self-avoiding closed loops
Wouter Meeussen
w.meeussen.vdmcc at vandemoortele.be
Fri Sep 11 10:15:42 CEST 1998
hi,
about:
self-avoiding closed loops on a hexagonal planar net.
For any binary string with a leading '1',
draw a path on a hexagonal net, turning right for '1', and left for '0' at each
junction.
If we consider paths to start at (0,0), and the leading '1' to represent the
vector ending in (1/2,Sqrt[3]/2),
then 1 , 1,1,1,1,1 is a closed hexagon turning right, and
1 , 0,0,0,0,0 is a closed hexagon turning left.
Altough the 'topological' paths are often equivalent (they can be made to
overlap by
translation, rotation and mirroring), the binary strings have a special
significance :
any self-intersecting path must have one of these binary sequences
'embedded' in its
binary representation.
Since the initial direction vector is arbitrarily choosen, but fixed, any
topological
figure will occur as many times as it has sides with *that* direction. This
is equivalent
to the translational symmetry. On the bit-string, it corresponds to
*rotating* it untill
an other '1' stands in front.
The three rotational positions of the topological figure also appear.
Mirroring of the topological figure:
the paths occur in pairs, if {1, string_of_binaries} occurs, then also
{1, string_of_binaries_with_(1 <-> 0)_exchange}.
****
So, there is a (strange) analogy between this "representation" of binary strings
and the more familiar NECKLACES.
****
the count of non-crossing closed loops on a planar hexagonal net is:
for n= 2, 4, 6, 8, 10, 12, 14, 16, 18, 20 binary digits or 'legs',
{0, 0, 2, 0, 10, 6, 26, 96, 390, 920 self-avoiding loops.
(timing: the last entry took 3.5 hours and the next would take 4 times as much).
the paths themselves are (archiving purposes):
{{},
{},
{32, 63},
{},
{528, 545, 578, 644, 759, 776, 891, 957, 990, 1007},
{2184, 2321, 2594, 3140, 3822, 3959},
{8481, 8514, 9264, 9288, 10167, 10320, 10337, 10385, 12093, 12126, 12143,
12432, 12449, 12482, 12578, 14061, 15222, 15225,
15287, 15311, 15606, 15738, 15803, 15804, 15995, 16094},
{34065, 34338, 34977, 35010, 35106, 35352, 35363, 35364, 35880, 35909,
37187, 37188, 37253, 37416, 37445, 37937, 37958, 37961, 38993, 39050,
40407, 40635, 41352, 41520, 41544, 41606, 41609, 41738, 42065, 42122,
43106, 43148, 43154, 44763, 44775, 44859, 45218, 45332, 46827, 46941,
47595, 47853, 47859, 47982, 48030, 48047, 48366, 48503, 49800, 49937,
50256, 50273, 50321, 50444, 50450, 50708, 51362, 51476, 52971, 53085,
53444, 53528, 53540, 55149, 55155, 55197, 56181, 56238, 56565, 56694,
56697, 56759, 56783, 56951, 57668, 57896, 59253, 59310, 60342, 60345,
60366, 60858, 60887, 61050, 61115, 61116, 62394, 62423, 62939, 62940,
62951, 63197, 63293, 63326, 63965, 64238}
rest not sent
}
wouter.
---------------------------------------------------------------------
Mathematica 3.0 program:
selfavoidingQ[k_Integer]:=
Module[{le,n=2,nogo={0},argu=0.,elem=10.,rel=10},
bin=IntegerDigits[k,2];le=Length[bin];
While[rel=!=0&&FreeQ[nogo,rel]&&n<=le,nogo={nogo,rel};
elem=Chop[elem+10. N at Exp[argu+=I Pi/3 (2 bin[[n]]-1)]];rel=Round[elem];
n++]; rel===0&&le+1===n ]
Timing[Cases[Range[2^n,2^(n+1)-1],_?selfavoidingQ]]
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