n-gon sequence

[Jon Perry]

If you draw a triangle, and then another inside it, with a common vertex,
how many times do these triangles intersect?

The answer could be 1, or if we compared endpoints of the edges of each
triangle, we get 4, which is what I used.

A simpler version is if a line passes through a vertex of a triangle - how
many times does the line intersect the triangle?

Jon Perry
perry at globalnet.co.uk
http://www.users.globalnet.co.uk/~perry/maths
BrainBench MVP for HTML and JavaScript
http://www.brainbench.com


-----Original Message-----
From: Brendan McKay [mailto:bdm at cs.anu.edu.au]
Sent: 15 October 2002 00:16
To: Jon Perry
Cc: Seqfan at Ext.Jussieu.Fr
Subject: Re: n-gon sequence


If no two triangles can intersect in more than 6 points, then
three triangles can't have more than binomial(3,2)*6 = 18 points
of intersection altogether.  I can't understand your example at all.

Brendan.


* Jon Perry <perry at globalnet.co.uk> [021015 03:15]:
> With further research, I got 20 intersections from 3 triangles - I did say
> it was without proof, as is my 20 being maximal.
>
> (Create a Star of David, and connect 2 opposite vertices up. The number of
> intersections here is 4).
>
> I also got the sequence for maximal intersections of 2 n-gons wrong.
>
> 3 and 4 are 6,8, but I got 18 and 20 intersection from two (non-self
> intersecting) 5- and 6- gons.
>
> But again these are without proof, so the bound might rise again.
>
> Jon Perry
> perry at globalnet.co.uk
> http://www.users.globalnet.co.uk/~perry/maths
> BrainBench MVP for HTML and JavaScript
> http://www.brainbench.com
>
>
> -----Original Message-----
> From: Brendan McKay [mailto:bdm at cs.anu.edu.au]
> Sent: 13 October 2002 14:53
> Cc: Seqfan at Ext.Jussieu.Fr
> Subject: Re: n-gon sequence
>
>
> * Jon Perry <perry at globalnet.co.uk> [021013 23:02]:
> > Is this sequence known? I've only worked out the first 2 terms of each.
> >
> > The maximum number of intersections of k n-gons.
> >
> > e.g. with triangles, the sequence opens 0,6 - my next term is 16, but
this
> > is without proof.
> >
> > For the general n-gon, the sequence always opens 0,2n
>
> Can you give a more precise definition please?  It is easy to
> draw 3 triangles so that each pair intersect in 6 points and
> all intersections are distinct, making 18 altogether.  So I
> guess you must mean something else.
>
> Brendan.