## 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:

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

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

thus

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.

(my lecture-note-blogging of winter 09 include some

remarks on \mapsto notation and much more

about linear fractional (“mobius”) transformations.)

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

rather than

.

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

with .

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

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

.

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)

(“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.

“for all”

“there exists”

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.

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.