Archive for the ‘MathEdZine’ Category

Photo on 4-23-15 at 6.30 PM

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.

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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 S \cong S^*.
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:

Photo on 2010-11-23 at 14.37

.
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.

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.


another microzine 8-pager
(one side of a single sheet
of typing paper, cut & folded).

pages 6 & 7 are this new version
of P^2({\Bbb F}_4)^*. 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 S = ({\Bbb Z}\times {\Bbb Z})_{\backslash 0} =...
to repair an *earlier* repair, done sloppily.

somewhere along the line i tacked on the “—{(0,0)}”
*without* adjusting the “zee-cross-zee” ({\Bbb Z}\times{\Bbb Z}).
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—\Bbb Q 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”
{{y_1}\over{x_1}} = {{y_2}\over{x_2}}
when x_1 y_2 = x_2 y_1).

oh heck. there’s that infinite-sloped line.
belongs to S/~, too. OK. modify the \Bbb Q.
let’s call it {\Bbb Q}^\infty, say. okay.
that’s it.
.

Photo on 2011-02-07 at 14.40

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).

Photo on 2011-02-07 at 14.44

this last one’s previously unpublished.

Photo on 2011-02-07 at 14.45

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.