(a more-or-less abandoned math
department, far away from any
place i have the authority to
lay my head down and think things
over. but!… with internet access.)

in the nature of the case, we graders
work while the students… um…
do whatever students do… “party”,
i suppose. so papers picked up on
friday after class have to be collected
by *somebody* (& returned by monday
morning; we’re trying to set a high
standard here after all). fair enough,
too.

it’s not like you don’t have to lie
to do *this* job right; getting us
to say things we don’t actually mean is
pretty precisely what money is *for*.
just this: there’s one *big* lie; the rest
is essentially about matters of taste.

and the “big lie”?
well. of course:
“make it look easy”.

no teacher (of anything) can afford
to be honest about what it’ll cost
to get good. *obviously* any rational
being will run like hell before enduring
anything *like* the tribulations that we
(every expert in any art) went through…
only realizing, much too late, that most
of it was for astonishingly foolish
reasons (“youth is wasted on the young”).

worth it, worth it, worth it.

& a long bus ride home.

(“worth it, worth it, worth it”: ramble of 8/09.)

and right here at work. i was able to log in
to this *math department* box… but can’t get
into the *university level* system (where stuff
like syllabi and grades are found and posted).
i’d been recently made to choose a new
(and messy) password, so first
i tried resetting that… and succeeded. the system
recognized me enough to ask me the three “mother’s
maiden name” style questions allowing this…
but then when i tried to get to the sites
where *actual work* can be done i learned,
to my chagrin, that *that* part of the system
couldn’t recognize my username at all.

well, i’m barely an employee. indeed i didn’t
have any duties assigned me until two days into
the quarter. so i figured maybe this had something
to do with it. too soon to tell, really: it took
about 15 minutes on the phone to discover that
(1) the guy at the help desk couldn’t help me &
(2) others are reporting similar issues.

i’m lucky to’ve gotten the e-mail assigning my
new grading duties at all… g-mail was working
this week evidently but my official campus e-ddress
was, emphatically, *not*. &… like i was saying
in my earlier post… we only get the web at “home”
(really my girlfriend’s house) by the on-and-off
grace of the vastly-indifferent cable company.

so it’s a problem.

my latest… i thought i was quarterly
when i issued it but maybe i’m semi-annual…
looks like this.
1876
2345

in the upper left are the “covers”:
these become the front and back
of the zine in folded form.

i discussed the title a few hours ago
(along with some other stuff).
suffice it to say here that our
subject is self-dual spaces.
*under* the title (in folded form;
above it here) are two
representations of the *simplest*
example: duality in P^2(F_2).

the space P^2(F_2) is itself
displayed (without a duality)
on p.5 (the lower right of the
photo) in the usual “triangle” way.
the colors are used to show
how these two objects…
*and* a certain subobject
of the space on pp. 7 & 8
(the 21-point P^2(F_4))…
can be considered “the same”.

the lower left (pp. 2 & 3)
are the desargues configuration.
i’ve discussed it recently
(here, for example)
and’m starting to hurry.

finally then (for now),
the cube-and-cross sections
bits are much the most vivid
visual motivator i’ve found
for the concept of “projectivity”.
the colors are surprisingly
helpful here. i’ll try to convince
you later. company coming. bye.

prehistory of MEdZ S ’11

the title is .
i pronounce this “ess is isomorphic
to the dual of ess” if i read it
symbol-for-symbol; one might also
read it (more freely) as
“S is a self-dual space”.
in some sense, this zine is about
spaces isomorphic to their own duals.

whatever that might mean.

in Some Finite Projective Spaces (01/07/11)
are several lectures-without-words shots
from earlier zines. including this one:

.
this was something of a breakthrough drawing for me.
(Self-duality of the Fano Plane, one might call it
[were it not a lecture-without-words].)
somehow i’d finally stumbled on a “visual” representation
for duality. this inspired a great deal of graphical
fiddling around by me. the same algebraic tricks
used in constructing the 7-point-to-7-line duality
for fano space could *also* be used on any n-point “space”
having n = 1 + q + q^2,
where “q” is some power-of-a-prime.
fano space is the case q = 2. so i did q=3 and q=5
(and the results are in the post i’ve linked to above)
and published ‘em. by february 1, i’d developed
a much cleaner-looking graphical presentation
for these (projecive-planes-of-small-order).
but the next real “breakthrough” ideas
came with the case of q=4. in this early draft

i went a little bit “deliberately weird looking”.
a later version of Self-duality of the Projective
Plane Over the Field of Four Elements
became the last display of the (black-and-white)
edition of Spring 2011 and i announced it
on april 17.

but now i have to backtrack. i haven’t done any of the math.
okay. enough for today.

PS. fall quarter finds me doing two sections
of Calc !V at big-state-u. i met the lecturer
wednesday and met my students thursday. whee!

meanwhile: dy/dan got a sweet publishing deal somewhere along the line.

In fact, consider the “canonical” set of five objects: {0, 1, 2, 3, 4}.

There are then ten ways to form pairs of these objects, as one sees from the following display:
04 03 02 01
14 13 12
24 23
34
.

(Note that by “pairs”, I refer to unordered pairs. For example, “40″ doesn’t occur in the display because this represents the same pair of objects as “04″. In fact, since {4, 0} = {0, 4} as sets, one could formalize the notion of “pairs” as I’m using it here by referring to sets-of-two-objects from our original five-set. Indeed, I’d do it this way despite the extra typing if I wasn’t afraid it would then look harder to read.)

