Squares & sticks
tivol at tethys.ph.albany.edu
tivol at tethys.ph.albany.edu
Fri Jun 17 11:15:38 EST 1994
Dear James,
For 21 cm sticks, where the squares are 25 cm on a side, the configura-
tion you drew in your last post is not possible. For the L-shaped case, say
A1, B1 and B2 share some sticks, A1 and B1 share some sticks and B1 and B2
share some sticks, but A1 and B1 do *not* share any sticks uniquely (unless
share some sticks, but A1 and B2 do *not* share any sticks uniquely (unless
a stick of zero thickness passes exactly through the corner shared by A1 and
B2). Thus, since every stick shared by A1 and B2 is also shared either by
B1 or A2, the appropriate configuration should be revealed if there are en-
ough sticks.
For the medium stick case (25 < L < 56 [=25*sqrt(5)]), the expectation
is that there will be some sticks shared by A1, B1 and C1, some by A1 and B1
uniquely and some by B1 and C1 uniquely, but none by A1 and C1 uniquely.
Furthermore, all sticks shared by A1 and C1 are also shared by B1, with no
possible alternative, which is different from the case for A1, A2, B1 and B2.
the fraction of sticks shared by 2, 3, 4(?),...squares should also help to
characterize the relationships among the squares.
The small stick case will have sticks entirely within one square, those
shared by two or three squares, and, for sticks of finite thickness, four
squares (if a stick lies on a corner); there are no other possibilities. The
medium stick case *may* have sticks lying within a single square (25 < L <
~51), will have sticks sharing two, three, or four squares (same caviat re
corners), and, in addition, may have sticks sharing four squares (where the
stick is nearly parallel eo an edge and overlaps squares on either end, e.g.
A1, A2, B2 and C2 or A1, B1, B2 and B3), five squares (~51 < L < ~56) where
the stick is parallel to a diagonal and slightly displaced from it, e.g. A1,
B1, B2, C2 and C3, or six or seven squares when the finite-thickness stick
lies on a corner or two corners--you can easily see this with a paper & pen-
cil, and I can't draw it well on the screen. The various distributions ob-
served should distinguish the possibilities if there are enough sticks.
I guess that the fact that all sticks shared by A1 and C1 are also
shared by B1 is an example of dependance, and that sticks shared by A1 and
B1 are independent of those shared by C4 and D4. From the above, it seems
to be necessary either to have full knowledge of *all* the squares shared by
various sticks--as opposed to only the knowledge that a particular pair
shares a stick, possibly with other square(s)--or to have several experiments
each of which has different length sticks.
All this reminds me of binding studies in ribosomes, where bifunctional
reagents of various lengths were found to bind subunits. The proximities of
the subunits, and, eventually, the structure of the assembly were found from
these data. Here, no reagent bound more than two subunits, so there is less
information available than in your case. If you remain stuck, you might try
the literature in this or related field(s). Good luck.
Yours,
Bill Tivol
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