## Archive for the ‘MathEdZine’ Category

here one has labeled the vertices of a cube

with (“euclidean”) 3-space co-ordinates.

the “origin”… we can think of it as

our “point of view”… is at 000.

[the “point” in 3-space usually

denoted (x, y, z) is here written

as “xyz”; we restrict our attention

to the values in {0, 1} for these

variables (since the (8) points

000, 001, 010, 011,

100, 101, 110, 111

then form the vertices of our cube).]

putting the primary colors (Red, Yellow, & Blue)

“on the x, y, & z axes (respectively)”, i.e., putting

Red—>001

Yel—>010

Blu—>100,

we may then conveniently put the *secondaries*

(Green, Orange, & Purple) at the “third vertex” of

the back left wall (“G = Y + B”),

the front left wall (“P = R + B”), and

the floor (“O = R + Y”)

(respectively).

the last vertex is the “ideal” point 111;

all the colors blend here to form Mud.

cool algebra ensues. but not till after dinner.

or you could look it up.

here’s a hex board

with seven icons on it;

each icon has seven colors;

the seven permutations of

colors-into-hexes (each icon

has seven hexes) can be

considered as the objects

of a cyclic group.

specifically

{

() = \identity

(1234567) = \psi

(1357246)= \psi^2

(1473625)=\psi^3

(1526374)=\psi^4

(1642753)=\psi^5

(1765432)=\psi^6

}.

(of course one has \psi^7=\identity,

etcetera… the group here is

essentially just integers-mod-7:

{ [0], [1], [2], [3], [4], [5], [6]},

with addition defined by

“cancelling away” multiples

of 7 (forget about the fancily

denoted “permutation structure”

displayed in defining \phi

and just look at the exponents).

the “cycle notation” used here

is *much* under-used, in my opinion.

but we’ll really only need it

for future slides.

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.

**introductory ramble**

i’ve been passing around the “spring 2011”

issue of MEdZ pretty freely and’ve posted

quite a bit *about* it.

but it isn’t very well-organized.

so in hopes of finding a curious reader

hoping to learn about the pictures in the zine

(MEdZ S’11 is in pictures-only “lecures

without words” format… an 8-page

micro-zine “unstaplebook” made by copying

all 8 images [including the front & back

covers of course] onto a single standard sheet

and cut-and-folded into tiny-booklet form;

this is the first issue in color and very much

my personal favorite [of the moment]), i decided

to poke around in the ol’ Archive for the ‘Graphics’ Category

(and so on) and write out some connecting remarks.

**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!

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.

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.

another microzine 8-pager

(one side of a single sheet

of typing paper, cut & folded).

pages 6 & 7 are this new version

of . i’ve used

the (more “obvious”) ordering

00 01 0a 0b

10 11 1a 1b

a0 a1 aa ab

b0 b1 ba bb

on the “finite points”;

last time i posted a drawing

of this space i used the weird

aa ab ba bb

a0 a1 b0 b1

0a 0b 1a 1b

00 01 10 11

.

it seemed like a good idea

at the time. part of the point

is that there *are* different

ways to go about putting in

a co-ordinate system.

but mostly i just wanted

to see how it would look.

i’ve just edited in a subscript-backslash-zero

to the top line…

which now reads …

to repair an *earlier* repair, done sloppily.

somewhere along the line i tacked on the “—{(0,0)}”

*without* adjusting the “zee-cross-zee” ().

a beginner-like blunder, i confess. onward! *more* mistakes!

(just get down in the dirt and *calculate*, by golly.)

anyhow, owners of *Math Ed Zine* #0.4— by name—

should please to adjust the appropriate page in their issues.

which, being interpreted, means that

the set of *rational numbers* (**Q**) can be

represented as the collection of *lines through

the origin* (in the usual *(x,y)*-plane),

having *rational slope*. The slope condition,

for a given line, is equivalent to the condition

that there be an *integer* pair lying on the line

(nonzero; it gets to be something of a pain…).

the algebraic process whereby S…

nonzero-integer-pairs…

“maps onto” **Q**

is called “factoring by a relation”.

the relation in this case is called “tilde” (~).

tilde is defined by

” (x_1, y_1) ~ (x_2, y_2)

MEANS THE SAME THING AS

x_1 * y_2 = x_2 * y_1″

(“cross-multiplication” is in effect;

tilde is the relation we want “because”

when ).

oh heck. there’s that infinite-sloped line.

belongs to S/~, too. OK. modify the .

let’s call it , say. okay.

that’s it.

.

here’s MEdZ #1 in both editions: last year’s “mini”

and this years “digest” sizes. the 2011 edition

features, along with “the hip-pocket vocab”

(a glossary for math-for-humanities), some

remixed drawings from some “micro” zines

(also from last year), along with some (new,

brief) handwritten commentary.

the seven-sections pictures look way better

cut together (so here *that* is).

this last one’s previously unpublished.

anyhow. it exists. there’s a even a (proof) copy in circulation.

mostly it’s just masters, though. i’ll be running off the first

big print run in the next few days i think. numbered & dated.