### blag 168

one has already learned the 7-cycle
sunday-monday-tuesday-wednesday-thursday-
-friday-saturday-(sunday-…)
as part of one’s cultural heritage
as an english speaker.

and it’s high time i *used* that fact
in studying “fano’s rainbow”
(aka “the seven-point [projective] space”).

and so to each of the
*positions* in the well-known
“three corners, three midpoints, one center”
representation of fano space,
i have assigned a *day of the week*.

now, the seven points of “rainbow space”…
mud, red, blue, green, purple, yellow, orange (, mud, red, …)
fall neatly… also a part of our cultural heritage… into
“three primaries, three secondaries, and one ideal”
(“mud” is of course the ideal).

so on sunday i put MRBGPYO into the big diagram
by (“arbitrarily”) “coloring the vertices”:
Red—> Lower Left
Blue–> Lower Right
Yellow–> Top.
the other “colors of the rainbow” can now be
filled in (in exactly one way) according to
the rule that
“rainbow lines fall on geometric lines”
(recall that the rainbow lines are
the “blends” RBP, BGY, and YOR,
the “blurs” RMG, BMO, and YMP,
and the ideal GPO.)

turning our attention to monday, i’ve permuted
the colored-triangles “forward one” in the order
m, r, b, g, p, y, o:
“mud goes where red was, red goes where blue was…
… orange goes where mud was”.

then i copied the shot from the zine…
wherein a certain lines-to-points correspondence
is laid out graphically…
into the fano-diagram called “sunday”.

the rest is chasing permutations around.
the question… in all its vagueness…
is “what happens to the geometric lines”
when we “move the colors” around the diagram
by “applying the MRBGPYO permutation”, as it
were, each day of the week.

and i’ve found some stuff out that i didn’t know.

and i’m calling it a day for now.

1. a points-to-lines correspondence
of our type… 7 of them are displayed
here… may be determined by choosing

(1st) which of the seven “lines” gets the “ideal” line
(1st) (the green-orange-purple “secondaries”);
(1st) there are (of course) 7 “choices” here—,

(2nd) which points of the chosen line get which
(2nd) colors; there are 6 choices for each of our
(2nd) “1st choices” (namely
(2nd) gop, gpo, ogp, opg, pgo, & pog)—, & finally

(3rd) which of the four *remaining* points
(3rd) (*not* on our chosen line) gets the
(3rd) “ideal” point (“mud” in our rainbow scheme);
(3rd) there are (again, obviously) 4 such.

and that’s it: any such choices—
the ideal line on any line
(in any order) and
the ideal point on any point
(not on the ideal line)—
will determine a points-to-lines “duality”
of the type i’ve been examining; more-
over, these are *all* of them.

there are 7*6*4 = 168
such assignments, in other words.
24*7, pleasantly enough:
a whole week of 24 hour days’
worth of dualizing maps.

• ## (Partial) Contents Page

Vlorbik On Math Ed ('07—'09)
(a good place to start!)