Infinities of Buds
Looking at any Bud’s attachment place on the cardioid, one can observe:
(for a Bud of Period p+q, in between adjacent Buds of Period p and q)
“Underneath” the Bud (between it and the cardioid) are two series of smaller and smaller Buds, one on either side.
Their Periods are (p+mq) and (q+mp) (for m = 1, 2. …).
Here we see Bud 3, flanked on the left by Bud 2, and on the right by Bud 4:
Along the Cardioid towards Bud 2, we can see Buds 3, 5, 7, 9, … – ie, 3+2m, m=0, 1, 2, … .
Towards Bud 4, we see Buds 4, 7, 10, … – ie: 4+3m, m=0, 1, 2, … .
(Note that Bud 3 is not “produced” by the combination of Buds 2 and 4 , since Bud 4 is not adjacent to Bud 2. Thus, 3 is not 2+4. What is it? 2+1. And so it’s Bud 1 that is the “co-parent, with Bud 2, of Bud 3. And so, we should expected the related series of Buds: simply 4, 5, 6, … – ie: 3+m, m=1, 2, 3, …)