1: The _division property_ - every nth term in order from the beginning reproduces the series or its inversion.

2: The _insertion property_ - consider an alternation between f and g; start with a seed such as g; surround it by f: fgf, insert and surround by g: gfgggfg, etc. - the series and its inverse are produced.

3: Concatenation property: Begin with seed gf; start from the left. Substitute gggf for g and gfgf for f. Add the substitutions in order on the right. Take each term in order. The series is produced. Or: begin with g, take seed f; substitute gg for f, gf for g. The series is produced.

4. For the zth term let m equal the largest integer such that z/2expm = integer. Then the term is g if m is even, f if m is odd.

There are other formulations and extensions; I've worked with this series for years, on and off. Now I think of it in terms of ordered linkages, equivalences, and identities. If one has all terms, say, g, then one has a number system to the base 1, i.e. g, gg, ggg, gggg, etc. Consider g - gg as Quine's n' operation. One might also use terms fgh, etc.

The inversion of the series, i.e. fgfffgfgfgfff etc., is equivalent to the series to the extent that the structure is based on a primary differntiation |f-g| = |g-f|.

The series, by the division property, is extraordinarily ordered, even more say than, say, simple alternation fgfgfg - where the division proper ty breaks down. If alternations are considered fff, fgfg, fghfgh, etc. then it's clear that g gg ggg gggg etc. represents the substance or null point of both series.

Note by substitution these are not couplings; an error anywhere in the terms will most likely propagate (by virtue of substitution construction).

So there are these relations: The calculus in Spencer Brown's Laws of Form; the halfgroupoid analyses in R. Hubert Bruck's A Survey of Binary Systems; materials such as the Sheffer stroke and its dual in relation to the propositional calculus (see also Wittgenstein's Tractatus and its mathematics); my own 'loosenings' of couplings and linkages and for that matter the formalism in my Structure of Reality (related to dynamic networks such as Petrie nets); and these ordered strings related to ruler topologies.

All of which tends towards a metaphysics of mathematics, or at least a metaphysical mathesis that must (I think) be resisted. On the other hand, there are useful metaphors (the dual of the Sheffer stroke as expulsion) and phenomenologies (difference between couplings and linkages) that may be of use thinking through (again for example) such things as classical causality; the foreclosings of bodies and worlds in a classical/modernist context in relation to virtual subjectivities; and postmodern geographies in general.

Finally, the ruler series is of great use, at least for me, in thinking through the related phenomenologies of _substance, order, thing, entity, extension, difference, and chain, etc._ Note in the example below of period 64, each line is identical except for the last symbol - and the _column_ of last symbols recapitulates the orderings within the _rows._

