[seqfan] Re: Arranging circles on a sphere -- A000055?
wild at music.mcgill.ca
Thu Aug 25 23:47:20 CEST 2016
On Thu, 25 Aug 2016, Richard J. Mathar wrote:
> For circles in the flat plane topology I have used that sort of counting
> extensively in http://arxiv.org/abs/1603.00077 .
Richard, this is very nice!
You might be interested to know that in the general case (i.e. A250001),
once you get to n=5, geometric concerns mean that not all
topologically-conceivable arrangements are actually circle-drawable.
My program enumerated 16968 conceivable arrangements of 5 pseudo-circles,
and Christopher Jones and I together have figured out how to show that 26
of these arrangements are not actually circle-drawable. We haven't written
anything up yet but I hope to get to it soon.
The spherical aficionados among you might like to know that the connected
5-arrangements (ones where you can reach any circle from any other circle
via a chain of intersections, which number 15528 among the 16968 planar
5-arrangements) collapse into 984 arrangements when reprojected onto the
surface of a sphere. Of these 984 spherical arrangements, all but 4 are
circle-drawable. (The four undrawable spherical arrangements reproject
into the 26 different undrawable planar arrangements.)
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