4D Billiards

[PatrickPowers]

What would a 4D billiard table be like? How about a triangular theme. Surface of the table a triangular prism and the balls in a tetrahedron, each ball contacting three others. A tetrahedron has either ten or twenty balls, not the standard fifteen. Two tetrahedra with a common base, like our nineball, gives 14. All three of these arrangements have all outer balls, no inner. An inner ball can't be struck by the cue ball without first touching an outer ball.

What is the minimal number of balls that would have an inner ball. To solve this I bought some pingpong balls and stuck them to each other. At first I tried sticking them together with Gajah rat glue, the stickiest substance on Earth. Used to trap rat, rat glue never dries, but it couldn't hold a pingpong ball still so it just made a very sticky mess. Next was superglue but it was too old and wouldn't set. How about black duct tape. That was ugly as could be but worked pretty well. I concluded that the 3D arrangement of balls such that there is an inner ball can be done with seven balls. The cue ball will still be at least 0.18 away from the inner sphere when it contacts an outer sphere. With six spheres the cue ball can reach the ostensibly inner ball while still about 0.10 away from any outer ball, so seven is the answer.

After tossing the rat glued balls in the trash and undoing the black tape I found some working super glue and made the seven ball arrangement. It's six balls at the vertexes of an octahedron with a central ball. More balls can be added to make 13 with one inner ball but everyone knows that's unlucky and the outer balls don't touch one another, so I say this would be unbilliards. Instead it is possible to symmetrically add eight more balls to the seven in the octahedral arrangement. Each of these eight balls is that minimal distance of 0.18 away from the inner ball. The result is fifteen balls with one inner ball and each outer ball touching three others. I'll make a model of that out of superglued pingpong balls and take a photo of it. Would the table then be a square prism or a triangular one? Could go either way, eh?

[PatrickPowers]

There are all sorts of possibilities. So far I haven't found any that are completely satisfactory. The main desiderata are : symmetrical, 15 balls. There is such an arrangement. Start with a central ball, surround it with osculating balls at the vertexes of a octahedron. So far that's seven balls. Add balls that are each touching three of the outer balls. That's eight for a total of fifteen. As a bonus there is an internal ball not on the surface. However it looks like an atomic nucleus, something I find unbilliards. So what are the alternatives?

Option Number of balls
Nucleus 15
Tetrahedral 10 or 19
Double Tetrahedron 14
Pyramid 14
Hexagonal Dense pack 13

Arrangements with even numbers of balls are unsuitable for the popular game of Eight Ball. The idea is that each player owns seven balls, either the striped balls or the solidly colored balls less the black eight ball. The goal is to sink all seven of your own balls then the eight ball to win. Pot the eight prematurely and you lose. This game requires an odd number of balls to be fair.

The nucleus seems the logical choice but gives me the creeps. It looks like it's about to undergo fission. Well it is, but, uh, you know.
The tetrahedron looks right but 19 balls seems like too much and ten too few. The double tet reminds of low-class unpopular nineball, the pyramid looks good but neither is 3D symmetrical and both have an even number of balls.

The dense pack is one ball surrounded by twelve other balls. Each ball contacts at least five others, a maximum, that's an advantage no other arrangement can match. There is an inner ball, which the tetrahedra and pyramid lack. It has an odd number of balls as close as possible to 15 without being actually equal to that number. It's hexagonal. I think I'd go for that, unlucky 13 notwithstanding. Then the game of eight ball becomes seven ball, the middle number.

Should the table be a triangular prism, square prism, or cylinder? Triangle goes with a tetrahedral arrangement, square with the pyramid, but we turned both of those down for spherical arrangements, for which a cylinder seems right. But it has a big problem. With a cylinder you have spots on the table that are hard to reach. Say you have a ball in the center. If you bounce it off the nearest wall then it returns to the center, no matter where in the 360 degrees you may send it. I say that's a deal breaker. Hmm, maybe 19 balls on a triangular prism table isn't so bad... Then eight ball becomes ten ball. That'll work.

Does everyone out there know about the Magic Eight Ball? https://magic-8ball.com/

[PatrickPowers]

Building models superglue works very well for white pingpong balls but for the orange ones it fails to gel. For orange pingpong balls one must instead use yellow contact cement. Should such things be considered obvious?

My favorite now is a truncated tetrahedron with 15 balls. It looks the most traditional. Start with a triangle with four rows instead of the canonical five. Add another layer of six balls on top of that for a total of fifteen. It means there is no head ball and no inner balls but I don't mind that.

The other candidate is take the 13 dense pack balls and add two more at mutual antipodes to get 15 balls. It looks exotic, highly untraditional, but better than the the spherical arrangements that are evocative of atomic nuclei. And it does have the advantages of having all of a headball, an inner ball, and maximal connectivity, all of which are traditional.