Counting n-gons

[Jonathan Post]

I'm impressed by Richard Mathar's exposition.  Dr. George Hockney should be
flattered that he's been corrected by Richard Guy! I should be dubious about
anything asserted to me by a baseball fan during a World Series game.

I've also played with self-avoiding walks, and wonder how Richard Mathar's
approach works for polygons embedded in the 3-D integer lattice, some
results of which are known in Knot Theory, both for short walks and for
sufficiently long walks.  For example, almost all self-avoiding walks in the
3-D integer lattice which start and end at (0, 0, 0) are knotted. Of course,
polygons embedded in the 4-D integer lattice cannot be knotted.

-- Jonathan Vos Post

On 10/25/06, Richard Guy <rkg at cpsc.ucalgary.ca> wrote:
>
> max area of 1,1,2,2 quad is cyclic, a kite
> formed by 2 right triangles with legs 1 & 2,
> area 2.      R.
>
> On Tue, 24 Oct 2006, Jonathan Post wrote:
>
> > Dr. George Hockney (ex-Fermilab, now JPL)
> > responded to my suggestion that
> > the representative of each equivalence class of
> > deformed polygons with the
> > same edge-sequences be that with largest area
> > that: (1)  this might not be
> > unique, and (2) that the maximum area (1, 1, 2, 2)
> > quadrilateral is the
> > (1+1=2, 2, 2) equilateral triangle of area sqrt 3,
> > where the two edges of
> > length 1 make an angle of 180 degrees, and asks if
> > that equilateral triangle
> > with a vertex in the middle of its side is really
> > a quadrilateral, or is
> > that another ambiguity in equivalence of polygons
> > under geometrical
> > similarity? He is also dubious that the
> > integer-edged polygonal knots in R^3
> > are really polygons, as I've stated.
> >
> > I agree with David Wilson's sequence.  What about
> > a sequence table of
> > maximal areas of all simple integer-sided k-gons
> > of perimeter n?
> >
> > -- Jonathan Vos Post
> >
> > On 10/24/06, David Wilson
> > <davidwwilson at comcast.net> wrote:
> >>
> >> And we should have a sequence that counts all
> >> simple integer-sided
> >> polygons
> >> of perimeter n. Depending on exactly what we are
> >> counting, the sequence
> >> might go:
> >>
> >> a(3) = 1, the 1-1-1 triangle.
> >> a(4) = 1, the 1-1-1-1 quadrilateral.
> >> a(5) = 3, the 1-2-2 triangle, 1-1-1-2
> >> quadrilateral, the 1-1-1-1-1
> >> pentagon.
> >>
> >> etc.
> >>
> >>
> >
>
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