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…

handwritingNum'Th'y1

one could simply cite the lemma
and get it over with. but that’s
not how i actually did the exercise…

let p be an odd prime &
let g & g’ be primitive
roots (mod p).

any Primitive Root, h, satisfies
(h^{{p-1}\over2})^2 = 1 (mod p);
since h^{{p-1}\over2}\not= 1 (mod p)
[this uses “h is primitive”],
we can conclude that
h^{{p-1}\over2} ~ -1 (mod p).

etcetera. forget it.
the handwritten stuff
is beautiful though.
\TeX 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).

Photo on 2014-03-14 at 18.56

ladies and gentlemen, PSL(2,7).

Photo on 2014-03-14 at 16.06

here in the middle are the seven colors
in “mister big \pi-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 P^2({\Bbb F}_2)“.
had we but world enough. and time. especially time.

Photo on 2014-03-14 at 11.29

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.

cosets



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.