I have Theta is 49.627° and Phi=+/-33.4258° and +/-146.5742° (the sum of both values is 180°). The 4th layer is symmetric relatively to the horizontal plane (the equator).
    - 3rd layer, the equator. Obviously Theta=90°, and there are 2 kinds of faces. I have 4 faces of kind C for which Phi=+/-65.1865° and +/-114.8135° (the sum of both values is 180°) and 2 faces of kind E (Phi=0° and 180°).

    Of course if you decide to put the faces of kind D at the pole, the new repartition of layer is 1+4+2+2+2+4+1, and if you put two C faces at the pole it is 2+4+4+4+2 with different Theta and Phi angles. So check carefully that you are not submitting the same solution with a different orientation.

    You can see it as my D14 that has faces oriented in the directions of the 6 faces and the 8 vertices of a cube where you would have replace two opposite faces by two pairs of faces (the kind C). By doing so, the faces corresponding to the vertices of the original cube are moved (the kind C) while the one corresponding to the faces of the cube that have not been doubled remain unchanged (kind A and D).

    The separation I was speaking before is an important parameter. This is the angle at the vertex of the cone that has as a basis one face and at the vertex the center of the die. If r2 is the distance from one face to the center and r1 the radius of the sphere then the separation is 2.arccos(r2/r1). Here the separation is 49,47° approximately (for a distance between two parallel faces of 20.00mm, the diameter of the sphere is 22.02mm).

    You have until September 30th to submit a design by answering this post. Tom will decide who the winner is if we have one. Note that by participating to this contest you accept that your design can be reused and modified by me or by Impact! Miniatures for our own usage, the compensation being the prize of the contest ($100 in prepaid gift coupon).

    Good luck! :)

  2. Magic
    Magic Well-Known Member
    Hi there,

    I have begun to receive some messages via PM.

    Here are two tips to better "see" the structure of your polyhedron without having to draw it, in case you work with the coordinates of the centers of the circular faces on the (unit) sphere or if you prefer the coordinates of the normal vectors of the faces.

    Having your vector (x, y, z), to caculate Theta and Phi, you can use these formula:
    - Theta = acos(z)
    - Phi = atan2(y,x)
    (do not forget to convert in degrees)

    Another thing that can be useful, is to make the dot product between all the vectors taken two by two, because it gives the cosinus of the angle between two vectors. Thus it is easy to see if some directions are perpendicular of if you find some angles similar to my solution.
    To calculate alpha the angle between vectors (x1,y1,z1) and (x2, y2, z2) use this formula:
    - alpha = acos(x1.x2+y1.y2+z1.z2)

    Let me know if you got some new D16!

  3. mctrivia
    mctrivia Well-Known Member
    my best answer so far is 48.440809062632 deg of separation. I will keep looking for better though.
  4. mctrivia
    mctrivia Well-Known Member
    49.6477401576 so I have finally beat you though barely. now to check all your other criteria.
  5. mctrivia
    mctrivia Well-Known Member
    so what classifies as evenly distributed? 51.194013858768deg spacing is best I have found. This as a physical averaging that should result in fair but not the nice cosmetic even spacing that yours shows.

    If this disqualifies me I have at least learned that different starting points can turn up better distributions. I will let my server farm run on this problem for a week and then draw up the best result what ever your ruling on the evenness of my design.
  6. Magic
    Magic Well-Known Member
    More than 51° is very good, congrats. The best known for 16 points (without the constraint of parallel faces) is 52.2443957° (see this page).

    I think the other important criterion is having an evenly distributed spherical area. I confess that, for me, this often means "as many symmetries as possible", but in a more rational approach it could also means "minimizing the largest round area".
    I guess a mathematical definition would be "Once your equal circles are packing the sphere, the maximum radius of any extra circle you could insert between them should be as small as possible".
    But at this point, trying to visualize the die is probably the best option.

    Note that if you have "quasi-symmetries" (like an angle at 89.57°) you could enforce them (by forcing the angle to be 90° exactly).

