D16 Contest [message #53828] Mon, 10 September 2012 05:56 UTC 


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 16sided die (this one was obvious )
 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!
So many things to design, so little time...



Re: D16 Contest [message #53862 is a reply to message #53828 ] Tue, 11 September 2012 06:06 UTC 


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



Re: D16 Contest [message #53876 is a reply to message #53862 ] Tue, 11 September 2012 15:06 UTC 


my best answer so far is 48.440809062632 deg of separation. I will keep looking for better though.
Follow me on twitter http://twitter.com/mctrivia or my blog at http://4ddice.blogspot.com/



Re: D16 Contest [message #53915 is a reply to message #53876 ] Wed, 12 September 2012 00:52 UTC 


49.6477401576 so I have finally beat you though barely. now to check all your other criteria.
Follow me on twitter http://twitter.com/mctrivia or my blog at http://4ddice.blogspot.com/



Re: D16 Contest [message #53918 is a reply to message #53915 ] Wed, 12 September 2012 04:04 UTC 


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.
Follow me on twitter http://twitter.com/mctrivia or my blog at http://4ddice.blogspot.com/



Re: D16 Contest [message #53926 is a reply to message #53918 ] Wed, 12 September 2012 06:15 UTC 


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 "quasisymmetries" (like an angle at 89.57°) you could enforce them (by forcing the angle to be 90° exactly).
Good luck!
So many things to design, so little time...



Re: D16 Contest [message #53956 is a reply to message #53926 ] Wed, 12 September 2012 21:45 UTC 


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.
Follow me on twitter http://twitter.com/mctrivia or my blog at http://4ddice.blogspot.com/



Re: D16 Contest [message #54035 is a reply to message #53956 ] Fri, 14 September 2012 18:47 UTC 


Wow! Your computation power is quite impressive...
I am curious to see how the "51.33°" die looks like.
Let us know your progress!
So many things to design, so little time...



Re: D16 Contest [message #54046 is a reply to message #54035 ] Fri, 14 September 2012 22:15 UTC 


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.
Follow me on twitter http://twitter.com/mctrivia or my blog at http://4ddice.blogspot.com/



Re: D16 Contest [message #54077 is a reply to message #54046 ] Sat, 15 September 2012 13:34 UTC 


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.

Attachment: best.pdf
(Size: 127.73KB, Downloaded 86 time(s))
[Updated on: Sat, 15 September 2012 13:36 UTC] Follow me on twitter http://twitter.com/mctrivia or my blog at http://4ddice.blogspot.com/



Re: D16 Contest [message #54088 is a reply to message #54077 ] Sat, 15 September 2012 16:02 UTC 


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...
[Updated on: Sat, 15 September 2012 16:04 UTC] So many things to design, so little time...




Re: D16 Contest [message #54091 is a reply to message #54077 ] Sat, 15 September 2012 17:38 UTC 


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 2662. Now the question is if this is "better" than the 14641 design (I should note for comparison that Magic's dice could be turned in a way that the design is 24442 (to my nonmath 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 2662, 14641 (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.
[Updated on: Sat, 15 September 2012 17:48 UTC]



Re: D16 Contest [message #54092 is a reply to message #54091 ] Sat, 15 September 2012 18:02 UTC 


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 nonplatonic 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.
Follow me on twitter http://twitter.com/mctrivia or my blog at http://4ddice.blogspot.com/



Re: D16 Contest [message #54093 is a reply to message #54091 ] Sat, 15 September 2012 18:17 UTC 


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 nonmath 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!



Re: D16 Contest [message #54094 is a reply to message #54092 ] Sat, 15 September 2012 18:26 UTC 


mctrivia wrote on Sat, 15 September 2012 18:02  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).
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.
 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!



Re: D16 Contest [message #54095 is a reply to message #54094 ] Sat, 15 September 2012 19:42 UTC 


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 2662 configuration (or more precisely a 224422, 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 A1A2 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.
[Updated on: Sat, 15 September 2012 19:56 UTC] So many things to design, so little time...



