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 X_i,m,l = #(PC_{m, l}ⁱ⁺¹) / #(PC_{m, l}ⁱ), Y_k,i,l = #(PC_{i+1, l}^k) / #(PC_{i, l}^k), and Z_k,m,i = #(PC_{m, i+1}^k) / #( PC_{m, i}^k).

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:

· X_i,m,l → λ_m,l

· Y_k,i,l → φ^2k

· Z_k,m,i → φ^k-1

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

For example, compare the table entries to the values λ_2,1 = 8 + √65 » 16.06225775 and λ_2,2 = (25 + 3√69)/2 » 24.95993579.