lectures without words #n: duality in 7-color space

April 10, 2015 in Lectures Without Words

a certain collection of three-point subsets of
{mud, red, blue, green, purple, yellow, orange}–

namely, the “ideal”,
{green, purple, orange}
(AKA “the secondaries”),

the “blends”,
{blue, green, yellow}
{red, purple, blue}
{yellow, orange, red},

and the “blurs”
{green, mud, red}
{purple, mud, yellow}
{orange, mud, blue} —

are called “lines” of 7-color space;
likewise the colors themselves are
called “points”.

the points of 7-color space can then
be made to correspond with the points
of “fano’s 7-point space”… which
is the smallest example of a so-called
“projective geometry”… in such a way
that the “lines” of color-space correspond
to “lines” of fano-space.

all this can be easily verified by comparing
the big “colored lines” diagram of fano space
at left with the uppermost “seven-color” space
to its right.

what we have here moreover is a certain
matching of our color-triple “lines”
in color space (the blends,
the blurs, and the ideal)
with the “colored lines” in fano space
shown in the big “triangle”: namely

Mud~~{green, purple, orange}

Red~~{blue, green, yellow}
Yellow~~{red, purple, blue}
Blue~~{yellow, orange, red}

Green~~{green, mud, red}
Purple~~{purple, mud, yellow}
Orange~~{orange, mud, blue}…

and “Mister Bigpie, oh” order
MRBGPYO, color coded.

to the right of the colored letters
i’ve “bent the line around” into
a circle… “mud” now follows “orange”
just as “red” follows “mud” and
so on.

returning our attention to the upper-right,
i’ve “applied the permutation” MRBGPYO
to the color-points of the first (higher
and to the left) triangle as follows:
the mud point goes where the red point was,
the red point goes where the blue point was,
…
the orange point goes where the mud point was;
as you can now easily see, this permutation
has “preserved the lines”. by this i mean
that in the second (lower and to the left)
triangle of this part of our display, each
“three-color set” of rainbow space lands on
a geometric line in good-old-fashioned fano space.

so this is a pretty cool phenomenon.

back at the “circle” diagram, i’ve highlighted
the “ideal” line (the “secondaries”) and i’ve
dotted-in one of the “blends” (namely {r, g, b});
i now claim that the other five “lines” of our
system are the other five triangles of the same
shape
(“choose a ‘first’ point, go forward one
for the ‘second’, then forward two for the
‘third’, and back to the first”; paraphrasing,
“up one, up two, come back”):
the seven such triangles are yet
another representation of our
seven-line “dual” space
(the Capital Letter color names
in the typographic display of
a few paragraphs ago).

got all that? good. because at least part
of the point here was that, finally, at the
lower-right i’ve calculated out (twice; the
small one was *too* small to content me so
i treated it as a first draft and redrew it)
what happens to the lines of the “original”
color-scheme [which, i hasten to add, is
somewhat arbitrary… “primaries at the
corners” etcetera] upon “applying the
MRBGPYO permutation” to its points.

at about this point, thinks become confusing
enough for me to want to start putting
*algebraic* labels everywhere and start
calculating in monochrome pencil “code”.
which i’ll spare you here.

because another part of the point is that
one simply has no *need* of *numerical*
calculations in most of this work (so far):
the “blending-and-blurring” properties known
(in my day) to every kid on the block
do much of the work for us (as it were).

here’s a years-back draft of this talk
from before i knew about this whole
“geometry’s rainbow” phenomenon.