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: 
whose cover more or less announced
implicitly that it was one of a series
called 
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.
(if i recall correctly), and in
high-art style, too (i used a
brush instead of a sharpie).
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 
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.
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.
a recent post in KTM asked why
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.
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.
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
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(z)+f(w) = f(z+w) trivially.
PS
s:=
[1^2, 2^2, 3^2, 4^2, …]=
[1,4,9,16,…]=
[n^2]_(n \in \Bbb N).
NB: 0 \in \Bbb N.
******************************************
S := x + 4x^2 + 9x^3 + …
(
******************************************
(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
OK
)
the livingston review 