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The Strange World of the Hausdorff Metric Geometry

 

XVII. Polygonal Chain Sequences, Asymptotic Behavior

The numbers in the polygonal chain sequences appear to increase quite rapidly. The successive quotients as shown in the following table indicate that the behavior of these sequences is asymptotically exponential. For the sake of convenience, let Xi,m,l = #(PCm, li+1) / #(PCm, li), Yk,i,l = #(PCi+1, lk) / #(PCi, lk), and Zk,m,i = #(PCm, i+1k) / #( PCm, ik).

 

i

6

7

8

9

10

Xi,2,1

16.06225775

16.06225775

16.06225775

16.06225775

16.06225775

Y2,i,1

6.854063862

6.854096407

6.854101155

6.854101848

6.854101949

Z2,2,i

1.618539787

1.617840851

1.618107769

1.618005808

1.618044753

Xi,2,2

24.95993579

24.95993579

24.95993579

24.95993579

24.95993579

Y3,i,2

17.95473888

17.94023899

17.94580726

17.94368472

17.94449609

Z3,3,i

2.619410257

2.617508525

2.618234731

2.617957317

2.618063275

 

In fact, these sequences are asymptotically exponential, and the limits of the ratios of consecutive terms are given in [21] as follows:

         Xi,m,l → λm,l

         Yk,i,l → φ2k

         Zk,m,i → φk-1

where φ is the golden ratio and λm,l is as defined on the previous page.

For example, compare the table entries to the values λ2,1 = 8 + √65 16.06225775 and λ2,2 = (25 + 3√69)/2 24.95993579.

 

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