hihi, all - i have found hundreds of 11-gons with area 21.5, with a computer program that looks at ones with longest edge at most 10 (in a few seconds), and thousands more, but none of smaller area, when the allowed edge lengths go up as high as 50 (in a lotta seconds: 13843.496u 18.417s), on commodity pcs with freebsd and linux (varying from 1.8GHz to 2.4GHz intel Pentium style machines) i've also looked at some larger polygons, to get some upper bounds on the minimum area, including (n = number of edges of polygon) (rad = maximum edge length, manhattan distance) (min = minimum area found) (number = number found with that same minimum area) (times = how long it took user cpu seconds, system cpu seconds, elapsed time, cpu % (2.2GHz celeron, 512MB main memory, running FreeBSD 5.1) (10ms clock tick, so that small times are unreliable) (. = same value as previous row) n rad min number times 10 20 14 203 57.200u 1.069s 1:03.54 91.6% . . . . 57.230u 0.835s 1:02.48 92.9% . . . . 55.306u 0.070s 1:00.19 91.9% . 30 . 430 261.873u 0.312s 4:41.90 93.0% . . . . 262.131u 0.328s 4:41.75 93.1% . 40 . 728 788.353u 1.101s 14:12.05 92.6% . . . . 788.329u 1.085s 14:08.73 93.0% . 50 . 1129 1920.953u 2.920s 34:41.34 92.4% 11 10 21.5 233 22.820u 0.000s 0:23.39 97.5% . 20 . 872 395.410u 0.010s 6:45.57 97.4% . . . . 394.030u 0.020s 6:45.59 97.1% . 30 . 1809 1823.860u 0.010s 31:12.17 97.4% . 40 . 3034 5403.410u 0.060s 1:32:29.02 97.3% . 50 . 4689 12985.860u 0.040s 3:42:13.26 97.3% 12 10 24 15 39.460u 0.010s 0:40.57 97.2% . 20 . 53 698.810u 0.000s 11:56.81 97.4% . . . . 695.760u 0.030s 11:54.40 97.3% . 30 . 112 3210.980u 0.030s 54:56.88 97.3% . 40 . 188 9484.200u 0.400s 2:42:30.10 97.2% 13 10 32.5 55 205.140u 0.010s 3:30.58 97.4% . 20 . 207 3964.860u 0.340s 1:07:50.53 97.4% . . . . 3946.270u 0.000s 1:07:31.88 97.3% . 30 . 423 18152.870u 1.390s 5:11:07.84 97.2% 14 10 40 68 709.800u 0.050s 12:09.59 97.2% . 20 . 248 14871.960u 0.080s 4:14:06.26 97.5% . . . . 14788.790u 1.100s 4:13:29.98 97.2% 15 10 51.5 121 3606.580u 0.000s 1:01:43.25 97.3% . 20 . 482 85559.430u 0.710s 24:22:23.00 97.5% 16 10 59 16 9207.270u 0.390s 2:37:48.86 97.2% 17 10 75.5 72 55945.880u 1.020s 15:56:06.60 97.5% 18 10 87 51 169100.770u 3.070s 48:08:23.04 97.5% 19 10 ?? ?? a long, long, time 8-) it will perhaps finish thursday, if i don't use the machine for anything else anyway, i obviously need a new program - i'm looking at something that will more quickly enumerate lattice points in wedges (the program is a simple backtracker that enumerates the points in the wedge between the line of the previous edge and the line towards the original point, centered at the current point - my program does this now by enumerating the entire lattice spiral and checking bounds, which isn't so bad as it might seem, since early on in the search, the wedges are fairly wide, and the spiral search is easy and fast enough - i want to switch over to a more direct enumeration at the end when the wedges are not very wide and other constraints can be applied) i imagine that it should be encouraging for the lower bound minded among you that the best value found for the minimum area is found already when the radius bound is 10 (and only n=17, among those with n<= 18, even needed to use a radius bound as much as 5 - that is, for each n<=18 except n=17, there was a polygon, with the minimal area that the program found for any bound i tried, with maximum edge length <= 4, but for n=17, they all had maximum edge lengths at least 5) it would be significantly helpful if there were a theorem upper bounding the smallest maximum edge length for a minimum area (convex lattice) n-gon (one for which the bound was a function of n that is not too large) - then this radius bounded search could be used to determine the precise minimum area more later, cal Chris Landauer Aerospace Integration Science Center The Aerospace Corporation cal@aero.org