Duality in the Context of Projective Geometry

Interchange elements such as points & lines edges, planes & faces & intersections & you can see them differently but some projective elements stay related in the same way: (***)


It gets rather lost in abstract algebra without the geometry graphics
https://en.wikipedia.org/wiki/Duality_(projective_geometry) . Except for maybe 2 classes when I took projective geometry at UCLA it was abstract algebra – didn’t look anything like GW’s works.
However it’s coming back again with virtual reality & graphics & games in the modern digital age.

that may, or probably is not, the same thing that i have been talking about under the hashtag #duality … i suspect that this sense of “duality” is just a particular mathematical definition.  it is interesting, however, if it is the same thing just in another context.

Actually, a duality has to have a connection otherwise it is just a couple of things. That connection or relationship or focus says something about other dualities of the same flavor. Me(I) can enjoy it without having to assert a difference – even yet. 
Ooogie what a thought – you & I are a duality – what kind?

usually when i think of #duality i am thinking of contexts like “the mind is in the body, and the body is in the mind” … or “you are in me, and i am in you” … stuff like that.

I go more general like to #two-ness i.e. ye not yee

yeah 2 is a different #edge.   what i am referring to as #duality may not have very much to do with 2.  neither does the sense of duality to which you refered above. 

Here is another sense … perhaps closer to the edge that i am stufying …

Duality is a connection between two things where the properties of one defines the properties of the other.

thought 22371

thinking about that one was what got me using this hastag again.  i have yet to totally digest that.

I feel this #duality #edge when i know that you can never have my exact qualia.  You cannot have my subjective experiences.  Oh sure you can have similar ones.  You can also listen to my descriptions of my subjective experiences and try to duplicat them for yourself … and shucks you can come pretty close i imagine.  But mine are mine … are on this side of a #duality that nature will not let you cross.  Some #spriitual stories will deny and contradic that.  So be it.  I am in one of those #boxes … and/or you are in one of those #boxes in which that is not null

I suspect that some hidden desire behind your Me(I) is at play here.

the hidden desire is to become aware of it … to understand it … even to be able to predict how others and myself crawl on the edge of this #duality … perhaps better to interact with others … perhaps better to talk with others ... better to share our #manifestations … nothing sinsiter at all … nothing #Egoo’ey at all.  … and #shucks, mark it is not even hidden … i wear it right on my face

was not implying it sinister

The fact of an Ego builds a duality . “or I am divided for love’s sake, for the chance of union.” LL-I,29

yeah that hangs true null null

you could say it the other way around too … “The #duality builds the #Ego” … not sure it matters which came first.  

It took many pages & references to decode what Peacock’s Law was until I found P.2737
… the Validity or Permanence of Equivalent FORMS.


GW mostly uses it for the geometry-algebra duality & it may be extended elsewhere – perhaps thinking & ideation itself.
The Ming tzu also comes to mind in shape & meaning. null Cross-ratio is one such instigation.

George Peacock 1791 – 1858

The principle here indicated by means of examples was named by Peacock the "principle of the permanence of equivalent forms," and at page 59 of the Symbolical Algebra it is thus enunciated: "Whatever algebraic forms are equivalent when the symbols are general in form, but specific in value, will be equivalent likewise when the symbols are general in value as well as in form."

For example, let , , , denote any integer numbers, but subject to the restrictions that is less than , and less than ; it may then be shown arithmetically that . Peacock’s principle says that the form on the left side is equivalent to the form on the right side, not only when the said restrictions of being less are removed, but when , , , denote the most general algebraic symbol. It means that , , , may be rational fractions, or surds, or imaginary quantities, or indeed operators such as . The equivalence is not established by means of the nature of the quantity denoted; the equivalence is assumed to be true, and then it is attempted to find the different interpretations which may be put on the symbol.


tag #PeacockLaw … and also via that tag … TZU & the Law of Peacock

Nice & GW extended well beyond algebra.

i think that #CrossRatio is related to the problem in my dream:  “Simple math problem from a dream ...

… and maybe even to a natural #duality 

Inquiry:  What is the difference between #analogy and “The principle of equivalence of forms” ??

tag #PeacockLaw

analogy (n.) Look up analogy at Dictionary.com
early 15c., "correspondence, proportion," from Old French analogie or directly from Latin analogia, from Greek analogia "proportion," from ana "upon, according to" (see ana-) + logos "ratio," also "word, speech, reckoning" (see logos). A mathematical term given a wider sense by Plato. Meaning "partial agreement, likeness or proportion between things" is from 1540s. In logic, "an argument from the similarity of things in some ways inferring their similarity in others," c. 1600.
perhaps it is the form thingy.  One can go as far as the Spirits of Form in RS on that one.