gfgggfgfgfgggfgggfgggfgfgfgggfgfgfgggfgfgfgggfgggfgggfgfgfgggfgg
gfgggfgfgfgggfgggfgggfgfgfgggfgfgfgggfgfgfgggfgggfgggfgfgfgggfgf
gfgggfgfgfgggfgggfgggfgfgfgggfgfgfgggfgfgfgggfgggfgggfgfgfgggfgg
gfgggfgfgfgggfgggfgggfgfgfgggfgfgfgggfgfgfgggfgggfgggfgfgfgggfgg
gfgggfgfgfgggfgggfgggfgfgfgggfgfgfgggfgfgfgggfgggfgggfgfgfgggfgg
gfgggfgfgfgggfgggfgggfgfgfgggfgfgfgggfgfgfgggfgggfgggfgfgfgggfgf
gfgggfgfgfgggfgggfgggfgfgfgggfgfgfgggfgfgfgggfgggfgggfgfgfgggfgg
gfgggfgfgfgggfgggfgggfgfgfgggfgfgfgggfgfgfgggfgggfgggfgfgfgggfgf
gfgggfgfgfgggfgggfgggfgfgfgggfgfgfgggfgfgfgggfgggfgggfgfgfgggfgg
gfgggfgfgfgggfgggfgggfgfgfgggfgfgfgggfgfgfgggfgggfgggfgfgfgggfgf
gfgggfgfgfgggfgggfgggfgfgfgggfgfgfgggfgfgfgggfgggfgggfgfgfgggfgg
gfgggfgfgfgggfgggfgggfgfgfgggfgfgfgggfgfgfgggfgggfgggfgfgfgggfgg
gfgggfgfgfgggfgggfgggfgfgfgggfgfgfgggfgfgfgggfgggfgggfgfgfgggfgg
gfgggfgfgfgggfgggfgggfgfgfgggfgfgfgggfgfgfgggfgggfgggfgfgfgggfgf
gfgggfgfgfgggfgggfgggfgfgfgggfgfgfgggfgfgfgggfgggfgggfgfgfgggfgg
gfgggfgfgfgggfgggfgggfgfgfgggfgfgfgggfgfgfgggfgggfgggfgfgfgggfgg
gfgggfgfgfgggfgggfgggfgfgfgggfgfgfgggfgfgfgggfgggfgggfgfgfgggfgg
gfgggfgfgfgggfgggfgggfgfgfgggfgfgfgggfgfgfgggfgggfgggfgfgfgggfgf
gfgggfgfgfgggfgggfgggfgfgfgggfgfgfgggfgfgfgggfgggfgggfgfgfgggfgg
gfgggfgfgfgggfgggfgggfgfgfgggfgfgfgggfgfgfgggfgggfgggfgfgfgggfgg
gfgggfgfgfgggfgggfgggfgfgfgggfgfgfgggfgfgfgggfgggfgggfgfgfgggfgg
gfgggfgfgfgggfgggfgggfgfgfgggfgfgfgggfgfgfgggfgggfgggfgfgfgggfgf
gfgggfgfgfgggfgggfgggfgfgfgggfgfgfgggfgfgfgggfgggfgggfgfgfgggfgg
gfgggfgfgfgggfgggfgggfgfgfgggfgfgfgggfgfgfgggfgggfgggfgfgfgggfgf
gfgggfgfgfgggfgggfgggfgfgfgggfgfgfgggfgfgfgggfgggfgggfgfgfgggfgg
gfgggfgfgfgggfgggfgggfgfgfgggfgfgfgggfgfgfgggfgggfgggfgfgfgggfgf
gfgggfgfgfgggfgggfgggfgfgfgggfgfgfgggfgfgfgggfgggfgggfgfgfgggfgg
gfgggfgfgfgggfgggfgggfgfgfgggfgfgfgggfgfgfgggfgggfgggfgfgfgggfgg
gfgggfgfgfgggfgggfgggfgfgfgggfgfgfgggfgfgfgggfgggfgggfgfgfgggfgg
gfgggfgfgfgggfgggfgggfgfgfgggfgfgfgggfgfgfgggfgggfgggfgfgfgggfgf
gfgggfgfgfgggfgggfgggfgfgfgggfgfgfgggfgfgfgggfgggfgggfgfgfgggfgg
gfgggfgfgfgggfgggfgggfgfgfgggfgfgfgggfgfgfgggfgggfgggfgfgfgggfgf
gfgggfgfgfgggfgggfgggfgfgfgggfgfgfgggfgfgfgggfgggfgggfgfgfgggfgg
gfgggfgfgfgggfgggfgggfgfgfgggfgfgfgggfgfgfgggfgggfgggfgfgfgggfgf
gfgggfgfgfgggfgggfgggfgfgfgggfgfgfgggfgfgfgggfgggfgggfgfgfgggfgg
gfgggfgfgfgggfgggfgggfgfgfgggfgfgfgggfgfgfgggfgggfgggfgfgfgggfgg
gfgggfgfgfgggfgggfgggfgfgfgggfgfgfgggfgfgfgggfgggfgggfgfgfgggfgg
gfgggfgfgfgggfgggfgggfgfgfgggfgfgfgggfgfgfgggfgggfgggfgfgfgggfgf
gfgggfgfgfgggfgggfgggfgfgfgggfgfgfgggfgfgfgggfgggfgggfgfgfgggfgg
gfgggfgfgfgggfgggfgggfgfgfgggfgfgfgggfgfgfgggfgggfgggfgfgfgggfgf
gfgggfgfgfgggfgggfgggfgfgfgggfgfgfgggfgfgfgggfgggfgggfgfgfgggfgg
gfgggfgfgfgggfgggfgggfgfgfgggfgfgfgggfgfgfgggfgggfgggfgfgfgggfgf
gfgggfgfgfgggfgggfgggfgfgfgggfgfgfgggfgfgfgggfgggfgggfgfgfgggfgg
gfgggfgfgfgggfgggfgggfgfgfgggfgfgfgggfgfgfgggfgggfgggfgfgfgggfgg
gfgggfgfgfgggfgggfgggfgfgfgggfgfgfgggfgfgfgggfgggfgggfgfgfgggfgg
gfgggfgfgfgggfgggfgggfgfgfgggfgfgfgggfgfgfgggfgggfgggfgfgfgggfgf
gfgggfgfgfgggfgggfgggfgfgfgggfgfgfgggfgfgfgggfgggfgggfgfgfgggfgg
gfgggfgfgfgggfgggfgggfgfgfgggfgfgfgggfgfgfgggfgggfgggfgfgfgggfgg
gfgggfgfgfgggfgggfgggfgfgfgggfgfgfgggfgfgfgggfgggfgggfgfgfgggfgg
gfgggfgfgfgggfgggfgggfgfgfgggfgfgfgggfgfgfgggfgggfgggfgfgfgggfgf
gfgggfgfgfgggfgggfgggfgfgfgggfgfgfgggfgfgfgggfgggfgggfgfgfgggfgg
gfgggfgfgfgggfgggfgggfgfgfgggfgfgfgggfgfgfgggfgggfgggfgfgfgggfgg
gfgggfgfgfgggfgggfgggfgfgfgggfgfgfgggfgfgfgggfgggfgggfgfgfgggfgg
gfgggfgfgfgggfgggfgggfgfgfgggfgfgfgggfgfgfgggfgggfgggfgfgfgggfgf
gfgggfgfgfgggfgggfgggfgfgfgggfgfgfgggfgfgfgggfgggfgggfgfgfgggfgg
gfgggfgfgfgggfgggfgggfgfgfgggfgfgfgggfgfgfgggfgggfgggfgfgfgggfgf
gfgggfgfgfgggfgggfgggfgfgfgggfgfgfgggfgfgfgggfgggfgggfgfgfgggfgg
gfgggfgfgfgggfgggfgggfgfgfgggfgfgfgggfgfgfgggfgggfgggfgfgfgggfgf
gfgggfgfgfgggfgggfgggfgfgfgggfgfgfgggfgfgfgggfgggfgggfgfgfgggfgg
gfgggfgfgfgggfgggfgggfgfgfgggfgfgfgggfgfgfgggfgggfgggfgfgfgggfgg
gfgggfgfgfgggfgggfgggfgfgfgggfgfgfgggfgfgfgggfgggfgggfgfgfgggfgg
gfgggfgfgfgggfgggfgggfgfgfgggfgfgfgggfgfgfgggfgggfgggfgfgfgggfgf
gfgggfgfgfgggfgggfgggfgfgfgggfgfgfgggfgfgfgggfgggfgggfgfgfgggfgg
gfgggfgfgfgggfgggfgggfgfgfgggfgfgfgggfgfgfgggfgggfgggfgfgfgggfgg

Pub. March 2000

DRC