Discussion
The bipole, tripole, quadpole etc. constitute a sequence
that is unique among simpler oscillators.
It is therefore natural to look for numeric patterns in the corresponding sequences
of cycle and limit matrices.
There are no easy targets here since the inverting of the matrices I-S generates rather complicated
calculations.
One can observe at least three sets of invariant behaviour:
-
The determinants of the cycle matrices have the following values Dnfor n-poles:
Dn = 1/9n , if 3 is not a factor of n+1, and = 0 otherwise.
(E.g. bipole and pentapole have cycle matrix determinants = 0.)
This can be proved by studying subdeterminants of the transition matrices.
- If cells 2 and 3 are black initially while the opposite corresponding cells are white,
the intermediate rotor cells get a linear sequence of asymptotic colour indices.
Bipole: 2/3, 1/3.
Tripole: 3/4, 2/4, 1/4
and similarly for other poles (n-poles apparently get the denominator n+1).
In the limit matrices this shows in the sums of columns 2 and 3, which get the above components.
This phenomenon is hardly surprising in view of the symmetry of the patterns.
- The differences of columns 2 and 3 (of the G* parts of the limit matrices) exhibit some more intricate invariants.
Let Δn be the vector col2 - col3 for the limit matrix of the n-pole
and let Δn,j be its jth component. Note that Δn has n components.
The following apparent invariants of the number n are suggested from empirical evidence:
Δn,n-2/Δn,n-1 - Δn,n-1/Δn,n = 1/4
Δn,n-3/Δn,n-2 - Δn,n-2/Δn,n-1 =1/60
Δn,n-4/Δn,n-3 - Δn,n-3/Δn,n-2 = 1/840.
These regularities are still unproved.