Secondary Filament

Robert P. Munafo, 2023 Mar 13.

In the nomenclature developed for the R2 naming system, a secondary filament is a connected set which is a subset of a primary filament and which is one of the "pieces" that remains after the lowest-period island is removed from that primary filament.

To illustrate by example, consider the "largest" primary filament, which is the one stretching from about -1.5437+0i out to the tip at -2+0i. This filament contains the largest island, R2F(1/2B1)S. If that island is removed, what remains is a lot of pieces that are each themselves connected sets, but are not connected to each other.

The Secondary Filaments of R2F(1/2B1)

the two largest
endpoints	external angle relative to R2F(1/2B1)S	filament subset suffix (abbreviated)	location of lowest- period island	period	full R2-name
-1.75+0i inwards to -1.5437+0i	0	FS[0]	-1.627+0i	5	R2F(1/2B1)FS[(0]S
from about -1.7903+0i outwards to -2+0i	1/2	FS[2]	-1.941+0i	4	R2F(1/2B1)FS[(1/2B1)t]S
two smaller ones:
endpoint that touches R2F(1/2B1)S	external angle relative to R2F(1/2B1)S	filament subset suffix (abbreviated)	location of lowest- period island	period	full R2-name
outwards from about -1.7603+0.0205i	1/4	FS[3]	-1.76924 + 0.05692i	7	R2F(1/2B1)FS[(1/3B1)t]S
outwards from about -1.7603-0.0205i	3/4	FS[2/3]	-1.76924 - 0.05692i	7	R2F(1/2B1)FS[(2/3B1)t]S
four smaller ones:
outwards from about -1.7490+0.0138i	1/8	FS[4]	-1.741584 + 0.025931i	10	R2F(1/2B1)FS[(1/4B1)t]S
outwards from about -1.7781+0.0077i	3/8	FS[2(3)]	-1.786869 + 0.019443i	10	R2F(1/2B1)FS[(1/2(1/3B1)B1)t]S
outwards from about -1.7781-0.0077i	5/8	FS[2(2/3)]	-1.786869 - 0.019443i	10	R2F(1/2B1)FS[(1/2(2/3B1)B1)t]S
outwards from about -1.7490-0.0138i	7/8	FS[3/4]	-1.741584 - 0.025931i	10	R2F(1/2B1)FS[(3/4B1)t]S
eight even smaller ones
and so on . . .

Filament Subset Operation

Of all the islands in a secondary filament, there is exactly one with lowest period. If that island is omitted, what remain are an infinite set of "tertiary" filaments. The filament subset operator (see R2 rules) gives names to the tertiary filaments, and it can be applied repeatedly to generate names for "quaternary", etc. filaments.

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This page was written in the "embarrassingly readable" markup language RHTF, and was last updated on 2023 Jun 02.

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