for five bucks you can get

a quarter’s worth of popcorn

at a kitsch shop at the mall

in the seven colors of the

MRBGPYO hexagon-plus-center

of the “color wheel”.

that, and a cup of coffee.

PS: i’m not altogether sure

that my Mud kernel isn’t actually

just a deep Purple. ah, well.

soon i’ll dissolve it all in

coffee and saliva and worse.

sic transit.

the pink point is “on its own polar”

(in the “polarity of P^2(F_4)” shown here)

& so the Big pink Circle “includes” one

of the little pink circles.

the other colored points of the display

*don’t* have this property… one sees

instead that the Orange Circle “includes”

a blue circle, & the Blue “includes” an

orange. note that the “maps” of 21-space

found in the Orange and Blue Circles

correspond to the *positions taken*

by the orange and blue circles

(respectively): the blue circles

“go across the top and then take the

rightmost point”—the Orange ‘position’,

if you will—and the oranges “go up the

right-hand-side and then take the very middle”

(i.e., the Blue position).

similarly the “turquoise” & “lime”

points (let’s say) are found

respectively on the lime & turquoise

lines.

before you recycle anything

scribble all over it and make copies.

so nothing goes to waste.

the medium is the massage

so i painted this with a brush

more or less as an exercise

in “graphic design”. if ever

the zine is reissued (it’s seen

here in its natural state, back in the brief

hardcopy-heyday of MEdZ),

this’ll be the cover.

never got drawn.

on the other hand,

*visual complex analysis*

is a marvelous work and a wonder.

(tristan needham, oxford u.~press)

not that i’ve looked it over *that*

carefully. a few minutes a couple

summers ago in a crowd and a few

more… or maybe a couple of hours…

with the PDF. online reading, yick.

http://needham_t_visual_complex_bhj.pdf.

another random shout-out.

this one’s a hard-copy.

*conics* (keith kendig,

2005, MAA).

suddenly i’m clearly seeing

lots of stuff i thought i’d

*never* understand. and in an

entertaining style, yet.

this guy’s *good*.

not just on , either…

but on *the big picture*:

itself.

(this turns out to be the natural setting

for understanding “conic sections”.

one *hears* this kind of thing all the time

but is seldom brought face-to-face with such

convincing *evidence*.)

vlorbik sez check it out.

five-string exercise for guitar

(lose the sixth string.)

tune the fifth up

a half-tone higher

(than its usual “A”,

to B-flat [or A-sharp;

B-flat to us today]):

X B♭ D G B E. so far.

now we’re gonna “barre”

four strings more or less

throughout the rest. in

“across the universe”

notation, playing

[[X,0,1,1,1,1]]— i.e.,

pushing down the first four

strings at the first fret—

in this tuning gives, as it

were, a “shifted-ordinary”

tuning: the familiar

A D G B E (learned by every

beginner) is “shifted up” to

B♭ E♭ A♭ C F.

the good news is, forget the

“accidentals”: the exercise

is to play on the high four

with the “A-string” (now in

some sense “really” a B♭)

droning away (or silenced…

but otherwise untouched by

the left hand) the whole time.

so new-school “D”:

[[X,0,1,3,4,3]].

this is (of course… ) just

“old school” D

[[X,0,0,2,3,2]]

“raised by one fret”.

but, alas, putting that second

“barre” down? (the first finger

is barring four frets “throughout”;

to get the “D” here, i have to

*also* barre three strings with

my third finger [and finally put

my pinkie in the middle of the bar,

one fret up.) that’s pretty brutal.

more good news: one already plays

[[X,X,3,2,1,1]] (regular tuning):

this is Beginner F (typically one’s

*first* “barre” chord… note that

only two frets must be barred).

well, new tuning *improves* on the

sound of that cord since we can now

loudly *play* that fifth string:

[[X,0,3,2,1,1]] (“new” tuning) gives

us an extra bass note, as it were

“for free”.

the real payoff, though…

or anyway my reason for having developed

this whole line of investigation…

the “A” chord and its variants

(as i think of ’em while playing

actual B-flats and *their* variants)

take the cool-sound-making form

[[X,0,3,3,3,1]]

[[X,0,3,3,3,3]]

[[X,0,3,3,3,4]]

where you’re just mashing down the

major-chord in the middle and can

drop in the boogie-woogie treble

with very small movements of one’s

left hand.

this A-form trick (e.g.) also works

in ordinary tuning at the fifth

and seventh frets and that where i

first figured it out.

the “raise the bass” trick lets me

play in the same style but lower

on the neck.

“algebra 1″ students in enormous numbers

are confronted every semester (or quarter)

with a survey-of-number-systems.

, and

…(to give them their standard symbols…

as the textbooks, more or less of course,

do *not* do [effectively; sometimes

they *gesture* at these symbols])…

denote the sets of

Natural numbers, integer numberZ,

rational (Quotient) numbers,

& (so-called) Real numbers.

the texts then go on to *ignore* their own “survey”.

