Archive for the ‘Graphics’ Category

i went down to the crossroads

at least one stranger “liked” last week’s
hell hound on my trail, so here’s another
guitar-neck diagram in the same format.

this one’s in “open G” tuning: DGDGBD
(“dog dug bed”). in key-neutral notation
one has 0-5-12-17-21-24 (“half steps” above
the lowest note). (the thing to memorize
here is probably the gaps-between-strings:
5, 7; 5, 4, 3 [“twelve & twelve”; there are
(of course) 12 half-steps… guitar frets
or piano keys for example… in an “octave”];
in the same notation, the “open D” tuning
from the previous slide would be “7,5;4,3,5”.)

adding “7” (and reducing mod-12 again) gives us
7-0-7-0-4-7; the point of this admittedly rather
weird-looking move is that the “tonic note”
for the open chord (“G” in the dog-dug-bed
notation) is placed at the “zero” for the system.
(“open D” is 0-7-0-4-7-0 in this format; the
tonic note falls on the high and low strings.)

my source informs me that open-G tuning is also
known as “spanish” tuning; a little experience
shows me that it’s even more convenient for

hell hound on my trail

here’s a diagram showing some of the notes for
a guitar tuned in “open D” tuning. i posted it
if f-book not long ago but of course one soon
loses all track of whatever is posted there.

three “inside strings” (the 3, 4, & 5; marked
here with a purple “{” [a “set-bracket” to me;
i’m given to understand that typographers call
it a “brace”])…

three “inside strings”, i say, can be played
as shown here to produce “the same” major chord
(the 0, 4, and 7 from a 12-note scale) in
three different “inversions” (namely, 7-0-4, 0-4-7,
and 4-7-0).

meanwhile, quasi-random bangings of the “outside”
strings—i like to use my thumb for these some-
times—particularly at frets 5, 7, & 12—will
blend in nicely and one needn’t worry much about
“muting” strings-not-played.

“I” is, alas, for “indian”.

the blister and still-forming scab is a trophy
from a recent encounter with some hot pigfat.
(the pigfat won.) i knocked off the fat with
the other hand within a fraction of a second
and very soon had the wound in cold water.
& so was able to pop the blister many hours
later quite painlessly… the juice squirted
out a hole too small to see and the burnt skin
went right on lying on top of the still-healing

our text is #2681: t.~izawa’s _my_ABC_book_ (1971).
digitization has put some weird distorto-mojo on
some of the lettering (as seen on my screen).
also you can barely see my penciled-in notes
and drawings and diagrams. they’re clear enough
in the actual object (or, of course, i’d ink
’em in). anyhow, for the record, i’ve drawn
12 notes of a piano keyboard at “X” along with
“EGBDF” and “FACE”; on the first two pages i’ve
copied out the whole alphabet (and indicated
the pagination… thus providing sort of a
A, B, C, and D as a cartoon. the rest is left
as an exercise.

views of P^2(F_3)

more (08/15).

longer and worse (10/16).

13-point projective space

the left-hand photo shows
a nine-point plane: an “ordinary
two-dimensional plane” over the
field with three elements (and its
label is, therefore, ${\Bbb F}_3^2$).

such a plane is ordinarily co-ordinatized as
(0,2) (1,2) (2,2)
(0,1) (1,1) (2,1)
(0,0) (1,0) (2,0):
the set of (x,y) such that
x & y are both elements of
the set {0, 1, 2}.
one could convey the same information more
concisely as
02 12 22
01 11 21
00 10 20.
it’s useful for our purpose here, however,
to consider our plane as belonging to a
*three*-dimensional space… (x, y, z)-
-space, let’s say… and as having a
*non-zero* “third” (i.e., “z”)-co-ordinate.
thus, in the photo, our plane is represented by
021 121 221
011 111 211
001 101 201.

the colors come into play in displaying the
solution-sets for various (linear) equations.
the reader can easily verify that the Green
equation—x=2— is “true” for the points of
the vertical line at the right… i.e., for
{ (2,0), (2,1), (2,2) } (old-school), i.e. for
{ 201, 211, 221} (“our” version).

likewise “y=x” (the Red equation) gives us
{(0,0), (1,1) (2,2)}… i.e.,the “Red line”
{001, 111, 221}.

