## Archive for October, 2016

### now… momentarily… with less spam

Blue Car, Blue Car (1998)
I finally got a learner’s permit
at age nineteen in Thousand Oaks,
where I’d been living like a hermit
on coffee, pot, and rum-and-cokes.
Now, back in school, a lack of patience
for classes during my vacations
had kept me out of Driver’s Ed.
But something had to give. I said,
“Of course I’d rather do things my way,
and walk, or thumb, or ride my bike.
But I can’t have things as I’d like.
This California’s one big highway!
It’s best to take things as they are.
I’d better learn to drive a car.”

And so my then-best-friend, Bob Shaffer,
agreed to bring me up to speed.
“I know a car that you could pay for.
I’ll teach you everything you need.”
“But what about repairs?” “Don’t panic!
This car was owned by a mechanic!
It’s in great shape! It runs just fine!
It sounded like that classic line:
“The only owner was some granny
who never drove”, but Bob was right.
I got the car that very night:
a sixty-three, push-button tranny,
Plymouth Valiant, not much rust.
It turned out worthy of my trust.

I made a hundred dollar payment
and owed another; then I’d bought
it. Breaking up the debt this way meant
I could pay with ease–I thought.
But then my boss at Howard Johnson’s–
whose every word was arrant nonsense–
said “Although it pains me, I
have got to let you go. Goodbye.”
(I thought I knew his secret reason:
I’d worked there for about a year,
and paid vacations cost them dear.
It’s always bellboy-shafting season.)
So even though I had enough
to make the payment, it was rough.

And so at last I started learning
how to drive. At least, I tried!
My second night, as I was turning
(way too fast and far too wid,
which should have been a minor error),
I saw a car and froze in terror,
making it a big mistake.
At last, too late, I hit the brake.
I’d caused a little fender-bender.
The other guy, whose car I’d hit
was more than fair, I must admit.
A small amount of legal tender
satisfied him–not too bad!
I called and got it from my dad.

The testing had me really worried
and, in fact, I failed it. Twice.
But then I got a guy who hurried
once around the block. How nice!
To earn the necessary rating
depended less on skill than waiting.
(I might have known from back in school
that grades are like that as a rule.)
I drove my Valiant to Laguna
to show my dad the famous dent.
He thought his money quite well spent;
he only wished I’d done it sooner.
He always hoped I’d leave the stage
of wayward youth and come of age.

But that’s another, longer, story
and not the one I came to tell.
I’m sticking to the task before me.
I think you’ll find it’s just as well.
Enough to say that now that twenty
years have passed, I’ve grown up plenty–
but still today, without a doubt,
I need a lot of bailing out.
Returning to my car: it never
once broke down, though there was once
I thought it had but like a dunce
I hadn’t checked the gas tank. Clever!
They’ll never make a fool-proof tool
as long as there’s a perfect fool.

Once I had the driving habit
I gave the car up as a loss.
I had a chance–and chose to grab it–
to move to Vegas with my boss.
It didn’t take a lot of thinking;
we’d sleep till noon, and then start drinking,
and work as little as we could
at cleaning carpets. Life was good.
But I was broke, as was my pattern,
and owed my landlord two months rent.
He got the car and off I went.
From then until I bought my Saturn–
that is, till fifteen years had passed–
that Valiant was my first and last.

At thirty-five, I got a fairly
well-paid job. But I lived far
enough away that I could barely
get around without a car.
I’ve never liked to deal with dealing–
a root canal is more appealing–
and so, I chose the one brand name
that cheated everyone the same
and wouldn’t make me drive a bargain
to drive a new car off their lot:
in short, a Saturn. Still, I got
a song-and-dance and empty jargon
about the warranty. Till Hell
completely freezes over, dealers sell.

I had a girlfriend, Betsy Baxter,
in Bloomington (my old home town).
I e-mailed, phoned (but never faxed) her,
and every weekend, drove around
five hundred miles in any weather
so we could have some fun together.
I married her, she finished school,
and then I learned I’d been a fool
for thinking that our love was thriving.
We hadn’t lived together long
when she decided we’d been wrong.
As long as it was mostly driving
back and forth from state to state
our marriage really worked out great.

Of course my car was bought on credit:
the price tag was eleven grand.
In sixty months, I’d finally get it
free and clear—or so I planned.
My enemies contrived to spoil it;
my whole career went down the toilet
(it might have lasted longer, but
I wouldn’t keep my big mouth shut).
I lived six months on unemployment
and credit cards whose interest rates
would ruin even William Gates.
Then, after all the fun enjoyment—
no job in sight—the bills came due.
So I went bankrupt. Wouldn’t you?

