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 #(PC_{m,l}^{k}) 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: PC_{2,2}^{3}; Bottom: : PC_{3,2}^{2}
If we fix two of the parameters k, m,
and l and let the other vary, we
obtain an infinite sequence of integers #(PC_{m,l}^{k}).
For k = 1 we have
#(PC_{m,l}) =
#(P_{m}) = L_{2m} and #(PC_{m,l}+S_{1}) = #(P_{m}+S_{1}) = F_{2m+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,
#(PC_{m,}_{1}^{k}) = R_{m,l} λ_{m,l}^{k}^{1}
+ R_{m,l} μ_{m,l}^{k}^{1},
where λ and μ (λ
> μ) are the eigenvalues of the matrix M = [m_{p,q}], with
m_{1,1} = L_{2m}F_{l}_{2} +
F_{2m+2}F_{l}_{1}, m_{1,2}
= L_{2m}F_{l}_{1} + F_{2m+2}F_{l}
m_{2,1} = F_{2m+2}F_{l}_{2}
+ F_{l}_{1}(F_{m}_{+2})^{2}, m_{2,2}
= F_{2m+2}F_{l}_{1} + F_{l}(F_{m}_{+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 
P_{2}^{i}(S_{1}) 
7 
113 
1815 
29153 
468263 
P_{i}^{2}(S_{1}) 

113 
765 
5234 
35865 
P_{2}^{2}(S_{i}) 
113 
176 
289 
465 
754 
P_{2}^{i}(S_{2}) 
7 
176 
4393 
109649 
2736832 
P_{i}^{3}(S_{2}) 

4393 
80361 
1425131 
25671393 
P_{3}^{3}(S_{i}) 
32733 
80361 
215658 
559305 
1469565 
The polygonal chains provide an infinite threeparameter family of previously unknown integer sequences, some of which can now be found in the OnLine Encyclopedia of Integer Sequences.