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

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.

April 10, 2015 at 3:36 pm

https://vlorbik.wordpress.com/2015/03/28/fanos-rainbow/