## Archive for the ‘“categories” is broken’ Category

i’ve given the equations of the (seven)

planes-in-ordinary-(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 old-fashioned —

or E^3(**R**) [euclidean 3-space]—

but to get the 7-point-space

— P^2 (F_2) [projective 2-space

over the 2-element 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 set-of-seven *equations*

(

or their planes in three-space,

or their color-triples,

or their color-triples-plus-000,

or the “lines” of P^2(F_2)…

these are all ways of saying

the same thing…

)

can *also* be given a 7-point

“fano space” structure.

and has been (in some sense),

here on the page, via the [X:Y:Z] notation.

(note that the set-of-seven *points*

already *has* such a structure

[i.e., any two distinct points

determine a unique 3-point 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.