Photo on 5-17-15 at 9.01 AM

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.

Photo on 5-13-15 at 10.48 AM

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

before you recycle anything
scribble all over it and make copies.
so nothing goes to waste.

Photo on 5-12-15 at 11.15 AM

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.

\Bbb C 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.

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 \Bbb C, either…
but on *the big picture*:
{\Bbb P}^2({\Bbb C}) 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”:

this is (of course… ) just
“old school” D
“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
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.

“G” form is another useful near-freebie.

Photo on 5-7-15 at 1.11 AM

i coulda had class.
(MEdZ 0.1.1 [2010])

Photo on 5-6-15 at 10.03 AM

“algebra 1″ students in enormous numbers
are confronted every semester (or quarter)
with a survey-of-number-systems.

{\Bbb N}, {\Bbb Z}, {\Bbb Q}, and {\Bbb R}

…(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”).

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
\{ {n\over d} : d \not= 0, d and n are integers \}“,
but certainly somewhere *nearby*
(if the deliberately-obfuscatory
high-theory “set-builder”
*must* somehow be included in our treatment).

namely, a table (showing possible numerators
and denominators & their Quotients), such that
{\Bbb Q} is “all the numbers in the table
(and their opposites)”.

(the little “\pm” 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’ {\Bbb Z} \times {\Bbb Z}“…
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.

Photo on 5-5-15 at 11.06 AM

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
\langle \phi, \psi \rangle.

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.)

Photo on 5-3-15 at 6.53 PM
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
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
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.
Photo on 5-3-15 at 8.33 PM

Photo on 4-28-15 at 1.21 PM

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=
\{ (x,y,z) | 0 \le x,y,z \le 1 \}
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 {\Bbb R}^3
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

thank you and good day.

F^3 in color

Photo on 4-25-15 at 4.24 PM #2

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…

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