hihi, all - I wrote:
Dan Hoey wrote (paraphrased):
...it's not a proof...
This is of course true. ...
...all of the minimum area polygons that my program found for even n are centrally symmetric (that is, the second half edge directions are the negatives of the first half ones). ... I am presently writing the code to use this observation in a different systematic search program.
This program has now run systematically through all the even n<=50 in less than an hour total (it got the same polygon answers for even n<=20 as my other program, in much less time): min min min longest n area perim edge 22 164 62 5 24 210 72 5 26 274 80 5 28 345 90 5 30 430 98 5 32 523 108 5 34 632 118 5 36 749 128 5 38 890 138 5 40 1039 148 5 42 1222 162 7 44 1412 176 7 46 1620 192 7 48 1847 202 7 50 2100 214 7 It should be noted still that the minimum longest edge and minimum perimeter do not necessarily occur with the same polygon, but they are minima over the minimum area polygons found of that size. Also, the minimum longest edge found and minimum perimeter can be less then the above entries, at a small cost in area (there is an n=22 polygon with perimeter 60 and longest edge 4, but it has area 167). Finally, the random search found these same minima for n<=30. more soon, cal Chris Landauer Aerospace Integration Science Center The Aerospace Corporation cal@aero.org