[seqfan] Re: partitions of a circle

[Ed Jeffery]

>* PS No progress by anyone on the question (number
*>* of ways to partition a circle into n parts), so I will
*>* mention that the V-Z paper is available from Tudor Zamfirescu's home page,
*>* http://tzamfirescu.tricube.de/, see item 62.
*>* The fifteen ways of partitioning a circle into 4 parts
*>* are shown on the second page.
*>* What is the next term?
*>*
*>>*
*>>* 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
*>>* --
*
>>> I don't know what the rules are as I am a little weak on my French, but from
>>> looking at the 15 diagrams in the paper, it appears to me that drawing a 'T'
>>> and drawing a 'Y' are not considered distinct. The second of the 15 circles
>>> in the paper has a T and one additional chord, but there is no circle with
>>> a 'Y' and one additional chord.

>>> Andrew



Trying to work through n = 5 using a CAD program, the rules are not clear
to me either. For n = 4, one can draw a lower-case Greek letter "pi" using
straight line segments extended to meet the circle. This version is not in
the list of figures, so it must have been taken by the authors to be
homeomorphic to at least one of the fifteen figures shown, but which
one(s)? This leads me to wonder, with great frustration: what is the
distinction between figures 5, 6, 9, 10, 11, 12, 13 and 15 (although it is
easy to see that they can be represented by distinct graphs)? Finally, it
seems that nontrivial reflections should be allowed (in the sense that a
derived figure cannot otherwise be obtained simply by rotation of the
original), in which case at least one possibility seems to be missing from
the list for n = 4.

Ed