& students get *worse-than-nothing* out of it.

in many cases, they’ll have seen this treatment

*many* times (and become *worse* prepared to

think about the rest of the course material

[about “graphing” and “factoring” and so on]

*every* time).

then there’re these exercises. wherein one is

required to say, about *each* of a (given) collection

of numbers, which of our “number-sets” include it

(and which don’t)… for instance,

“pi” is real but not natural, integer, or rational

“-11″ is integer, rational, and real, but not natural

etcetera. right on the front page of the final…

the same damn *problem*, verbatim, for years,

in at least one instance known to me… and yet

substantial subsets of *every* class of students

miss these incredibly-easy-if-the-mere-vocab-

ulary-is-understood “exercises”.

because the one thing they know for sure

after all this time is “i will never understand

what any of these words mean” (so “just show me

how to ‘do’ the problems”).

whereas.

rational number arithmetic is routinely taught

to children in functional schools (and families).

with a certain amount of effort, of course.

but it’s “easy” enough if it’s done *clearly*.

so here’s something that *ought* to be in the text.

not necessarily *instead* of

“ and are integers “,

but certainly somewhere *nearby*

(if the deliberately-obfuscatory

high-theory “set-builder” thingum

*must* somehow be included in our treatment).

namely, a table (showing possible numerators

and denominators & their Quotients), such that

is “all the numbers in the table

(and their opposites)”.

(the little “” icon at upper-left

is meant to indicate that, e.g. “-3/2″

(the “opposite” [or “additive inverse”] of “3/2″)

should be considered to be “in the table”…

this is a nuance, improvised for the zine

[and if i were giving this lecture today

at the blackboard, i’d very likely draw

out another part of the table and include

some negative rational numbers explicitly].)

there it is. we’re looking right at it.

*now* we can talk about it.

(and, say, how various “decimals”

do or don’t “give us” a

“number on the table”.)

(secretly, of course, “we” know this means

“a number that can be represented as [an]

integer-over-rational-number”… but

it does no good to keep on *saying so*

once we know darn well we’re being

“tuned out”.)

now. about those dotted-line circles.

(… get the student to talk…)

hey, “lowest terms”. wow, cool.

now about this funky *graph* over here.

same idea, different picture.

each “lowest terms” ratio appears

as the *nearest point to zero*

on a “line” having the “slope”

represented by that number.

(by “nearest” [point to zero],

one here means “nearest on the

‘integer lattice’ “…

but of course one will not

[necessarily] *say so* explicitly;

this is… or was… a “lecture

without words” more-or-less precisely

*because* the real work is getting

the *student* to do a lot of the

talking [and pencil-moving];

“nearest *on the diagram* is sufficient

for our purpose [a clear view

of the set-of-rational-numbers,

in case you’ve forgotten].)

you’ve gotta give ’em hope.

here’s the page i munged yesterday, four times bigger

(twice in each dimension, duh). one (more) plainly

sees here that points of an arbitrary hemisphere

(in the top drawing of the lower right panel)

can be identified with longitude-and-latitude pairs

.

such co-ordinates depend on “choosing

a central meridian” (this was done for

usual planetary co-ordinates by passing

the central meridian through greenwich).

on the drawing (and the planet), we’re

*given* a cutting-of-the-sphere. but

on a more abstractly-given sphere

(in some other context) we might need

to consider *how* the sphere is to be

cut into hemispheres (and *which* hemi-

sphere is to be “drawn” [or what have

you… “studied”]).

on our globe-of-the-world model, this would

amount to selecting a different “north pole”

(notice that this gives us an “equator”,

as it were, “for free”).

since the (actual) north pole has a *fixed*

position, we might consider shifting our

“point-of-view” as if we were some satellite,

far enough away to see half the surface

of the “planet” we want to co-ordinatize.

when we look down *from the north pole*,

the “equator” for this point-of-view

is then (of course) the *actual* equator

(the great circle equidistance from the

two poles).

when our satellite is over some *other*

“point” (of our beloved mother earth),

what we’ll see “looking down” is essentially

a *polar projection* of (half of) the surface.

our co-ordinate frame on this model would then

appear as (1) a collection of concentric circles

(the analogues of “latitude”… the angle measured

“up north” [on the north pole model] having been

replaced by the “distance in” [toward the new center])

and (2) a collection of radii (the “meridians”…

measuring the “angle from greenwich” on the

globe and the “angle from the top of the

camera view” (say) in the space-travel version.

all this is perfectly straightforward

& beginners pick it up in a few minutes

of lecture if they’ve got any experience

with maps-of-the-world.

the tricky bits are the reason for all the

“identification diagrams” littered about

the rest of the page. and about these,

i propose to say but little. today.

heck, very little. “identify antipodes.”

here’s a song while i fix breakfast.

ana ng at yootube. (ad-ridden, natch.)

up top, upside-down, i did

some noodling around with

the trusty 4-color pen,

without adding much of any

value i think, and clobbering

some of what was there already.

in the lower half, upside-up,

i’ve colored in two pages of

the microzine from 2010.

perhaps this will refresh your memory.

