## Archive for the ‘Cool Tricks’ Category

### for hannah on the back of an envelope

almost-symmetric desargues’ theorem.
(7-color version)

5 “flat” and 5 “tall” triangles
arranged as the 10 “intersection points”
of a (so-called) pentagram.

thus far the black-and-white “underlying
diagram” (which i probably should have
photographed before coloring it in and
erasing it… it was the best b&w version
i’ve done so far, i think… oh well…).

the b&w *already* “tells the story” (if you
know how to look): the ten “triangles” are
in the ten “positions” on the “big picture”
diagram in such a way that, mutatis mutandis
(read “line” for “triangle” & “point” for
“position, to wit), one has the well-known
points-&-lines “duality” exhibited in
purely *graphical* form: not only a
“lecture without words”, but a “lecture”,
if you will, without *code itself*.

so it’d be a pretty awesome picture, done right,
&’d make a good art project for somebody patient
enough to make actual measurements & stuff.

as would the colorized version.
so, briefly.

the letters {a, b, … , j}
are posted in “positions” in such a way
that the when the “triangles”
{abc, ade, afg, bdh, … , ijk}
are “placed in” the appropriate positions
(as is done here)
one has each letter-triple in the position
of its *dual* letter (“a” to “ijk”, eg.;
one should probably look at some other
now replace all the letters with colors
according to the scheme pictured at right.
(whose connection with the “fano plane”
[& “vlorbik’s 7 color theorem”] is solid
but we don’d have to go into it here.)

happy neighborhood of yr b’day.
thanks for your part in, you know,
teaching me to read & bringing me
back from the dead and all that.

PS the *actually* symmetric version has
the ten “lines” modeled as *diagonals
of an dodecahedron* (the “diameters”
connecting opposite vertices; there
are of course 20 such…);
the “asymmetric” (textbook) version
has a *marked* point (and, of course,
a “marked” dual line…); never mind…

### if only this power could be used for *good*

$ii = \begin{pmatrix} \begin{pmatrix} 0 & 2 \\ 1 & 2\end{pmatrix}& \begin{pmatrix} 1 & 1 \\ 0 & 1\end{pmatrix}& \\ \begin{pmatrix} 1 & 0 \\ 2 & 1\end{pmatrix}& \begin{pmatrix} 2 & 2 \\ 1 & 0\end{pmatrix} \end{pmatrix}$
$i = \begin{pmatrix} \begin{pmatrix} 1 & 2 \\ 1 & 0\end{pmatrix}& \begin{pmatrix} 2 & 1 \\ 0 & 2\end{pmatrix}& \\ \begin{pmatrix} 2 & 0 \\ 2 & 2\end{pmatrix}& \begin{pmatrix} 0 & 2 \\ 1 & 1\end{pmatrix} \end{pmatrix}$

$iii = \begin{pmatrix} \begin{pmatrix} 0 & 1 \\ 2 & 2\end{pmatrix}& \begin{pmatrix} 1 & 0 \\ 1 & 1\end{pmatrix}& \\ \begin{pmatrix} 1 & 2 \\ 0 & 1\end{pmatrix}& \begin{pmatrix} 2 & 1 \\ 2 & 0\end{pmatrix} \end{pmatrix}$
$iv = \begin{pmatrix} \begin{pmatrix} 1 & 1 \\ 2 & 0\end{pmatrix}& \begin{pmatrix} 2 & 0 \\ 1 & 2\end{pmatrix}& \\ \begin{pmatrix} 2 & 2 \\ 0 & 2\end{pmatrix}& \begin{pmatrix} 0 & 1 \\ 2 & 1\end{pmatrix} \end{pmatrix}$

$II= \begin{pmatrix} (mgy) (nrp) & (ybr) (pog) & \\ (mro) (ngb) & (mpb) (nyo) \end{pmatrix}$
$I= \begin{pmatrix} (bo) (mpgnyr) & (mn) (ygbpro) & \\ (yp) (mbrnog)& (rg) (mopnby) \end{pmatrix}$
$III = \begin{pmatrix} (mbp) (nor) & (mor) (nbg) & \\ (yrb) (pgo) & (myg) (npr) \end{pmatrix}$
$IV = \begin{pmatrix} (rg)(mybnpo) & (yp)(mgonrb) & \\ (mn)(yorpbg) & (bo) (mryngp) \end{pmatrix}$

### Virtual MEdZ #1.0

the tetrahedral group at left: A_4, to the group-theory geeks.

