D16 Contest

Magic, that D16 2D4 looks great!

As promised, this is my D18, designed with maximum separation ("packing") and parallel faces. The angle is 47.9821 deg, compared to 45deg of Magic's D18 based on the squares of Rhombicuboctahedron.



You may play with the applet here to see the symmetries (PackAnti, 18) https://www.aleakybos.ch/sph_codes.htm

There is a chance that someone may find an even better angle, as the design was done by a numerical search with some analytical optimization once the symmetries were guessed. If you find a better design, please let me know.

Cheers, Alea Kybos

PS. McTrivia, I put quite some effort into optimizing my simulation, as I do not have access to a huge server farm like you. A brute force approach for a D2N would use 6N cartesian or 4N spherical coodinates, whereas I brought it down to 2N-3 spherical coordinates and each evaluation of the objective function uses 2N-3 sine and cosine operations and O(N^2) multiplications. Still a lot...

Hi AleaKybos,

Thank you for this new polyhedron!
I played with the applet and I have a concern: all seems symmetric relatively to two perpenticular planes except two faces... I am wondering why these skewed faces would give a better result than faces that would be parallel to the axes...

In this drawing, if the top and botom faces were horizontal, I think it would help.


What do you think?

Oh, actually, it makes sense.
Look at the dice without these two skewed faces and with them...
They perfectly fit inside the empty space!
Very interesting: it proves that the intuition that symmetry is always better is wrong in some cases.


[EDIT] Here is the link to the model: https://www.shapeways.com/model/745108
The animation where the die rotates around the Z axis helps to understand how the faces are located and why the skewed position for th last two ones is the best...

Magic, thank you very much for the design! Yes, the "poles" are shifted compared to the other layers. I therefore rotated the die by 90 deg by setting the pole half way between points 0 and 1 and costructed the die from there.

Here comes another confirmation of your statement "the intuition that symmetry is always better is wrong in some cases". After having designed the D18, I ran my tools again over your D16, and, oops!, after some minutes a "better" solution popped up. I have kept my PC busy over the last 36 hours and I still keep getting better solutions, although progress is 4 digits after the comma... Current status is 49.63899330 deg compared to 49.62703013 deg.

My explanation is that in your design there are 2 rattlers (points that do not have minimum distance to any neighbor), as explained in my earlier post. Now there seems to be an even better solution if those neighbors are shifted closer towards the rattler and to distribute the "gained angle" over the rest of the configuration, thereby increasing the minimum distance (separation). The final version may thus no longer have the same symmetries. I uploaded the current best solution in my applet 16 Points, PackAnti), there you see the configuration (coords and angle are in the report window).

Anyway, the difference to the contest winner is tiny, only about 0.02%, i.e., way below production tolerances. So this analysis is purley "alealogic" ... I will keep you posted.

This improvement is really interesting indeed (from the theorical or "alealogic" point of view ).
Unfortunately, I am afraid that we will not notice the difference with the naked eye. That's a pity.

On my side, I have worked on another D16 with tetrahedral symmetry (not with numbers on faces, though), and I had to deal with "rattlers" too. You will find the story here.
I did not checked yet, but I am sure we can find at least one of them with your program...

The D18 discovered by Alea Kybos is now resized, hollowed, numbered and available for sale...

Alea Kybos was kind enough to send me these pictures of the D18 he designed and ordered in metal.
Here are two pictures:


The result in metal is fantastic.

And on these pictures the asymmetry of the "18" face is clearly visible.

Thank you Alea Kybos!

The discussed D16, as well as some other Sphere Dice, is about to be mass produced: check this thread!