The Strange World of the
Hausdorff Metric Geometry
In 2006, Lisa Morales, Dan
Schultheis, and I explored a new family of finite configurations, the polygonal
chains. A polygonal chain #(PCm,lk) is constructed by chaining together k
copies of m polygonal configurations with string configurations of
length l. Two examples are shown below.
Figure 11
Top: PC2,23; Bottom: : PC3,22
If we fix two of the parameters k, m,
and l and let the other vary, we
obtain an infinite sequence of integers #(PCm,lk).
For k = 1 we have
#(PCm,l) =
#(Pm) = L2m and #(PCm,l+S1) = #(Pm+S1) = F2m+2.
Using
a lot of induction and some techniques from linear algebra in [22], we found
that for m ≥ 2, k ≥
2, and l ≥ 1,
#(PCm,1k) = Rm,l λm,lk-1
+ Rm,l μm,lk-1,
where λ and μ (λ
> μ) are the eigenvalues of the matrix M = [mp,q], with
m1,1 = L2mFl-2 +
F2m+2Fl-1, m1,2
= L2mFl-1 + F2m+2Fl
m2,1 = F2m+2Fl-2
+ Fl-1(Fm+2)2, m2,2
= F2m+2Fl-1 + Fl(Fm+2)2
and
, and .
In these formulas we let F-1 = 1. The table
below shows a few entries in these polygonal chain sequences.
i |
1 |
2 |
3 |
4 |
5 |
P2i(S1) |
7 |
113 |
1815 |
29153 |
468263 |
Pi2(S1) |
|
113 |
765 |
5234 |
35865 |
P22(Si) |
113 |
176 |
289 |
465 |
754 |
P2i(S2) |
7 |
176 |
4393 |
109649 |
2736832 |
Pi3(S2) |
|
4393 |
80361 |
1425131 |
25671393 |
P33(Si) |
32733 |
80361 |
215658 |
559305 |
1469565 |
The polygonal chains provide an infinite three-parameter family of previously unknown integer sequences, some of which can now be found in the On-Line Encyclopedia of Integer Sequences.