what i was looking for

was somehow to represent the

*lines*-of-the-base-space-are-

-*points*-of-the-“dual”-space

phenomenon with a *single picture*.

naturally i used the smallest space

that “worked”: P^2(F_2). in the first

draft (at left), i extended the lines

of the “base space” (namely, the “points”

{001, 010, 011, 100, 101, 110, 111};

these are probably too small to read

here; it matters but little for my

purposes today)…

extended the “lines” of the “inner

triangle” space, i say…

and put a big bubble on each

extended line (and a bubble in

the middle). these “bubbles”.

of course, represent the *points*

of the dual space (identified,

as you will recall, with the lines

of the base space).

in the colorized version, i’ve made the

[point-of-dualspace]—>[line-of-base-space]

correspondence more explicit… or more

vivid. or something. i hope.

i was pretty pleased with this (lefthand)

drawing. but it was the next one that

convinced me i was really onto something.

so i jammed it all together and ran off copies.

then i made another zine with a page about

P^2(F_2) that i liked even better and lost

track of this one.

i colorized that newer one some time later…

the *second* really good idea of this project.

and there it lay for a while.

lately i’ve been colorizing madly, though.

new today… a *slightly* good idea…

are the seven “planet” icons.

this came close on the heels of having

decided to put *days of the week* in…

another. (the *really* good idea here

[if any] was to look around for sets-

-of-seven having some “natural” order:

the symbols in {0,1,2,3,4,5,6} are

already getting overworked and it

comes quite in handy to have a few

well-known ordered sets lying around.)

the planets-to-days correspondence is

partly well-known (“sun”day, “moon”day,…

“satur[n’s]”day) in our ambient culture

in the english-speaking world. i think.

anyway it’s out there.

so when i get to talking about ordering

the *colors*… as MRBGPYO…

i’ll have sunday-monday-…-saturday[-sunday-…]

as an ordered set of seven

“places to put information”.

(one that isn’t geometric *or* algebraic

*or* chromatic… another part of

“concept space” to work with.)

the “planets” thing is a bonus.

the tie-in with astrology is all to the good;

whatever *else* i’m doing here, i’m the last

to deny that something *mystical* is going on.

the source i copied the icons from

even had a “seven ages of man” graphic.

i had *that* in my “seven lists of

seven things” sermon a few months ago.

but so far, i haven’t connected

shakespeare to rainbow space.

stay tuned.

i’ve given the equations of the (seven)

planes-in-ordinary-(x, y, z)-space

that pass through the various

“color triples” of rainbow space—

the blends, the blurs, and the ideal.

(upon putting the “primaries” [*not* one

of our triples] {R, Y, B} onto the vertices

(100, 010, and 001) of a cube

[namely, the “unit cube” I^3=

…

if you wanna get all *technical*…].)

somewhat awkwardly, *one* of our planes

(up in the upper right somewhere) does

*not* pass through (0,0,0)

(or 000 as it’s also called here).

what gives?

mod 2 arithmetic, is what.

the equations in the display “work”

in good old-fashioned —

or E^3(**R**) [euclidean 3-space]—

but to get the 7-point-space

— P^2 (F_2) [projective 2-space

over the 2-element field]… if

you wanna get all *technical*…—

we have to “work mod 2″.

and, believe it or not, 0 = 2

on this model. (so we pick up

(0, 0, 0) as a solution to

“x+y+z = 2″

[which it now becomes more

convenient to write as

“x + y + z = 0 (mod 2)”

]).

and that’s essentially it.

this “dualization” i’ve been

going on about for years, now?

here it is.

the set-of-seven *equations*

(

or their planes in three-space,

or their color-triples,

or their color-triples-plus-000,

or the “lines” of P^2(F_2)…

these are all ways of saying

the same thing…

)

can *also* be given a 7-point

“fano space” structure.

and has been (in some sense),

here on the page, via the [X:Y:Z] notation.

(note that the set-of-seven *points*

already *has* such a structure

[i.e., any two distinct points

determine a unique 3-point line]).

(details suppressed with great effort…

part of the point here is that we

*don’t* need the [“linear algebra”]

formalism [usually learned, oddly

enough, in “calculus” classes (if at

all)]—“dot products” and so on—

to achieve our dualization: we

can just *draw* the doggone thing

and check directly that our structure

“works”.)

thank you and good day.

the “rotation model”

suggested by the lower-left

(addition-mod-3) table really

only “works” for addition

(insofar as it lacks any [obvious]

*multiplicative identity*);

so too for the “eight-hour shift”

version (what would it *mean*

to “multiply” by “adding 8

hours on a 24-hour clock”?).

and working in color presents

quite a few technical difficulties.

so quite the done thing

is to use something typo-

graphically “nice” like

{[0]_3, [1]_3, [2]_3 }…

or even {0, 1, 2}, when we get

right down to actually *calculating*…

when *presenting* this kind of thing.

but the eye-appeal of colorful or

geometrically-suggestive notations

can actually make things, well,

*easier to see*…

so we beat on…