## Archive for March, 2014

i’ve typeface-ized the “formula” stuff

but the point here is the english.

tonight i encountered the passage

Since there are (p-1)/2 quadratic residues & 1^2, 2^2, …, [(p-1)/2)]^2 are all the residues, we need to show that the quadratic residues modulo p are all distinct…

and, after much wailing and gnashing

of teeth, decided that the best spin

i could put on it would be

to *omit* the first “the*

and to replace the second “the”

with “these”:

Since there are (p-1)/2 quadratic residues & 1^2, 2^2, …, [(p-1)/2)]^2 are all residues, we need to show that these quadratic residues modulo p are all distinct…

(which “works” in its context

as the original passage certainly

does *not*).

they should give medals for this kind

of copyediting. this is *hard work*.

not that it does anyone any *good*,

mind you…

let p be an odd prime &

let g & g’ be primitive

roots (mod p).

any Primitive Root, h, satisfies

(mod );

since (mod )

[this uses “h is primitive”],

we can conclude that

(mod ).

etcetera. forget it.

the handwritten stuff

is beautiful though.

is too hard.

it’s not so bad in the real editor,

of course.

oh, ps. (gg’)^[(p-1)/2]

is now seen to be congruent

to 1… and so is not a

primitive root (which, on

the day, was to’ve been shown).

ladies and gentlemen, PSL(2,7).

here in the middle are the seven colors

in “mister big -oh” (from ohio) order:

MRBGPYO

(mud, red, blue, green, purple,

yellow, orange). i’ve drawn the “line”

(which appears as a triangle) formed by

“marking” the purple vertex and performing

the “two steps forward and one step back”

procedure: one easily verifies that

{G, O, P} is a line as described in

the previous post (“the secondaries”).

all to do with “duality in “.

had we but world enough. and time. especially time.

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.

here’s the same group…

call it G… appearing as the

lower-left hex board

from a set of four.

we intend to display an accurate

visual representation of the

simple group (S) of order 168.

our group (G) is one of many

7-element subgroups (this

particular one is selected

because \psi is easily memorized).

what we *have* displayed

on this sheet of paper is G

together with three of its

right-cosets:

{G = G1, Ga, Gb, Gc}.

it so happens that

{1, a, b, c} form a group

in their own right (a V_4

or “klein 4-group”).

however, we *don’t* get

a group structure on the cosets

out of this.

i’ve drawn one more sheet

looking quite a bit like this one

(4 more cosets of G).

when i’ve drawn four more,

that’ll be S itself (and a

colorful treat for eyes and brains).

here’s a hex board

with seven icons on it;

each icon has seven colors;

the seven permutations of

colors-into-hexes (each icon

has seven hexes) can be

considered as the objects

of a cyclic group.

specifically

{

() = \identity

(1234567) = \psi

(1357246)= \psi^2

(1473625)=\psi^3

(1526374)=\psi^4

(1642753)=\psi^5

(1765432)=\psi^6

}.

(of course one has \psi^7=\identity,

etcetera… the group here is

essentially just integers-mod-7:

{ [0], [1], [2], [3], [4], [5], [6]},

with addition defined by

“cancelling away” multiples

of 7 (forget about the fancily

denoted “permutation structure”

displayed in defining \phi

and just look at the exponents).

the “cycle notation” used here

is *much* under-used, in my opinion.

but we’ll really only need it

for future slides.