Complete Hexabolo Constructions
All patterns use all hexaboloes. Therefore only a
limited symmetry (one axis) is possible because of the
odd number of
odd pieces . So I constructed a repetitive pattern (the
3 houses) and many convex patterns. A hexagon, a heptagon and a octagon with
one reflection axis are shown here. You can find the only symmetric pentagon
and the the only quadrangle among other
extreme constructions.
The list of all convex polygons with odd numbers of parallel h-edges
is very long. If you omitt all items with more than 120 h-edges or s-edges,
you will get 1853 different polygons. In the range between 120 to 140 h- or s-edges
there are eight polygons and I think they are impossible to build,
but it isn't quite sure.
The symmetric convex polygons are listed below. The numbers are the length of
the sides starting with h-units (right down), then s-units (right),
then h-units (right up) and so on.
pentagons
1: 0 6 0 52 3 0 3 52 | 110s + 6h = 118,49
hexagons
1: 0 15 0 19 3 9 3 19 | 62s + 6h = 70,49
2: 0 12 3 16 3 12 0 22 | 62s + 6h = 70,49
3: 0 14 0 22 1 12 1 22 | 70s + 2h = 72,83
4: 0 13 1 21 1 13 0 23 | 70s + 2h = 72,83
5: 0 6 11 4 11 6 0 26 | 42s + 22h = 73,11
6: 9 0 12 2 12 0 9 8 | 10s + 42h = 69,40
7: 0 11 0 27 3 5 3 27 | 70s + 6h = 78,49
8: 0 8 3 24 3 8 0 30 | 70s + 6h = 78,49
9: 0 10 0 30 3 4 3 30 | 74s + 6h = 82,49
10: 0 7 3 27 3 7 0 33 | 74s + 6h = 82,49
11: 0 2 11 12 11 2 0 34 | 50s + 22h = 81,11
12: 1 0 16 2 16 0 1 32 | 34s + 34h = 82,08
13: 0 1 13 9 13 1 0 35 | 46s + 26h = 82,77
14: 0 3 7 23 7 3 0 37 | 66s + 14h = 85,80
15: 0 7 0 45 1 5 1 45 | 102s + 2h = 104,83
16: 0 6 1 44 1 6 0 46 | 102s + 2h = 104,83
17: 2 0 5 40 5 0 2 46 | 86s + 14h = 105,80
18: 0 3 3 49 3 3 0 55 | 110s + 6h = 118,49
(-- 0 2 3 60 3 2 0 66 | 130s + 6h = 138,49 --)
heptagons
1: 8 4 11 0 11 4 8 6 | 14s + 38h = 67,74
2: 2 4 13 0 13 4 2 22 | 30s + 30h = 72,43
3: 6 2 6 16 7 0 7 16 | 34s + 26h = 70,77
4: 2 6 2 28 5 0 5 28 | 62s + 14h = 81,80
octagons
1: 2 13 2 13 7 3 7 13 | 42s + 18h = 67,46
2: 2 8 7 8 7 8 2 18 | 42s + 18h = 67,46
3: 4 9 9 1 9 9 4 11 | 30s + 26h = 66,77
4: 4 6 9 4 9 6 4 14 | 30s + 26h = 66,77
5: 8 3 11 1 11 3 8 7 | 14s + 38h = 67,74
6: 2 10 2 18 5 4 5 18 | 50s + 14h = 69,80
7: 2 7 5 15 5 7 2 21 | 50s + 14h = 69,80
8: 1 11 1 21 4 5 4 21 | 58s + 10h = 72,14
9: 1 8 4 18 4 8 1 24 | 58s + 10h = 72,14
10: 6 1 7 15 7 1 6 17 | 34s + 26h = 70,77
11: 2 3 9 11 9 3 2 25 | 42s + 22h = 73,11
12: 2 2 7 20 7 2 2 30 | 54s + 18h = 79,46
13: 2 3 5 25 5 3 2 31 | 62s + 14h = 81,80
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