Re: D16 Contest [message #54108 is a reply to message #54095 ] Sun, 16 September 2012 02:13 UTC 


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
Follow me on twitter http://twitter.com/mctrivia or my blog at http://4ddice.blogspot.com/



Re: D16 Contest [message #54119 is a reply to message #54108 ] Sun, 16 September 2012 15:04 UTC 


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
Follow me on twitter http://twitter.com/mctrivia or my blog at http://4ddice.blogspot.com/



Re: D16 Contest [message #54120 is a reply to message #54119 ] Sun, 16 September 2012 15:42 UTC 


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



Re: D16 Contest [message #54121 is a reply to message #54120 ] Sun, 16 September 2012 15:53 UTC 


a triangle with 20mm and 4.61mm sides has 13deg angle.
Follow me on twitter http://twitter.com/mctrivia or my blog at http://4ddice.blogspot.com/



Re: D16 Contest [message #54122 is a reply to message #54121 ] Sun, 16 September 2012 16:17 UTC 


Yep. But here, the distance from face to center is 10mm only (20mm is from face to face).
So many things to design, so little time...



Re: D16 Contest [message #54124 is a reply to message #54122 ] Sun, 16 September 2012 16:43 UTC 


there we go that works out now.
So far it looks like your arrangement may very well be the ideal.
Follow me on twitter http://twitter.com/mctrivia or my blog at http://4ddice.blogspot.com/



Re: D16 Contest [message #54146 is a reply to message #54124 ] Sun, 16 September 2012 23:46 UTC 


after ten billion computations I am 99.99999% sure yours is the best possible solution Magic. Will have to get one of these dies.
[Updated on: Sun, 16 September 2012 23:47 UTC] Follow me on twitter http://twitter.com/mctrivia or my blog at http://4ddice.blogspot.com/



Re: D16 Contest [message #54147 is a reply to message #54146 ] Sun, 16 September 2012 23:50 UTC 


mctrivia wrote on Sun, 16 September 2012 23:46  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!



Re: D16 Contest [message #54155 is a reply to message #54147 ] Mon, 17 September 2012 06:13 UTC 


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



Re: D16 Contest [message #54393 is a reply to message #54155 ] Sat, 22 September 2012 16:19 UTC 


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 B1B4) all have minimum distance to the North pole and also to one of their two neighbors on the same layer (45 and 67). Layer 3, the equatorial plane, is interesting. Of the 6 points, there are two pairs with minimum distance to each other (e.g. 310) and to two points in the upper layer (35 and 107). 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
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 (45 or B1B2), it can be shown that
cos(Beta)=(cos(Theta)cos^2(Theta))/(1cos^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



Re: D16 Contest [message #54395 is a reply to message #54393 ] Sat, 22 September 2012 17:16 UTC 


aleakybos,
Great post ... very interesting reading even for a nonmathematican.
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!




Re: D16 Contest [message #54886 is a reply to message #54608 ] Thu, 04 October 2012 17:50 UTC 


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 4antiprism, 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?)



Re: D16 Contest [message #54930 is a reply to message #54886 ] Fri, 05 October 2012 11:50 UTC 


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.
Follow me on twitter http://twitter.com/mctrivia or my blog at http://4ddice.blogspot.com/



Re: D16 Contest [message #54960 is a reply to message #54930 ] Sat, 06 October 2012 09:12 UTC 


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 rewrite 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 massproduced by Tom from Impact! Minatures !
I'll keep you updated!
So many things to design, so little time...



Re: D16 Contest [message #55054 is a reply to message #54155 ] Tue, 09 October 2012 00:33 UTC 


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.



Re: D16 Contest [message #55056 is a reply to message #55054 ] Tue, 09 October 2012 00:50 UTC 


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




Re: D16 Contest [message #55118 is a reply to message #55106 ] Wed, 10 October 2012 02:23 UTC 


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.



Re: D16 Contest [message #55305 is a reply to message #55118 ] Sat, 13 October 2012 17:48 UTC 


As you wish. Il will make this doubleD4 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.
So many things to design, so little time...



Re: D16 Contest [message #55418 is a reply to message #55305 ] Tue, 16 October 2012 05:07 UTC 


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 Rhombicuboctahedron. 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.




