slight change in format

to (a) the finite Points
(of P_2; these are
lines-of-finite-slope
in F^3) and (b) the
“Line at infinity”
consisting of the
*vertical* F^3 lines,
we have now added
(c) the “Point at infinity”
(in the lower-right-hand
corner of the diagram).
voila.

here it is again obfuscated by graphics

note the “pac-man topology” effect.
the three finite Points (clusters of
nine dots *above* the dotted line)
running down the middle column
*all* depict “lines of slope one”
but this is only obvious when one
has become used to “wrapping
around” at opposite edges.

note how the dots in the middle Point
proceed “up and to the right”.
now look at the Point below this one.
starting at the left-hand edge, we
again see the up-and-right move
but then “vanish off the top of the
screen” to “reappear at the bottom”
(like pac-man or many another
video icon… likewise, when a point
“vanishes off the right edge”,
it “reappears” at the left…).
from an algebraic point of view,
all this is a consequence of
“working mod 3″.

the zero slopes along the left edge
and the “infinite” (vertical) slopes
along the bottom are of course
much easier to make out.

just remember [2]+[2]=[1] and we’re all set

pp. 2 & 3 of the newly-minted .

the (linear) equations above the line
will turn out to correspond to “finite points”
of the projective space of the title;
the algebraically-alert will easily confirm
that these are the lines (in )
having “finite slopes”. the x = A
lines can then be thought of as
having *infinite* slope (and represented,
just as in R^2 [ordinary real 2-space],
by *vertical* lines). here come the
next couple pages.

(the icon by my finger, for now,
indicates only that these three
latest points “have something
in common”… a hint of things
to come.)

origin of species

MEdZ 0.9.1 pp. 2, 3.

there’s no place like home… there’s no place like home…

the camera on madeline’s computer still works.

“now here’s something you’ll *really* like”… “that trick *never* works”

i’m actually kind of reluctant to post this;
it’s probably the best idea i’ve had since
i started making up lectures without words
and now it’ll be easy to steal. you saw it
here first.

the Big triangle is made up of seven Little triangles.
each Little has seven Points. moreover the Points
of each Little (considered as subobjects of their Little)
are arranged in the same pattern
as the Littles themselves are (considered
as subobjects of Big).

pick any Little and call it L1.
there are three dark points in L1.
find the three Littles that correspond
to these three Points.

(for example, let L1 be the lower-left Little;
the three Littles i refer to now run along
the right side of Big [just as the dark Points
run along the right side of L1]).

the three Littles in question are then
precisely those in which the Point…
P1, say… that corresponds to L1
(considered as a subobject of Big)
is dark. i’ll be here all week.

(it’s better with the handwaving.)

perhaps *this* will refresh your memory

pages 4 & 5 of MEdZ #0.9.

trying again dammit

Photo on 2010-11-23 at 14.05

Originally uploaded by vlorbik

the camera built into legion (= my mac-book)

broke a few months back (or the software

decided to mess with me… what amounts

to the same thing). so i haven’t been posting

photos. but my office at Big State U has a mac

and it turns out i still remember how (to make ‘em

and to post ‘em in this rather clunky way).

what we have here… but what *belongs*

in MEZB (i *told* you it was clunky)… is

a picture of the theorem

(“the two-dimensional projective space constructed

on the field with five elements is isomorphic to its own

dual space”).

i put it in a zine recently (along with diagrams for

and ;

is left as an exercise).

stop me if you’ve heard this

pages from MEdZ #0.8…
the “K_4 — K_n remix”
issue. this provides a
wordless introduction
to the standard-binary-
-into-’MEdZ-notation’
algorithm (and its inverse).

everybody gets this pretty quick
that tries in my experience so
it’s a lot of fun to talk about.
which is what the “lectures
without words” series is *for*
in some sense. anyhow,
i’ve been having a blast
thrusting these on people
and then banging out whatever
little mini-lectures i can get ‘em
to sit still for.

this is some of the simplest stuff there is.
in my pedagogy it’s *obvious* that you’ll
appeal again and again to the simplest
things so you might as well get used to
at least a few of ‘em.

