Archive for the ‘Lectures Without Words’ Category

photo-on-2-5-17-at-9-45-am

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
noodling-about-on-three-strings. that is all.

photo-on-1-9-17-at-12-26-pm

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.

Photo on 4-24-15 at 10.18 AM

yet another sketch from the
“lectures without words” run
of MEdZ. here improved with
colored inks (and spoiled by
flouting the “without words” rule).

“binary arithmetic” is here exploited
to assign *number values* to the corners;
the symbol “xyz” chosen from
000, 001, 010, 011, 100, 101, 110, 111
corresponds on this model to
4x + 2y + z.

(this follows the usual “place-value”
conventions typically used in the
context bases-other-than-ten
[in base ten, the same symbol “xyz”
would denote 100x + 10y + z].)

the “front face” of our cube (for example)
is now {000, 001, 100, 101}.
these number triples share the feature
y=0…
and are the *only* triples with this feature.

now, we can think of “y=0” as meaning
“don’t move in the y direction at all”
(the “y direction” here is “front to back”…
going [as it were] from the 000 point “back”
toward the 010 point is the only way to get
a y=1… that’s why “y=0” gives us the “front face”.

but the point 010 is not itself a “direction”…
so another notation is introduced: [0:1:0].

the diagram shows (or hopes to) that similarly
[1:0:0] “is perpendicular to”
the left-hand face {000, 001, 011, 010} and
[0:0:1] \perp {000, 100, 010, 110}.
(excuse me my “joy of \TeX” here;
\perp has what i hope is its obvious
meaning.)

anyhow… there’s real work to be done
(getting to campus and back; the hardest
part of the job some days)… that’s *almost*
it for today.

it remains only to remark that
[0:1:1], [1:0:1], [1:1:0], and [1:1:1]
can also be considered as “perpendicular”
to the other four rainbow-space “lines”
(certain cross-sections of the cube
on the 3-D model)… giving us a
full-blown *algebraic* model of
fano-space duality.
[exercise. hint: binary arithmetic.]

feed me!

Photo on 2014-07-12 at 10.42

Photo on 2014-07-12 at 10.47

Photo on 2014-03-14 at 18.56

ladies and gentlemen, PSL(2,7).



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.


yesterday’s more verbose version.


ten poles, ten polars,
and ten pairs-of-triangles:
ten ways to use one drawing
(from MathEdZine, of course)
to illustrate one theorem.

(desargues’ theorem at w’edia.)

performing calculations using
actual blobs of color rather than
alphabetical symbols *standing*
for colors is time-consuming
(and demanding of special tools)…
but one literally “sees” certain things
*much more readily* than with
symbol manipulation.

hey, i’m a visual learner.


at long last. this has been sitting
on the paperpile very-nearly-finished
for quite a while.

i started the “lectures without words”
series early on with 0.1: \Bbb N.
whose cover more or less announced
implicitly that it was one of a series
called \Bbb N  \Bbb Z \Bbb Q  \Bbb R \Bbb C. and that
was, like, five quarters ago.
and they’re only 8 micro-size pages.

a couple days ago i inked the graphs
and the corresponding code (the stuff
under the dotted line had *been* inked
and the whole rest of the issue was
entirely assembled). and zapped it off.

and yesterday i passed ’em around
at the end of class (to surprisingly few
students given that i’ve got freshly-
-graded exams). it went okay.

