[seqfan] Re: partitions of a circle
Neil Sloane
njasloane at gmail.com
Thu May 10 21:21:40 CEST 2012
Jon, Ignore the last sentence - sorry, I did not
realize you had already included the n=3 figs!
On Thu, May 10, 2012 at 1:52 PM, Jon Wild <wild at music.mcgill.ca> wrote:
> Dear Neil,
>
> This sequence seemed familiar to me and I just found and scanned a closely
> related sequence I drew some time ago in an old notebook (I never submitted
> it to the oeis).
>
> My rules were: n straight, non-intersecting line segments are drawn
> successively in a circle, such that both end points of each segment lie
> either on the perimeter of the circle or on a previously drawn line. No
> endpoints may coincide. (Thus no "V"s.) My sequence is reduced for mirror
> symmetry. For 0 to 4 added lines, I counted 1,1,2,9,63 distinct
> configurations. (Under these rules n lines always result in n+1 pieces of
> circle, of course.)
>
> http://music.mcgill.ca/~wild/**circlePieces.pdf<http://music.mcgill.ca/~wild/circlePieces.pdf>
>
> In the scan linked here, the 9 configurations for n=3 appear down the
> left-hand margin. For each 3-config, I found all the 4-configs you can
> produce by adding one line segment, except those that had already been
> produced from an earlier 3-config. I was quite methodical so I'm reasonably
> confident in these results.
>
> Jon Wild
>
>
>
> On Mon, 7 May 2012, Neil Sloane wrote:
>
> Dear Sequence Fans, There is a paper:
>> *Valette, G.*; *Zamfirescu, T.* Les partages d'un polygone convexe en
>> 4 polygones
>>
>> semblables au premier. (French) *J. Combinatorial Theory Ser. B* *16
>> *(1974),
>> 1--16. MR0331217 <http://www.ams.org/**mathscinet-getitem?mr=0331217<http://www.ams.org/mathscinet-getitem?mr=0331217>>
>> *(48
>> #9551),*
>> *see **http://www.sciencedirect.**com/science/journal/00958956/**16/1,*<http://www.sciencedirect.com/science/journal/00958956/16/1,*>
>> *which studies the ways to divide a polygon into 4 congruent pieces.
>>
>> *
>>
>> But they begin by looking at a simpler question:
>> the number of ways to divide a circle into 4 pieces: there are 15 ways,
>> according to their rules. Similarly, I think there are 4 ways to divide a
>> circle into 3 pieces.
>> So there is a sequence 1, 1, 4, 15, ... I can't tell yet if it is in the
>> OEIS. I scanned in their
>> illustration of the 15 partitions into 4 pieces, and I will be happy to
>> send it to anyone
>> who wants to try to help find the next couple of terms.
>>
>> The 4 ways to cut a circle into 3 pieces are:
>> 1. draw 2 parallel chords in a circle
>> 2. draw a T in a circle
>> 3. draw a Y in a circle
>> 4. draw a V in a circle
>>
>> Neil
>> --
>> Dear Friends, I will soon be retiring from AT&T. New coordinates:
>>
>> Neil J. A. Sloane, President, OEIS Foundation
>> 11 South Adelaide Avenue, Highland Park, NJ 08904, USA
>> Phone: 732 828 6098; home page: http://NeilSloane.com
>> Email: njasloane at gmail.com
>>
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>
>
>
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--
Dear Friends, I will soon be retiring from AT&T. New coordinates:
Neil J. A. Sloane, President, OEIS Foundation
11 South Adelaide Avenue, Highland Park, NJ 08904, USA
Phone: 732 828 6098; home page: http://NeilSloane.com
Email: njasloane at gmail.com
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