Archive for the ‘Lectures Without Words’ Category

ten poles, ten polars,
and ten pairs-of-triangles:
ten ways to use one drawing
(from MathEdZine, of course)
to illustrate one theorem.

(desargues’ theorem at w’edia.)

performing calculations using
actual blobs of color rather than
alphabetical symbols *standing*
for colors is time-consuming
(and demanding of special tools)…
but one literally “sees” certain things
*much more readily* than with
symbol manipulation.

hey, i’m a visual learner.


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

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
(-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.

the rest is left as an exercise.

a recent post in KTM asked why
{6 \choose 2} = {6 \choose 4}.
i know that one.

the “graph” in the upper right is K_4…
the “complete graph on 4 vertices”…
and has *6* (so-called) edges.

its “complement” (upper left) has *none*
of the edges. obviously there’s one
(so-called) *complete* graph having
all six possible edges and one “empty”
graph having *none* of the edges.

move down to the next couple lines:
the graphs having exactly *one* edge
match up in one-to-one fashion
with the graphs having *five* edges
(because “including five (edges)”, in this context,
is the same as “leaving out *one* (edge) out”.

so on for two-edge graphs.
each is the “complement”
(via the inclusion-exclusion principle
as hinted at a moment ago)
of a *four*-edge graph.

posting; gotta go to class.

Photo on 2010-11-27 at 11.47

Photo on 2010-11-27 at 11.48

Photo on 2010-11-23 at 14.37

Photo on 2010-11-23 at 14.35

Photo on 2010-11-23 at 14.05

Photo on 2010-11-27 at 16.25 #2

Photo on 2010-11-27 at 16.25

Photo on 2010-11-27 at 16.24

above we have pp 6, 7, 8, 1;
below are pp 2, 3, 4, 5. you fold
it up into a little booklet (a “micro-zine”,
i sometimes say… a “mini” [also known
as a “quarto”] is four times this size
[twice as large in each direction]).
here’s section 1.1
of my textbook (and much the most-hit
post of this blog): “the set of natural numbers”.
and here’s a shot of
this and some other MEdZes
in their “folded” state.
stand by for some more links.
who knows maybe some direct
explanation even.

flickr shot

all these MathEdZines…
## 0.1, 0.1.1, 0.2, 0.3,
0.6, 0.7, and 1
(together with the un-
numbered nanozine version
of \Bbb Z)…
are going with us to SPACE
a few hours from now.
where, with any luck, i'll trade 'em
all away for some really cool comics.

z:= x+yi
w:= a+bi

zw = (xa-yb) + (xb+ya)i

u:= xa-yb
v:= xb+ya

f: \Bbb C \rightarrow M_2(\Bbb R)
f(p+qi) := \begin{pmatrix} p & q \\ -q & p\end{pmatrix}  (p, q \in \Bbb R)

f(z)f(w) = \begin{pmatrix} x & y \\ -y & x\end{pmatrix} \begin{pmatrix} a & b \\ -b & a \end{pmatrix} =

=\begin{pmatrix} xa-yb & xb+ya \\  -ya-xb & xa-yb \end{pmatrix} = \begin{pmatrix} u & v \\ -v & u\end{pmatrix} = f(zw)

f(z)+f(w) = f(z+w) trivially.


\begin{pmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{pmatrix} \begin{pmatrix} \cos\psi & \sin\psi \\ -\sin\psi & \cos\psi \end{pmatrix} =

=\begin{pmatrix} \cos\theta\cos\psi - \sin\theta\sin\psi & \cos\theta\sin\psi+\sin\theta\cos\psi \\ -\cos\theta\sin\psi -\sin\theta\cos\psi & \cos\theta\cos\psi - \sin\theta\sin\psi \end{pmatrix}=

=\begin{pmatrix} \cos(\theta+\psi) & \sin(\theta+\psi) \\ -\sin(\theta+\psi) & \cos(\theta + \psi) \end{pmatrix}

[1^2, 2^2, 3^2, 4^2, …]=
[n^2]_(n \in \Bbb N).

\Bbb N =: “\Bbb N”
NB: 0 \in \Bbb N.

