Archive for May, 2011
here’s another picture of a dualization
of the projective plane on the field
of four elements: .
in the first of two tweaks since the last version
published here, i reversed
the positions of the roots of x^2 + x +1…
i’ve been calling ’em alpha and beta…
reversed their positions, i say,
for the vertical axis (as opposed
to the horizontal:
0a 1a aa ba
0b 1b ab bb
01 11 a1 b1
00 10 a0 b0
as opposed to the earlier
0b 1b ab bb
0a 1a aa ba
01 11 a1 b1
00 10 a0 b0
this had the pleasing effect that
*all* the “lines” now had naked-eye
symmetry. the earlier drafts had
some twisty-looking “lines” once
the alphas and betas got involved;
this was in some part due to the
artificial “symmetry breaking” that
i’d indulged in by *putting alpha
first*. the new version had beta
close to the origin just as often
as alpha… which better describes
their relation in .
anyhow, the second, more radical
“tweak” involved swapping (0,0)
and (1,1)… and causing “lines”
actually *looking like lines* to
appear as *broken* lines
(but simultaneously causing the
alpha-points and beta-points
to behave better still).
this looks like a pretty good board
for pee-two-eff-four tic-tac-toe.
(or, ahem, “21-Point Vlorbik”…
he who blowet not his own horn,
that one’s horn shall never be blown).
Tic Tac Toe
Official Rules (all rights reserved).
“The Board” is the 21-line
array of letters:
two players alternate turns.
in the first turn the first player
chooses any letter from A to U
and “colors” all five copies of
that letter on the board;
the second player then chooses
any *other* letter and “colors”
all five copies (in some other
color… X’s and O’s can be made
to do in a pinch if colored pencils
players continue alternating turns,
each coloring all five copies of
some previously uncolored letter
at each turn.
play ends if one player… the winner…
has colored *all five letters* of any row.
otherwise play continues until all (21)
letters are used.
if one player… the winner… now has more
“four in a row” lines than the other, so be it;
otherwise the game ends in a draw.
i actually played this for the first time
this morning on the bus with madeline:
a four-to-four draw.
in the thus-far-imaginary computer version,
upon selection of a letter, five dots…
all the same relative position in their
respective “crosses”… light up and
“five in a row” becomes “an entire
cross lights up”. this might be worth
learning to write the code for.
better if somebody else did it, though,
i imagine. i just want a “game designer”
credit and a small piece of every one sold.
composition of linear fractional transformations
compared to two-by-two matrix multiplications.
in other words,
let f(x) = (Ax+B)/(Cx+D) and
let g(x) =(ax+b)/(cx+d) and
consider the function (“f\circ g”,
i.e. f-composed-with-g). recall
(or trust me on this) that
[f\circ g](x) = f(g(x)); i.e.,
functions compose right-to-left
(“first do gee to ex; then plug in
the answer and do eff *to* gee-of-ex”…
first g, then f… alas. but there it is).
so we have
whereas one also has
so the matrix-multiplication equation
can be obtained from the function-composition equation
merely by applying an eraser here and there.
the last display of MEdZ #0.4
is this wordless version of
cantor‘s epoch-making result
that there are “more” (in the sense
of cardinality, a concept
i outlined in the last post) elements
in (the Real Numbers)
than in (the Naturals).
and a pretty picture it is, too…
to at least *one* set of eyes.
but there’s rather a glaring mistake.
every self-publisher… including
creators of “handouts” for classes…
probably soon learns that certain
mistakes only become visible to
their authors after producing
a large number of copies.
anyhow, it’s always been so for me.
in this case.
i should’ve started the fourth line up
then the next line should begin
or i could just edit
out of the whole mess altogether…
a fix with the charming property
that i could perform it with
white-out alone (no marker).
also the third line should begin
what the heck i was thinking
writing out an implication
with no antecedent i’ll never know.
that one i’ll need a marker for.
at long last. this has been sitting
on the paperpile very-nearly-finished
for quite a while.
i started the “lectures without words”
series early on with 0.1: .
whose cover more or less announced
implicitly that it was one of a series
called . and that
was, like, five quarters ago.
and they’re only 8 micro-size pages.
a couple days ago i inked the graphs
and the corresponding code (the stuff
under the dotted line had *been* inked
and the whole rest of the issue was
entirely assembled). and zapped it off.
