Wikipedia:Reference desk/Archives/Mathematics/2016 June 1

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June 1

Packing pac-men...

I was wondering what percentage of the plane could be packed for each possible partial circular disk (identical in size) running from n=0 to an entire circle (2Pi). At 2Pi, the answer is the standard circle packing, .9069, and as the measure approaches 0, the covering approaches 1. (The slices alternate direction and pack into columns with squiggly sides that get straighter as the arc gets smaller. I'm wondering whether at some point below 2Pi, say about 1.8Pi, the percentage covering is less than that of the full circle as the amount uncovered in each circle goes up, without a neighboring circle being able to cover much of it. Not sure of the minimum there, but half circles (Pi) would be able to pack at least as well as circles...) Ideas?Naraht (talk) 03:47, 1 June 2016 (UTC)

Interesting question. Simplifying assumption: the best way to pack two pac-men is by placing the top of one mouth flat against the bottom of the other mouth. Assume that the radius is 1, and let $\alpha$ be the radius of the missing wedge (the mouth). Now, if I'm visualizing my triangles correctly, when you pack two pac-men together like this, the distance between centers is $2\cos(\alpha )$.

So two full circles occupy a 2 by 4 rectangle and have total area 2pi, while two pac-men occupy a 2 by $(2+2\cos(\alpha ))$ rectangle and have total area $2\pi -\alpha$. The two circles have a better ratio of filled area to rectangle area; to see this, compare the two expressions, cross multiply, replace cosine with its second-order approximation and then simplify, and this becomes the claim that $\pi \alpha ^{2}/2<\alpha$, which is true for sufficiently small $\alpha$.

Of course, this isn't the end of the story, since in an optimal packing, these rectangles will overlap for different pairs. But that effect should be smaller for pac-men than for full circles. So my handwavey claim is that for sufficiently small positive $\alpha$, the packing is worse than for full circles.--2406:E006:45D:1:146D:8FF:3DC1:AE2C (talk) 04:47, 1 June 2016 (UTC)

Sort of surprised this didn't get more comments. With considered thought, I'm not even sure that the function plane coverage = f(disk_percentage) is even continuous, there may be places where things get bizarre. For example, a partial disk that is exactly one third of a disk will have a coverage equal to the full disk, but what happens with something just a bit bigger.Naraht (talk) 17:12, 3 June 2016 (UTC)

"For example, ..." I don't think this is right. I believe that the other construction you mentioned (glue sectors together along their radii, alternating whether curves are "up" or "down", then layer stacks like this together) is close to 94% density when sectors are 1/3 of a circle. On the other hand, I agree that it is not clear that the function is continuous. --JBL (talk) 20:38, 3 June 2016 (UTC)

More concretely, the density of the packing of sectors where you make strips of alternate up-down sectors sharing their radii, then layer these strips in the densest possible way, is ${\frac {t}{\sin t\cdot ({\sqrt {4-\sin ^{2}t}}-\cos t)}}$, where t is half the central angle of the sector. (Here the sector is at most half a disk.) At t = π/2 we get the same as the dense packing of circles, and for slightly smaller t we actually get worse than this, but for every t < 1.33257 we get something denser from this packing (and the density is monotonically increasing to 1). --JBL (talk) 21:46, 3 June 2016 (UTC)

Fractional Game Theory

I can't imagine it, but mathematics is full of surprises. Is there such in the literature to date? I know it won't be here. Number of players a non-integer or fuzzy intermediate value.Julzes (talk) 10:41, 1 June 2016 (UTC)

In combinatorial game theory games can have non-integer or fuzzy values, but the number of players is an integer (usually 2). Gandalf61 (talk) 14:01, 1 June 2016 (UTC)

In MMORPGs, players vary by level of activity. Some are online several times a day, while others are on once a week, or less. That might qualify. StuRat (talk) 15:33, 1 June 2016 (UTC)

As far as I know, no one has developed a theory of MMORPGs, or if they have Blizzard hasn't told anyone. But it seems to me that game theory with a large but undetermined number of players is a description of economics. --RDBury (talk) 01:48, 2 June 2016 (UTC)