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…
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
since 
[this uses “h is primitive”],
we can conclude that
etcetera. forget it.
the handwritten stuff
is beautiful though.
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) (w’edia).
here in the middle are the seven colors
in “mister big 
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.
the livingston review 