Now let’s call the objects of this ten-set
{01, 02, 03, 04, 12, 13, 14, 23, 24, 34}
the points of a “space” called D (for Desargues). “D” itself is then this set of points, together with a collection of certain (unordered) triples of points. The triples themselves are called the lines of D.

Here are the lines of D.
{01, 02, 12}
{01, 03, 13}
{01, 04, 14}
{02, 03, 23}
{02, 04, 24}
{03, 04, 34}
{12, 13, 23}
{12, 14, 24}
{13, 14, 34}
{23, 24, 24}
.

To form a line, take any three objects from {0, 1, 2, 3, 4} and form all three possible pairs.

One now has a geometry-free construction of the “space” I’ve been examining here in the blog. In particular, each point appears in exactly three lines (just as each line contains exactly three points).

I intended more of this today. But, to my delighted surprise, duty calls: I’m meeting a new tutee in a couple hours and have to go catch the bus.

yesterday’s more verbose version.

the MEdZ logo indicates where
the ten lines of the desargues diagram
fall. one has ten such lines.
also ten points. each line
can be considered as a set
of three points; similarly each
single *point* belongs to three *lines*.
in fact, we have a “duality” here…
theorems about points-and-lines
remain true when the words
“point” and “line” are interchanged.

for example, in a self-dual space
having the property… which this one
does *not*… that “any two points
determine a unique line”, one would
also have “any two lines determine
a unique point” (and one makes
adjustments to plain-english like
“any two points *lie on* a line”
becoming-replaced-with
“any two lines *meet at* a point”
or what have you).

it turns out in desargues-space
(let’s say, in D) each line has
three parallel lines, all meeting
at a point. the point is said to be
the pole of the line and
the line is said to be the polar
of the point. a choice of pole-and-polar
for the diagram is a *polarity*.

ten polarities are displayed in color here.
each white dot represents the pole
for its diagram; the polar is colored
with the three “secondary” colors
(green, purple, and orange).
the three lines through the pole
have matching “primary” colors:
a red line, a blue line, and a yellow line
(if you will).

it turns out that the primary-color points
can be arranged… in exactly one way…
into *two* red-yellow-blue “triangles”
(whose “edges” are along lines of D).

now we come to the payoff.
the two triangles are said to be
“perspective from the pole”
(“p”, say; call the polar line “l”
while we’re at it if you please):
one imagines shining a light
held at p through the vertices
of one triangle to produce the
other triangle… like a slideshow.

and what happens is that now
*either* red-blue line will “hit”
the purple point… and *either*
red-yellow line will hit the orange
point… and *either* blue-yellow
line will hit the green point:
the colors “mix like pigments”.
recall that the secondaries…
the “mixed” colors… all fall on
the polar of p. recall that this
is a *line* of D.

when, as in this case, the three
points-of-intersection for the three
corresponding-edge-pairs
of a pair of triangles happen to
lie on a single line, the triangles
are said to be “perspective from
the line”. in our RBY metaphor,
perspectivity from a line means
we color the vertices of the triangles
and form the secondary colors
by intersecting the lines.
perspectivity from a line means
that the secondaries all line up.

now
if P is a “space” (a set of points
together with certain subsets
called lines) satisfying certain
axioms (those of a projective
space), then we have
desargues’s theorem
any two triangles perspective
from a point are perspective
from a line; any two triangles
perspective from a line are
perspective from a point.

(d’s theorem is *almost* true in
the ordinary euclidean plane…
but alas special cases must be
written in to account for
parallel lines. parallels are
banned from projective spaces
which makes ‘em easier to work with
algebraically but harder to visualize.
luckily “ordinary” planes can be
made to “sit inside” projective planes
so we can recover all of euclidean
geometry in a more-convenient-for-
-abstract-symbol-manipulation form.

PS
i still haven’t “solved”
ten-point reverse TTT,
by the way. but it’s very likely
only a matter of time.
somebody skilled in computer coding
could probably knock it out in a few hours.

a by-hand reduction
of my desargues-space poster,
with one less layer of redundancy
(and the colors of the triangles
reversed; also some cosmetic tweaks).

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.

and this is true even for
solomon garfunkel on common core standards
(PDF; from the notices).

they screw you up, your uncle sam.
they ought not mean to, but they do.

the three directions associated
with “arrows’… 001, 010, and 100
in the binary code… are here
colored yellow, red, and blue
(it didn’t reproduce as well as
one would’ve liked).

these so-called “primary” colors
(for pigments; for colored lights
the rules are different) blend
along the appropriate cross-sections
of the diagram (the easy-to-see
ones with the arrows attached)…
*or* the sides of the triangle:
yellow and red form orange,
yellow and blue form green,
red and blue form purple.

blending *opposite* colors…
the primary/”secondary” pairs
yellow/purple, red/green, blue/orange…
gives a muddy brown (“mahogany”
sez crayola) and again the appropriate
cross-sections of the cube (and lines
of the “fano plane” triangle) associate
the appropriate “blends” with the
pairs-of-pigments combining to form them.

finally, there’s the weird cross-section
(the 3-5-6 arc in the fano diagram)
consisting of all three secondaries.
on the cube diagram, this appears
as a tetrahedron-through-the-origin
rather than a plane-through-the-origin.

colored pencils (with erasers) rock.
we now return to zeerox-friendly B&W.