On 11/30/06, Fred lunnon <fred.lunnon@gmail.com> wrote:
I've now got a search program running which copes with equal lengths and possible imaginary faces. I'm not sure it's hoovering up every proper planar integer chart, but there are a lot more than previously --- so far approx 120 up to edge- sum 85, but none more than singly planar.
Fred Lunnon
This search program has trawled every (I hope) planar chart with integer edges and 4 proper triangles up to edge-sum 111, at which point I'm calling it a day (or two) for now. It found 250 planar hexads, quite a few of which show also a second "flawed" permutation also planar, but with one linear "face"; however _no_ genuine double charts were detected. In a remarkably large proportion of cases the vertices ("cities") are also concyclic. There are also many cases of multiples of smaller hexads; rhombuses and rectangles (Pythagorean); parallelograms; trapeziums with diagonals equal to one side; all of which can probably be organised into orderly families without excessive effort. The list is available on request. Since I can't meet Ed's challenge, I'll substitute a (gentler) problem of my own ---- Four cities, no three lying on a straight line, are linked by straight roads, each stretching an integer number of miles. The first road is 2 miles long; how long are the other five? Fred Lunnon