S := x + 4x^2 + 9x^3 + …

(S = \sum_{i = 1}^\infty i^2x^i)
(1-x)*S = x + 3x^2 + 5x^3 + 7x^4 + …
(1-x)^2*S = x + 2x^2 + 2x^3 + 2x^4 + …
(1-x)^3*S = x + x^2

S = (x+x^2)/(1-x)^3
x = 1/10

S = .1 + 4*.01 + 9*.001 + 16*.0001 + …
S ~ .150892
S = (.1+.01)/(.9)^3
S ~ .150892

I Relaunch(01/05) as MathEdZineBlog at the site…… of my quit-in-a-fit-of-frustration Vlorbik On Math Ed of ’07 to ’09. I offer subscriptions to the then-purely-imaginary MathEdZine (and, though of course this isn’t revealed in the post itself, I get no takers. I am not at all surprised. Heck, I’m not at all discouraged, even. “Disappointed” might not be going too far.) In Where Did That Amazing Blogroll Go? (01/05) I offer a backhanded apology for having deleted the blogroll for the relaunch; this was probably the feature I’d’ve found most attractive about VME if somehow I’d found one just like it done by somebody else before I’d ever started blogging. A fellow mathblogger with better net chops than mine… “sumidiot”… had already modified it for the new feedbased formats anyhow, so my work was finished (“feeds” are part of what I’d been so frustrated at when I’d quit, I’ll go ahead and mention). Also WDTABG? was the first in what promises to be a pretty steady stream of links lists. Probably I shouldn’t drop links here in this post for all of ’em… here’s the WordPress Archive for Jan ’10.

In Songs For Beginners: Pascal’s Triangle (01/05) I hinted at one of my ideas for the first issue of the zine. “Lectures Without Words” is a phrase I’d been dropping here and there already… for example in this post from Open A Vein (my “homepage” blog) back in December. I have indeed gone on to create four zines on this theme and have convinced myself I could probably do one a week or so… on the model of the hugely-popular-though-nobody-knows-about-it cartoon diaries of recent years… indefinitely and love doing it. Not that I will. I’ve got plenty of other ideas about math, and ed, and zines. Unless this is some monster hit, I’ll move on pretty quick I imagine. One never knows. Where was I.

in lovely dark and deep (1/07)
i slip into my “ramble” style for some
discussion of early drafts of issue zero.
i crossposted this
in order, i think, to cause a link to show
up in twittr or facebook or some other
“social” site that i’d caused to be linked
but had lost track of (this problem has
of course grown even worse in the
ensuing weeks).

then in comment thread reprint (1/08)
i continued my very-intermittent conversation
with the frighteningly-successful dan meyer
about sizzle and steak; i followed up immediately
with “continued from…“, yet another
in a never-ending line of luddite ravings.
what am *i* gonna do with it?

Vlorbik On Math Ed(1/13) is the title of this post; its contents are actually a list of links to selected posts from the blog of that name… much like this one though much better organized. I’ve revised it substantially from the version of its posting date. Usually I avoid doing this but in this context it looks like plain common sense to go ahead and over-ride that “house style” rule.

prehistory of MEdZ, part i (1/19)
tells, rather, the prehistory of the ten page news
(volume iii), with some math ed thrown in.
editorial license.
prehistory of MEdZ, part ii (1/21)
is a dismal failure wherein i tried to put
a photo in and it was there briefly and
even drew a comment from JD (one of
VME‘s most faithful readers;
thanks jonathan). further fiddling…
specifically this photo-post (1/25)…
clobbered the picture; to heck with it.
i’ve gotten a little better at moving
pictures around with flickr and tumblr
but it’s infuriatingly “intuitive” and
i can still barely work any of it at all.
the picture that stuck… not at all by
the way… is of madeline’s scanner/printer.
i’ve made some mighty cool zines with
this thing already but i’ll probably
have to beat it to death with a goddamn
hammer before too much longer.
when it can be made to work at all…
at random as far as i can tell…
it makes gorgeous copies but
gohd knows what they cost and
anyway i can’t stand the aggro
and when it breaks soon i’m done.

so somebody comment already