the algebraization of the rainbow (remix)
i’ve given the equations of the (seven)
planesinordinary(x, y, z)space
that pass through the various
“color triples” of rainbow space—
the blends, the blurs, and the ideal.
(upon putting the “primaries” [*not* one
of our triples] {R, Y, B} onto the vertices
(100, 010, and 001) of a cube
[namely, the “unit cube” I^3=
…
if you wanna get all *technical*…].)
somewhat awkwardly, *one* of our planes
(up in the upper right somewhere) does
*not* pass through (0,0,0)
(or 000 as it’s also called here).
what gives?
mod 2 arithmetic, is what.
the equations in the display “work”
in good oldfashioned —
or E^3(R) [euclidean 3space]—
but to get the 7pointspace
— P^2 (F_2) [projective 2space
over the 2element field]… if
you wanna get all *technical*…—
we have to “work mod 2”.
and, believe it or not, 0 = 2
on this model. (so we pick up
(0, 0, 0) as a solution to
“x+y+z = 2”
[which it now becomes more
convenient to write as
“x + y + z = 0 (mod 2)”
]).
and that’s essentially it.
this “dualization” i’ve been
going on about for years, now?
here it is.
the setofseven *equations*
(
or their planes in threespace,
or their colortriples,
or their colortriplesplus000,
or the “lines” of P^2(F_2)…
these are all ways of saying
the same thing…
)
can *also* be given a 7point
“fano space” structure.
and has been (in some sense),
here on the page, via the [X:Y:Z] notation.
(note that the setofseven *points*
already *has* such a structure
[i.e., any two distinct points
determine a unique 3point line]).
(details suppressed with great effort…
part of the point here is that we
*don’t* need the [“linear algebra”]
formalism [usually learned, oddly
enough, in “calculus” classes (if at
all)]—“dot products” and so on—
to achieve our dualization: we
can just *draw* the doggone thing
and check directly that our structure
“works”.)
thank you and good day.

1
Pingback on May 18th, 2015 at 2:05 am
[…] Vlorbik discusses the algebraization of the rainbow […]
April 28, 2015 at 2:41 pm
MEdZ version etcetera
https://vlorbik.wordpress.com/2015/04/24/3isleftasanexercise/
April 28, 2015 at 3:05 pm
https://vlorbik.wordpress.com/2013/02/17/teardownthewall/
(of course the whole mess embeds into 21point space.
but for *that*, i still need the algebra.)
April 28, 2015 at 3:30 pm
shorter & prettier:
https://vlorbik.wordpress.com/2015/04/23/oldmoleskinsketchnewlyinked/