My two divorces, uncontested,
For this, though, common sense suggested
the banks would go for all I had.
My wife, the former Shauna Kearney,
referred me to a good attorney
who looked at my accounts with me
and, taking out a hefty fee,
said, “Well, this car’s worth too much money.
Depreciation’s cut the cost,
but not enough. And so, you’ve lost
because you’ve saved. It’s funny
but that’s the way things sometimes are.
You’ll keep the rest; they’ll get the car.”

### 13-point projective space

the left-hand photo shows
a nine-point plane: an “ordinary
two-dimensional plane” over the
field with three elements (and its
label is, therefore, ${\Bbb F}_3^2$).

such a plane is ordinarily co-ordinatized as
(0,2) (1,2) (2,2)
(0,1) (1,1) (2,1)
(0,0) (1,0) (2,0):
the set of (x,y) such that
x & y are both elements of
the set {0, 1, 2}.
one could convey the same information more
concisely as
02 12 22
01 11 21
00 10 20.
it’s useful for our purpose here, however,
to consider our plane as belonging to a
*three*-dimensional space… (x, y, z)-
-space, let’s say… and as having a
*non-zero* “third” (i.e., “z”)-co-ordinate.
thus, in the photo, our plane is represented by
021 121 221
011 111 211
001 101 201.

the colors come into play in displaying the
solution-sets for various (linear) equations.
the reader can easily verify that the Green
equation—x=2— is “true” for the points of
the vertical line at the right… i.e., for
{ (2,0), (2,1), (2,2) } (old-school), i.e. for
{ 201, 211, 221} (“our” version).

likewise “y=x” (the Red equation) gives us
{(0,0), (1,1) (2,2)}… i.e.,the “Red line”
{001, 111, 221}.

now for some high-theory. by Algebra I, one has
a well-developed theory of Lines (in the co-ordinate
Plane). the usual approach there is to use the
(so-called) Slopes. the (allegedly intuitive) notion
of “rise over run” allows one to calculate—for any
*nonvertical* line—a number called the Slope (of that
line). vertical lines are said to have “undefined”
slopes. one might also say that they have an “infinite”
slope… though this invites confusion and is usually
best left unmentioned.

y = Mx + B
x = K
are then our “generic” *equations of a line*.

any particular choice of numbers M & B will
correspond to the a set of solutions lying
along a (nonvertical) line having the slope
of M (an passing through (0,B)… the so-
-called “y-intercept” of the line); each vertical
line (likewise) is represented by some particular
choice of K.

now. having different “forms” for vertical and for
nonvertical lines can be devilishly inconvenient,
so, also in algebra I, one sometimes instead uses
the “general form” for an equation of a line in the plane:
Ax + By = K
(with A & B not both zero).
likewise (but typically *not* in algebra-i)
Ax + By + Cz = K
(with A, B, & C not all zero)
is the equation of a *plane* in *three*-dimensions.

thus far i’ve been as precise as i know how.
but now i’m going to start waving my hands around
and making leaps-of-faith all over the place.
in the second photo, four new “points” have been
added into our framework (namely
{010, 120, 110, 100}—the “line at infinity”).

the upshot… without the details… of this move is
that one now has an algebraic theory of “Lines” in a
“Plane” containing precisely our 13 “Points”. more-
over, this theory is “structurally” very similar to
“ordinary” linear theory. in particular, we dealing
with solutions to
Ax + By + Cz = 0—
the “K=0” case of the “general form” for (the 3D case
of the “ordinary” theory).

the Green equation—which must now be written without
its “constant term” (x = 2 is “the K=2 case” of x = K)—
becomes x – 2z = 0;
similarly, rather than (the three-point “line”
of ${\Bbb F}_3^2$) “y = 1” (concentric black-and-
-white circles), the (“homogeneous”—for us, right now,
this can be taken as meaning “having no constant term”)
equation is “y – z = 0” (and, again, we pick up a “new”
point at 100).

when the smoke clears… which won’t be here and now…
we’ll have a *very nice* geometry. just as in “ordinary”
space, two distinct points determine a unique line.
but… *unlike* “ordinary” space, it’s also true that
(*any*) two lines determine a unique point.

the same “trick”—homogeneous co-ordinates in finite
fields—converts any plane having p^2 as its number
of points to a *projective* plane having
p^2 + p^1 + p^0
as its number of points. thus there are PP’s having
7, 13, 31 = 5^2 + 5 +1, 57 = 7^2 + 7 + 1, ….
as their number-of-points. there are also some others.
but the margin is too small.

• ## (Partial) Contents Page

Vlorbik On Math Ed ('07—'09)
(a good place to start!)