and yesterday i passed ’em around
at the end of class (to surprisingly few
students given that i’ve got freshly-
-graded exams). it went okay.
i did right away
(if i recall correctly), and in
high-art style, too (i used a
brush instead of a sharpie).
wasn’t much later.
i have plenty of notes for ,
too, and could knock out a version
on any day here at the studio
(given a couple hours and some
peace of mind) that’d fit right in.
anyhow, what we have here are,
first of all, obviously, a couple graphs
and a bunch of code. here, at risk
of verbosity, is some line-by-line
in the upper left is
part of the graph of
the linear equation
namely the part whose x’s
(x co-ordinates) are between
-1 and 1.
and my students (like all
deserving pre-calculus graduates)
are familiar with *most* of the
notations… and *all* the ideas…
in this first line.
that funky *arrow*, though.
well, i can’t easily put it in here
(my wordpress skills are but weak)
but i’m talking about the one
looking otherwise like
and in the actual *zine*, it’s
a Bijection Arrow.
something like ” >—->>”.
as explained (or, OK, “explained”)
*below* the dotted line.
where *three* set-mapping “arrows”
are defined (one in each line;
the in each line
denotes “is equivalent-by-definition to”).
the Injective Arrow >—> denotes that
f:D—->R is “one-to-one” (as such
functions are generally known
in college maths [and also in
the pros for that matter; “injective”
and its relatives aren’t *rare*,
but their plain-language versions
still get used oftener]).
“one-to-one”, defined informally,
means “different x’s always get
different y’s”. coding this up
(“formally”), with D for the “domain”
and R for the “range” (though i
prefer “target” in this context
when i’m actually present to
*explain* myself) means that
when d_1 and d_2 are in D,
and d_1 \not= d_2
(“different x’s”), one has
f(d_1) \not= f(d_2)
likewise the Surjective Arrow —>>
denotes what is ordinarily called
an “onto” function:
every range element
(object in R)
“gets hit by” some domain element.
and of course the Bijective Arrow >—>>
denotes “one-to-one *and* onto”.
there’s quite a bit of symbolism
here that’s *not* familiar to
typical college freshmen.
the Arrows themselves.
and the seldom-seen-even-by-me
“such that” symbol that, again,
i’m unable to reproduce here
still, i hope i’m making a point
worth making by writing out
these “definitions without words”.
anyhow… worth doing or not…
it’s out of the way and i can return
to the main line of exposition:
the mapping from (-1, 1) to (0,1)
in the top line of my photo here
is a “bijection”, meaning that it’s
a “one-to-one and onto” function.
for *finite* sets A and B,
a bijection f: A —-> B
exists if and only if
# A = # B…
another unfamiliar notation
i suppose but readily understood…
A and B have
*the same number of elements*.
we extend this concept to *infinite*
sets… when A and B are *any*
sets admitting a bijection
f:A—>B, we again write
#A = #B
but now say that
A and B have the same cardinality
(rather than “number”;
in the general case, careful
users will pronounce #A
as “the cardinality of A”).
i’m *almost* done with the top line.
i think. but there’s one more notation
left to explain (or “explain”).
the “f:D—>R” convention i’ve been
using throughout this discussion
is in woefully scant use in textbooks.
but it *is* standard and (as i guess)
often pretty easily made out even
by beginners when introduced;
one has been *working* with
“functions” having “domains”
and “ranges”, so fixing the notation
in this way should seem pretty natural.
but replacing the “variable function”
symbol “f” by *the actual name
of the function* being defined?
this is *very rare* even in the pros.
the rest is left as an exercise.
yang’s Ohio ARML page at CSCC.
i volunteered at a practice session yesterday.
mostly logistical stuff, more or less of course.
but with, anyway, a few opportunities to talk
about math and life with people not inclined
to dismiss me outright as soon as… or even
*before*… i even open my mouth to speak.
rare enough in these dark days i assure you.
i’ll do it again in a few weeks.
my only prior math-contest experience was
taking the notorious “putnam” as a college senior.
i answered one question easily right away
and spent the rest of the day trying and failing
to get a handle on any of the other 11.
that was fun too. i should remember better
who the coaches were. andrew lenard
is the only one i can name for sure.