    Good luck!

  7. mctrivia
    mctrivia Well-Known Member
    i seem to be maxing out at 51.3372888793 deg. pretty close to best known. Interesting but it sure takes a lot of computational power the way I am doing it. It takes me approximately 50000 iterations of my physics engine to compute one possible outcome(about 10 sec). I have my server trying 12 parallel attempts, running 24/7. In 1 week I will have done 36,288,000,000 iterations.

    I have only found a 4deg improvement in total, and in the last 24 hours have not even gone up 0.2 deg.
  8. Magic
    Magic Well-Known Member
    Wow! Your computation power is quite impressive...
    I am curious to see how the "51.33°" die looks like.
    Let us know your progress!

  9. mctrivia
    mctrivia Well-Known Member
    51.3613415955 is best I can get. Yes my server farm is pretty powerful. last i checked had over 1000 computer running in it. Will design the model and see what it looks like.
  10. mctrivia
    mctrivia Well-Known Member
    51.36 deg separation. not as pretty as yours though. i think yours will be the most cosmetically appealing solution. THe dual looked so pretty guess the minor differences get exaggerated on the sphere.

    Attached Files:

    • best.pdf

      File size:
      127.7 KB
    Last edited: Sep 15, 2012
  11. Magic
    Magic Well-Known Member
    I must say I was hoping that some symmetries would appear...
    Can you upload it on Shapeways so we can better see the model in the 3D view, please?
    I am unsure but I think there is a strip of 8 faces touching each other around the equator, four faces on the top and four faces on the bottom.
    In this case, the 8 faces are the ones that are determining the "separation". If they were all on the equator forming a regular octogon, it would not be very efficient. We have to move some of them up and some of them down.
    The best configuation I see is the following one ("+" means up and "-" means down for each vertex of the octogon):

    Perhaps we should try to initialize the system with such a configuration and see if we end up with something interesting...

    Last edited: Sep 15, 2012
  12. mctrivia
    mctrivia Well-Known Member
  13. McTrivia,

    Wow I didn't know you could make turnable image PDFs. I found that amazing good to really look over what you had made. I'm wishing Shapeways had this option now for each product ... would save me a lot of time trying to figure out the designs I'm looking over!

    Magic ... after spinnning McTrivia's version around ... the pattern in it looks to me like it is 2-6-6-2. Now the question is if this is "better" than the 1-4-6-4-1 design (I should note for comparison that Magic's dice could be turned in a way that the design is 2-4-4-4-2 (to my non-math mind trying to compare and contrast the dice designs)).

    McTrivia ... I've followed a lot of your designs on Shapeways. Your talent is impressive as well.

    So let me get your thoughts. If you were personally picking a mass production version of the D16 ... would you favor the 2-6-6-2, 1-4-6-4-1 (which both have numbers on the faces) ... or Magic's verison with the numbers on the edges:

    My concern is this ... from our last KickStarter ... we had a lot of folks call out that even sided dice should have numbers on the faces ... they were fine and understood this not being the case for more true odd sided dice.

    I will admit in getting acceptance of the dice by the masses ... being able to say that 36.3 billion computations were done with the physics computer to get the optimal shape. Goes a long way to arguing the fairness equation.

    I will say ... I personally will be happy that if the two of you come to a consensus on what is the right die to make for the masses ... that I am very willing to listen to that advice.

    The end goal of our next project is to make a DCC set that has a uniform color scheme and is spherical in nature for the whole set. The D16 is one of those we need to get nailed down for the project to work.

    Last edited: Sep 15, 2012
  14. mctrivia
    mctrivia Well-Known Member
    What is best is a matter of what is the use. Both i suspect are fair designs. Magic's strict mathematical approach comes up with some beautiful intelligently designed dice this being in my opinion one of the best non-platonic designs. Myself I am using chaos and particle physics to come up with a design that should be fair and have the widest possible separation of axes(sets of faces). The advantage of my approach is in 10 sec I can come up with the dual of any even number sided die and with multiple iterations fine if not the best very close to it separation possible. Magic's method may not be ideal separation but will almost always come up with a more cosmetically appealing approach.