today i learn that the “i” of “the _i_ching_”
can mean “simple” as well as “change”.
now “ching” means something like “scripture”
(my source actually said “sutra”;
another way to say it would be
“classic text”). so on this model
the “sixty-four ways”… 000000
up to 111111… are “simple scripture”.

anyhow, an *introduction* to
simple scripture…
there are actually 4096 “readings”
possible for th’ _i_ching_, not just
64… and they’re not all equally
likely… so things get complicated
pretty quick (as you can generally
expect math problems to do).
just thought you should know.

the robot and the guy are doing battle.

it said there’d be some thunder at the well

i drew the i ching yesterday.
there are sixty-four “hexagrams” here.
each one consists of six “lines”;
each “line” is a yin (“broken”)
or a yang (“unbroken”). one counts
from bottom to top so that for example
_ _
_ _
_ _
___
___
___
has yang lines in its first,
second and third positions
(the “lower trigram”).

the arrangement of the hexagrams here
is my own; i copied it from a similar
arrangement of sixty-four objects
having exactly the same
“six things, with each ‘thing’
taking one of *two* possible
values” form… namely an
arrangement of “subgraphs”
of a certain “graph” having
six “edges”. (i don’t know why
i’m getting all “quotemark happy”
here exactly… maybe it’s the
*opposite* of “scarequotes” i’m
aiming at: by marking technical
terms in this way, i mean to indicate
that their technical sense is
here *recognized as such* [but
without actually wishing to
bother ourselves with their
technical *meanings*]…
“don’t worry about this;
it’s from another part of
a different course [but
worth mentioning here
just the same]“. oh never
mind.)

so much of the “work” of this “exercise”
i set myself will have been, if we want
to think of it this way, *calculation*:
translating each *graph* of the original
drawing i was working from into a
*number* (using binary arithmetic;
the zine i took the original drawing
from [K_4, MathEdZine #0.8] is
essentially *about* binary arithmetic)
and then translating this number
(from 0 to 63… starting at zero
makes things much easier) into the
“hexagram” notation.

of course i could also have just learned
directly which lines on the hexes
corresponds to which line of the graphs,
and spared myself lots of calculation
(by paying the price of having to briefly
memorize subsets of {1,2,3,4,5,6}
a whole bunch of times instead…
or maybe “wordstrings” like
“yes-no-no-yes-yes-no”).
but my way looked easier for me.

then i went ahead and drew ‘em all
over again on lined paper. and in
a much more natural order: eight
rows of eight, 0–63. the book i then
turned to *also* had eight rows
of eight but in a different order…
and so i went ahead and looked
up all the correspondences so i’d
have the names… and the numbers
(in the “standard” order for the
writings-of-the-sages associated
to the various hexagrams). and
here i used a “mixed” strategy…
the trigrams have chinese *names*
so as i checked back and forth
i’d subvocalize along the lines
“chi’en over chen” more often
than “seven over zero” since
(as it turns out) i can recognize
even nonsense syllables (in
latin letters) with
less effort than it takes to
“read” a trigram as a number
from 0 to 7.

and i got to thinking about
drilling and killing. now
obviously if i were doing
this kind of thing all day
on a deadline or something,
i’d want a quick-and-easy
*routine* and soon would
arrive at one, too. i’d
slip into a bit of a zone
where i’d devote most of
my conscious effort to
controlling the movements
of my hand to make better
lines and let my inner robot
do the names-and-numbers bits.
i even did some of this yesterday.

but only for a little bit at a time;
this takes *concentration* and
is, pretty specifically, no fun. no.
i haven’t *got* a deadline (alas)
and sure as heck ain’t gonna
do this all day, anyhow not
yesterday i wasn’t. not a bit.
i was also looking up some of
the hexagrams… the one i
typed out a moment ago shows
the “female” sign over the “male”
and denotes “peace” (and when
the positions are reversed,
“stagnation”)… for example,
and thinking back over certain
classroom work in my past and
fantasizing forward about working
with classes of students again.
it’ll never replace DEAD BEEF
but i’d love to give an “i ching
as intro to binary logic” lecture
before they finally haul me away.
to everything there is a season.
a time to gather stones for casting.