fano’s rainbow

the 7 points, as usual:
{m, r, b, g, p, y, o}.

Let P = {mud, red, blue, green,
purple, yellow, orange}.

wipe all traces of ROYGBIV
away for now and consider
the “colors of the rainbow”

the virtue of this renaming-
-and-reordering is that i can
now present the seven “lines”
of fano space using the color
scheme (*without* reference
to geometric or numeric data).
specifically, as
“the blends”, “the blurs”, and
“the ideal”, where
{
{red, blue, purple},
{blue, green, yellow},
{yellow, orange, red}
}
is the set of “blends”,
{
{mud, red, green},
{purple, yellow, mud},
{orange, mud, blue}
}
is the set of “blurs”, and
{green, purple, orange}
is the “ideal”.

the lines then fall in pleasant places.
in two ways that matter to me rather a lot.

we can jam {red, blue, yellow}…
the so-called “primaries” in my scheme…
onto the x-, y-, and z- axes of some
three-dimensional vector space
via
red —> (1,0,0)
yellow —> (0,1,0)
blue —> (0,0,1)
(say… one of course has five
other ways to assign colors to
axes).

*or*, as shown above, we can jam
the colors onto the “corners” of
the well-known fano diagram
representing the two-dimensional
projective space over the field
of two elements.

together, the colors give me
a very convenient way to talk about
certain correspondences between
the situations (7 non-zero
corners of an algebraic “cube”,
on the one hand, and
7 points of fano space
on the other): certain “planes”
of the cube become “lines” of
fano space, for instance…
with, of course, the green
*plane* in 3-space (say) associated
to the green *line* in fano space.

on and on it goes, this thing of ours.