    Personally if I was in the need of a D16 for just a general use then I would pick Magic's and I suspect most would. I can see mine be a cool and different addition to a board game along with several other chaotically designed dice.
  15. So for someone wanting a quick image comparison ... here is McTrivia's die from views similar to the ones Magic has above.


    I will note ... I can see how the overall rounded space is less with this version than Magic's. I can also now see that it is done at the cost of looking equally spaced from the 3 different views that Magic has of his version.

    Tough one to call. The uniform look when cut in half on 3 axis of Magic's version has an appeal to non-math folks like me ... but I also like knowing the round space is minimal from McTrivia's.

    I will be interested to hear from you guys on your future thoughts on this.

    Thank you both for looking at this project.

    Tom @ Impact!
  16. You have a good point with this McTrivia. My problem is that I love chaos so your design has an appeal to me.

    What we could definitely do is put it to a vote when the KickStarter hits the point of funding the D16 and give the backers the choice between the two versions. Explain that one is equal dimensioned on 3 axes and that the other is designed with chaos physics to have the minimual amount of rounded space (but I think not equal on any axes? (that I could see)). I'm happy to produce the one that the backers vote for.

    If you make the change that Magic requested ... I would really be interested in seeing what comes of that modification.

    Tom @ Impact!
  17. Magic
    Magic Well-Known Member
    Hi Tom and McTrivia (and all the others :)),

    I am not at home currently, but I managed to access an old version of the software I wrote to optimize dice.
    Taking Tom's notation, I've done a 2-6-6-2 configuration (or more precisely a 2-2-4-4-2-2, because you can add some more degrees of freedom by letting 2 of the 6 faces of the second layer being a little bit shift relatively to the others) in the following way:
    - A1, A2 on top
    - B2, B5 on an axis perpendicular to A1, A2
    - B1, B3, B4, B6 touching the equator, so that B3 is symmetric to B4 relatively to the A1-A2 axis and B1 is symmetric to B6
    The best I get is 44.62° of separation. The more recent version of my software will probably give a better result, but still this is a very poor value for the seperation.
    And unfortunately I cannot visualize it currently.
    More news from me tomorrow night, I guess...

    PS: I did not notice this was a 3D PDF ;)
    It is very handy indeed.

    Last edited: Sep 15, 2012
  18. mctrivia
    mctrivia Well-Known Member
    Bad news. found an error in my physics engine which invalidates all my previous results. have started rerunning the engine but down to only 48.405834944358
  19. mctrivia
    mctrivia Well-Known Member
    magic are you sure you have 49deg separation? I did not recreate your point cloud to check but instead used your sphere dimensions.

    As I calculate it a sphere 22.02mm with faces 20mm apart results in a face 4.6065mm in radius. this gives an angle from center to edge of the circle of 12.97deg. so your minimum separation angle is only 25.94deg

    if my math is correct that makes my 48.405834944358(current best after fixing math error), 1.86 times better spaced.

    Please check these discrepancies. I did check 0.01mm error factor in your numbers only increases to 26deg
  20. Magic
    Magic Well-Known Member
    4.6065mm for the radius of the face is correct, but I don't undersand how this leads to 12.97°.

    The calulation I did is the following one:
    - distance from center to face = 10.00mm (20.00 / 2)
    - radius of the sphere = 11.01mm (22.02 / 2)
    - separation = 2.arccos(10.00/11.01) = 49.627°
    You got the same result wih 2.arcsin(4.6065/11.01).

    Note that for the sphere, I always use a radius slightly smaller than the exact value to be sure that the faces are proprerly separated even if the sphere is only an approximation of sphere.
    So 49.467° is the actual value used for the model, but from my calculations the theorical value should be rather 49.627°.