\Bbb Z i did right away
(if i recall correctly), and in
high-art style, too (i used a
brush instead of a sharpie).
\Bbb Q wasn’t much later.
i have plenty of notes for \Bbb C,
too, and could knock out a version
on any day here at the studio
(given a couple hours and some
peace of mind) that’d fit right in.

anyhow, what we have here are,
first of all, obviously, a couple graphs
and a bunch of code. here, at risk
of verbosity, is some line-by-line
commentary.

in the upper left is
part of the graph of
the linear equation
y=(x+1)/2…
namely the part whose x’s
(x co-ordinates) are between
-1 and 1.

and my students (like all
deserving pre-calculus graduates)
are familiar with *most* of the
notations… and *all* the ideas…
in this first line.

that funky *arrow*, though.
well, i can’t easily put it in here
(my wordpress skills are but weak)
but i’m talking about the one
looking otherwise like
(-1, 1) \Rightarrow (0,1).
and in the actual *zine*, it’s
a Bijection Arrow.
something like ” >—->>”.

as explained (or, OK, “explained”)
*below* the dotted line.
where *three* set-mapping “arrows”
are defined (one in each line;
the :\Leftrightarrow in each line
denotes “is equivalent-by-definition to”).

the Injective Arrow >—> denotes that
f:D—->R is “one-to-one” (as such
functions are generally known
in college maths [and also in
the pros for that matter; “injective”
and its relatives aren’t *rare*,
but their plain-language versions
still get used oftener]).

“one-to-one”, defined informally,
means “different x’s always get
different y’s”. coding this up
(“formally”), with D for the “domain”
and R for the “range” (though i
prefer “target” in this context
when i’m actually present to
*explain* myself) means that
when d_1 and d_2 are in D,
and d_1 \not= d_2
(“different x’s”), one has
f(d_1) \not= f(d_2)
(“different y’s”).

likewise the Surjective Arrow —>>
denotes what is ordinarily called
an “onto” function:
every range element
(object in R)
“gets hit by” some domain element.

and of course the Bijective Arrow >—>>
denotes “one-to-one *and* onto”.

there’s quite a bit of symbolism
here that’s *not* familiar to
typical college freshmen.
the Arrows themselves.
\forall “for all”
\exists “there exists”
\wedge logical “and”
and the seldom-seen-even-by-me
“such that” symbol that, again,
i’m unable to reproduce here
in type.

still, i hope i’m making a point
worth making by writing out
these “definitions without words”.

anyhow… worth doing or not…
it’s out of the way and i can return
to the main line of exposition:
the mapping from (-1, 1) to (0,1)
in the top line of my photo here
is a “bijection”, meaning that it’s
a “one-to-one and onto” function.

for *finite* sets A and B,
a bijection f: A —-> B
exists if and only if
# A = # B…
another unfamiliar notation
i suppose but readily understood…
A and B have
*the same number of elements*.
we extend this concept to *infinite*
sets… when A and B are *any*
sets admitting a bijection
f:A—>B, we again write
#A = #B
but now say that
A and B have the same cardinality
(rather than “number”;
in the general case, careful
users will pronounce #A
as “the cardinality of A”).

i’m *almost* done with the top line.
i think. but there’s one more notation
left to explain (or “explain”).

the “f:D—>R” convention i’ve been
using throughout this discussion
is in woefully scant use in textbooks.
but it *is* standard and (as i guess)
often pretty easily made out even
by beginners when introduced;
one has been *working* with
“functions” having “domains”
and “ranges”, so fixing the notation
in this way should seem pretty natural.

but replacing the “variable function”
symbol “f” by *the actual name
of the function*
being defined?
this is *very rare* even in the pros.
alas.

the rest is left as an exercise.


a recent post in KTM asked why
{6 \choose 2} = {6 \choose 4}.
i know that one.

the “graph” in the upper right is K_4…
the “complete graph on 4 vertices”…
and has *6* (so-called) edges.

its “complement” (upper left) has *none*
of the edges. obviously there’s one
(so-called) *complete* graph having
all six possible edges and one “empty”
graph having *none* of the edges.

move down to the next couple lines:
the graphs having exactly *one* edge
match up in one-to-one fashion
with the graphs having *five* edges
(because “including five (edges)”, in this context,
is the same as “leaving out *one* (edge) out”.

so on for two-edge graphs.
each is the “complement”
(via the inclusion-exclusion principle
as hinted at a moment ago)
of a *four*-edge graph.

posting; gotta go to class.