hihi, all - minimum area convex lattice polygons well, this problem has really caught my attention and held it (i've been dreaming about graph paper and polygons) - every time i decide to stop working on it i think of something else that will speed up the programs, give better results, or contribute towards proving that this method works the extra normalization i mentioned before did not work usefully: polygons normalized to have vertices (0,1), (0,0), and (1,0) often have longer edge lengths, and the larger number of fractions needed to find longer edges greatly overshadows the savings on selection i used this unsurprising (but unanticipated 8-() observation to find yet another faster algorithm for even n: the program makes all of the n/2 selections from the set of 2*nf fractions representing all possible edges in one half of a centrally symmetric polygon (using the original set of possible edges for the first corner and their +90 degree rotations for the second corner), and it does not assume that (0,0) -> (1,0) is the first edge (only that (0,0) is the first vertex and all y-values are non-negative) i also realized that any polygon edge can be transformed into (0,0) -> (1,0) by a rigid motion and some SL(2,Z) element, so i changed the reduction program and reduced almost all minimum area polygons that my various programs found down to a unique polygon for each size (i'm still not convinced that i am finished with this reduction) - only n=7, 11, 15, 20, and 22 stil have two distinct minimum area polygons under these transformations for n<=50 summary of results so far min number of min min longest sl(2,z) unique n area perim edge polygons (if not 1) 3 0.5 4 2 4 1.0 4 1 5 2.5 8 2 6 3.0 8 2 7 6.5 12 3 2 8 7.0 12 2 9 10.5 16 3 10 14.0 18 3 11 21.5 22 3 2 12 24.0 24 3 13 32.5 28 4 14 40.0 30 3 15 51.5 34 4 2 16 59.0 36 3 17 75.5 42 5 18 87.0 44 4 19 106.5 50 5 20 121 52 4 2 21 144.5 60 5 22 164 62 5 2 23 24 210 72 5 25 26 274 80 5 27 28 345 90 5 29 30 430 98 5 31 32 523 108 5 33 34 632 118 5 35 36 749 128 5 37 38 890 138 5 39 40 1039 148 5 41 42 1222 162 7 43 44 1412 176 7 45 46 1620 192 7 47 48 1838 208 8 49 50 2088 218 8 i still can't figure out what B&T mean by ``the minimizing polygon'', so i have no useful comment about that paper yet more soon, cal