Archive for May, 2011


here’s another picture of a dualization
of the projective plane on the field
of four elements: P^2(\Bbb F_4)^*.

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 F_4.

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).

P_2(\Bbb F_4) Tic Tac Toe
Official Rules (all rights reserved).

“The Board” is the 21-line
array of letters:

ABCDE
AFGHI
AJKLM
ANOPQ
ARSTU
BFJNR
BGKOS
BHLPT
BIMQU
CFKPU
CGJQT
CHMNS
CILOR
DFLQS
DGMPR
DHJOU
DIKNT
EFMOT
EFLNU
EHKQR
EIJPS
.
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
aren’t available).

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.

consider
\lbrack x \mapsto {{Ax +B}\over{Cx+D}} \rbrack \circ \lbrack x \mapsto {{ax +b}\over{cx+d}} \rbrack\,.

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 (“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
\lbrack x \mapsto {{Ax +B}\over{Cx+D}} \rbrack \circ \lbrack x \mapsto {{ax +b}\over{cx+d}} \rbrack
= \lbrack x \mapsto { {A{{ax +b}\over{cx+d}} + B}\over{C{{ax +b}\over{cx+d}} +D} }\rbrack
= \lbrack x \mapsto { {A(ax+b) + B(cx+d)}\over{C(ax+b) + D(cx+d)}}\rbrack
= \lbrack x\mapsto { {(Aa+Bc)x + (Ab+Bd)}\over{(Ca+Dc)x + (Cb+Dd)} }\rbrack\,.
thus
\lbrack x \mapsto {{Ax +B}\over{Cx+D}} \rbrack \circ \lbrack x \mapsto {{ax +b}\over{cx+d}} \rbrack =\lbrack x\mapsto { {(Aa+Bc)x + (Ab+Bd)}\over{(Ca+Dc)x + (Cb+Dd)} }\rbrack\,.

whereas one also has
\begin{pmatrix} A & B \\ C  & D\end{pmatrix} \begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} Aa +Bc & Ab + Bd \\ Ca+Dc  & Cb + Dd \end{pmatrix}\,.

so the matrix-multiplication equation
can be obtained from the function-composition equation
merely by applying an eraser here and there.

(my lecture-note-blogging of winter 09 include some
remarks on \mapsto notation and much more
about linear fractional (“mobius”) transformations.)

bug report


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 \Bbb R (the Real Numbers)
than in \Bbb N (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
\exists b = .b_0 b_1 b_2 b_3 ...
rather than
\exists b = .b_1 b_2 b_3 b_4... .
then the next line should begin
b_0 \not= a_{00}, b_1 \not= a_{11}....

or i could just edit a_0
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
with \exists S \Rightarrow a_0 = ....
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: \Bbb N.
whose cover more or less announced
implicitly that it was one of a series
called \Bbb N  \Bbb Z \Bbb Q  \Bbb R \Bbb C. 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.

\Bbb Z i did right away
(if i recall correctly), and in
high-art style, too (i used a
brush instead of a sharpie).
\Bbb Q wasn’t much later.
i have plenty of notes for \Bbb C,
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
commentary.

in the upper left is
part of the graph of
the linear equation
y=(x+1)/2…
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
(-1, 1) \Rightarrow (0,1).
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 :\Leftrightarrow 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)
(“different y’s”).

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.
\forall “for all”
\exists “there exists”
\wedge logical “and”
and the seldom-seen-even-by-me
“such that” symbol that, again,
i’m unable to reproduce here
in type.

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.
alas.

the rest is left as an exercise.

m’aidez

math stood tall: edusolidarity recap at JD’s.

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.