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Packing Ellipses Into a Circle - Preprint of WriteUp

Posted 5 years ago
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Frank, I like this paper.

You're probably looking for comments, I just have some minor ones.

1) In the discussion of previous work, it is probably fine that you refer to other papers for historical review, but it could be more clear how ellipse packing fits in. Is it for the technical challenge, refining tools and techniques? Is it part of a general trend toward practical packing with general shapes? I get the sense it is one or the other or both, but it could be something else that you have in mind.

2) Maybe I am too predictable, but I always wonder how things fit in to the point of view from Stephen Wolfram's "A New Kind of Science" The problem of packing circles of two radii gets mentioned in a few places. The context is one of the myriad possible responses to his central premise that simple rules are useful/common/etc., in particular, one wonders how optimized things fit in. We know that engineered systems can be complex without being simple, and engineering can be conceptualized as an attempt at optimization. When one packs a large number of circles, the overarching pattern is roughly hexagonal (and so in some sense are predictable). On the other hand packing different shapes can exhibit complexity. The question here is it really the case that some simple optimization situations (like having two different shapes) can make patterns which are in some sense unpredictable (and one might say the ellipse packings still involve too few ellipses to make a guess).

Another question is what sort of meta-patterns can appear. This could mean when are there tilings which closely approximate large optimal packings in a similar way that hexagonal packings approximate large circle packings. (Of course tilings have their own set of undecidable questions.)

Todd, thanks for your comments. I'll forward them to my co-authors. My joking explanation of why I'm doing ellipse packing is that a sarcastic remark from Daniel Lichtbau caused me to realize I'd been stuck on packing circles for too long, so I decided to move on to something more complex. There's some truth to that statement. Also, I'm hoping that the approach used in the paper, using Lagrange multipliers to set up constraints for quantities that cannot be symbolically defined, has more general application. As to applications, some years ago a friend in the medical profession told me that it could be useful for radiation treatment of cancers to find ellipsoids that have small overlap except in the region of the cancer. I haven't yet looked into that. As to part 2 of your comments, I haven't yet been able to pack a large enough number of ellipses to look for patterns. Right now I've just started looking at packing ellipses into regular polygons, as that seems like a logical extension, although no applications come to mind.

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