## Archive for the ‘Lectures Without Words’ Category

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.

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.

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] {000, 100, 010, 110}.

(excuse me my “joy of ” here;

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!

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: .

whose cover more or less announced

implicitly that it was one of a series

called . 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.

i did right away

(if i recall correctly), and in

high-art style, too (i used a

brush instead of a sharpie).

wasn’t much later.

i have plenty of notes for ,

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

.

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 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.

“for all”

“there exists”

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

.

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.