### one step forward, two steps back: vlorbik’s seven-color theorem

okay. not so much a theorem as a simple brute *fact*…

a perfectly *obvious* fact but one that i managed

to overlook for years. (this kind of thing

happens all the time of course.)

**Directions

let “clockwise” be the “positive” direction

(for this post only; the usual [trig class]

convention is to use the opposite orientation).

let “up” be “up” and let “down” be “down”.

let “day” and “night” be undefined.

**Seven-Color Space

seven-color space has seven colors.

as follows.

there are three Primaries: Red, Yellow, & Blue.

PC = {R, Y, B}.

there are three Secondaries: Green, Purple, and Orange.

SC = {G, P, O}

(of course O\not=0; this is obvious

as i type but milages vary: your

internet ain’t like mine).

there is one Ideal: Mud.

IC= {M}.

**Points

the set

PS = {R, Y, B, G, P, O, M}

is called the Underlying Set.

its elements are called “points”

of the space (or of U). of course,

its elements are *also* called “colors”.

**Lines

certain three-element subsets of U

are singled out and given the name “Lines”.

specifically, the lines are

{R, Y, O}, {R, B, P}, {Y, B, G}

(the “blends”… red and yellow paints

mix together to form orange, for example),

{R, G, M}, {Y, P, M}, {B,O, M}

(the “blurs”… pairs of “opposite” colors

of paint [like red and green] form a

Muddy neutral non-color), and

{G, P, O}

(the “secondaries”; we have encountered

this set before as SC).

for calculations, we will of course suppress

the set braces; we may conveniently denote

the set of lines for seven-color space as

LS = {RYO, RBP, YBG, RGM, YPM, BOM, GPO}.

(the understanding here is that the XYZ

stands for the *unordered* triple

{X, Y, Z} (= {X, Z, Y} = … = {Z, Y, X});

the *set* of colors and *not* their order

is what makes a line a line.)

The Standard Permutation

like *any* seven objects, the colors of CS

can be listed in any of 5040 (= 7*6*5*4*3*2*1)

orderings. it’s convenient to fix a *particular*

ordering from very early on in the discussion.

we have chosen

0: Mud

1: Red

2: Blue

3: Green

4: Purple

5: Yellow

6: Orange

as our Standard Permutation:

M_R_B_G_P_Y_O

(“mister big pie, oh!”).

The Regular Heptagon

(vlorbik’s seven-color theorem)

label the vertices of a regular heptagon

(clockwise) with the colors in the standard

permutation.

the circle-like nature of the heptagon

induces a “cyclic” structure on the colors,

which we can now think of as

M_R_B_G_P_Y_O_M_R_B_G_P_Y_O_M_R_B_G_P_Y_O…

“circling around forever”.

mark any vertex.

M_R_B_G_P_Y_*O*_M_R_B_G_P_Y_*O*_M_R_B_G_P_Y_*O*_…

(i have “marked the orange vertex”).

we can now compute one of the *lines*

[one that includes the “marked” color]

by going… this is “vlorbik’s theorem”…

Two Steps Forward and One Step Back

from the marked color:

“two steps forward” from O, for example,

gives us R (read to the right), and “one

step back” gives us Y; sure enough we get

one of the “blend”s: {R, Y, O} is one of

the Lines of seven-color space.

rotating the whole heptagon

(through an angle of 2\pi/7

or any integer multiple thereof)

permutes the colors in such a way

that lines are taken to lines.

only 168 of the 5040 permutations

of {R, B, Y, O, P, G, M}

share the property that “lines

are taken to lines”. i’ve just

drawn them all: announcing

Math Ed Zine #2. send money.

March 28, 2015 at 11:46 am

https://vlorbik.wordpress.com/2014/03/14/the-poster-will-look-more-finished-the-zine-is-another-story/

at least one page is wrong.

eventually i’ll draw all 168 again

if universe spares me long enough…