now for some high-theory. by Algebra I, one has
a well-developed theory of Lines (in the co-ordinate
Plane). the usual approach there is to use the
(so-called) Slopes. the (allegedly intuitive) notion
of “rise over run” allows one to calculate—for any
*nonvertical* line—a number called the Slope (of that
line). vertical lines are said to have “undefined”
slopes. one might also say that they have an “infinite”
slope… though this invites confusion and is usually
best left unmentioned.

y = Mx + B
x = K
are then our “generic” *equations of a line*.

any particular choice of numbers M & B will
correspond to the a set of solutions lying
along a (nonvertical) line having the slope
of M (an passing through (0,B)… the so-
-called “y-intercept” of the line); each vertical
line (likewise) is represented by some particular
choice of K.

now. having different “forms” for vertical and for
nonvertical lines can be devilishly inconvenient,
so, also in algebra I, one sometimes instead uses
the “general form” for an equation of a line in the plane:
Ax + By = K
(with A & B not both zero).
likewise (but typically *not* in algebra-i)
Ax + By + Cz = K
(with A, B, & C not all zero)
is the equation of a *plane* in *three*-dimensions.

thus far i’ve been as precise as i know how.
but now i’m going to start waving my hands around
and making leaps-of-faith all over the place.
in the second photo, four new “points” have been
{010, 120, 110, 100}—the “line at infinity”).

the upshot… without the details… of this move is
that one now has an algebraic theory of “Lines” in a
“Plane” containing precisely our 13 “Points”. more-
over, this theory is “structurally” very similar to
“ordinary” linear theory. in particular, we dealing
with solutions to
Ax + By + Cz = 0—
the “K=0” case of the “general form” for (the 3D case
of the “ordinary” theory).

the Green equation—which must now be written without
its “constant term” (x = 2 is “the K=2 case” of x = K)—
becomes x – 2z = 0;
similarly, rather than (the three-point “line”
of ${\Bbb F}_3^2$) “y = 1” (concentric black-and-
-white circles), the (“homogeneous”—for us, right now,
this can be taken as meaning “having no constant term”)
equation is “y – z = 0” (and, again, we pick up a “new”
point at 100).

when the smoke clears… which won’t be here and now…
we’ll have a *very nice* geometry. just as in “ordinary”
space, two distinct points determine a unique line.
but… *unlike* “ordinary” space, it’s also true that
(*any*) two lines determine a unique point.

the same “trick”—homogeneous co-ordinates in finite
fields—converts any plane having p^2 as its number
of points to a *projective* plane having
p^2 + p^1 + p^0
as its number of points. thus there are PP’s having
7, 13, 31 = 5^2 + 5 +1, 57 = 7^2 + 7 + 1, ….
as their number-of-points. there are also some others.
but the margin is too small.

(POG)(RYB)

that 7-space has seven-way symmetry is obvious
(rotate the drawing by 2\pi/7—about 51.43\degrees—).

but the *three*-way symmetry isn’t so obvious.
here it is in its 7-color wonder.

we’ve chosen to fix the “Mud” point at the top.
we then chose the “secondaries” line
{Green, Purple, Orange} and permuted;
the “primaries” permute accordingly;
voila.
*************************************************
a 2-way symmetry can be displayed by “swapping”
each primary with its “opposite”:
(RG)(BO)(YP).

i call this one “reflection in the Mud”.

(one may also… of course… reflect
in any of the other colors;
for each (there are three)
“line” through a given color,
interchange the positions for
the other two colors.
the lines-on-blue are
{bgy, bmo, bpr}, so
“reflection in the Blue” has
(GY)(MO)(PR)
as its permutation-notation.)

vlorbik’s 7 color theorem. yet again.

or, the fano plane presented symmetrically.

each of the three triangle-edges
found along any of the “long lines”
(joining vertex-to-vertex
on the biggest 7-point “star”)
is a “line” of rainbow-space.

check it out. the “points” are
Mud Red Blue Green Purple Yellow Orange
the lines are MRG, RBP, BGY, GPO, PYM, YOR, OMB.

here’s a version from last year.
this new one’s much cooler.

zoom

while you still can.

• (Partial) Contents Page

Vlorbik On Math Ed ('07—'09)
(a good place to start!)