up top, the “yrb” labeling of the vertices of a cube,
with the bit-string digital code and a 2-D projection.
the seven-color theorem… concerning the simple group
of order 168
& MRBGPYO… is hinted at.

under that, as one can *kind of* read on the blurry photo,
is “desargues theorem in color” — ten “points”
(one Mud, two Yellow, two Blue, two Red, one Green,
one Purple, one Orange) in abstract “space”.
the best version… i’m not technically up to drawing it…
is to put the colors on the ten diagonals of a dodecahedron.
next best is the five-point star version taking up
the biggest part of the file-folder.

next to that on the right: the vertices of the cube
colorized again. pretty much the same way if memory serves.
the points-to-lines “duality” is colorized better here, i think.

at the bottom, several versions of the seven-point star version
of the MRBGPYO theorem… and other stuff about heptagons.

there’re some books in there, too. that’s it for today.

### boxing day

boxing day. these shots show shelves
partly festooned in seasonal decor.
i’ll be tearing it down now. meanwhile
the rest of the house is, anyhow, more
orderly than it’s been in quite some time.
and would deserve photographing &
posting online if it weren’t so much damn
trouble. poor me. everybody else can
do everything on their phones. and
drive cars and stuff. having a wonder-
ful time. wish you were here.

### why is yellow green?

(because violet blue red… but that’s
off-topic [and off-color]…)

three green-peppers; none of ’em green.
it’s even beautiful in its package and,
of course, even more so the more they’re
played with.

as shown here, the tops have been cut off
and served (with their middles cut out)
with ranch. the next thing that happened
was that a “ring” was cut off at the top
of each and the whole set got bagged and
put away.

the rings, in their turn, were opened up
into long slices and split down their
middles, the long way, with a steak knife.
finally (thus far), diced fine and stirred
into a chicken salad (along with some
carrots, also chopped fine, and, obviously,
some chicken—one big american breast).
add mayo to taste; mix; serve. (serving
suggestion: ritz crackers. we’ve got
lettuce & tomato, though, so actual sand-
wiches aren’t out of the question.)

### mel bay upside down: 7 major chords

ABCDEFG
(seven chords seven ways, part mercury)

ten for maria.

1 – There isn’t enough user-generated content or “making your own math artifacts.”

equations, most likely, first.

but wait. zero-th.
by-hand copies of the *symbols*
for the material at hand.
“the student learns essentially
nothing until the student’s
pencil makes marks on the page”
is a pretty good first approximation
a lot of the time… or anyhow,
i’m far from the only teacher
given to *saying* stuff like this.

i’ve got plenty to say, too, *about*
this but i’m hoping for a list of ten
in under 2^12 characters (for a little
longer; i’ve begun to despair already
at least a little though if you want
to know the truth).
“unions” should look different from “u” ‘s
as an example more or less at random.

*our medium is handwriting.*

first-and-a-half.
out-loud discussion of and…

second.
those equations. written
at leisure without the
instructor (or fellow student).

third.
similar or exact versions of such equations,
repeated, or, much better of course,
improvised, in a “public” setting
with small or, slightly better i
suppose, large *groups* of fellow
students. oral presentation of
the sentences themselves is not
only okay here but much to be
preferred (the board should not
be littered with sentences).
the “correctness” of the sentences
should nonetheless be at issue
throughout the presentation.
said “correctness” is to refer
explicitly to “code”…
utilizing (hey! ed jargon!)
the symbols from our step zero.

it does not escape my attention
that the “artifacts” created by
the student presentations i here
imagine are scribbles of chalk
on a board, soon erased. so be it.

leaving some out…

sixth
yick, computer code.

seventh
student-designed exercises,
exam templates, lesson plans…

eighth
songs and other verse, games,
comics and other graphics,
something to astonish even me.

ninth
blogs.

tenth
fanzines.

Sue VanHattumMarch 7, 2010 at 7:08 PM

Owen wrote:
>”the student learns essentially
nothing until the student’s
pencil makes marks on the page”

Maybe for higher math, but not at all for young kids. The mathematical issues they’re working on don’t usually require pen(cil) and paper.

My son is thinking so much about what I’d call place value these days. “60 and 60 is 120, right?” “Yep.” No writing – at home, anyway. Lots of mathematical thought.

I’ve long been puzzled by your emphasis on “the code”. Maybe someday I’ll get it… :^)

AnonymousMarch 8, 2010 at 6:57 PM
My son is thinking so much about what I’d call place value these days. “60 and 60 is 120, right?” “Yep.” No writing – at home, anyway. Lots of mathematical thought.

I’ve long been puzzled by your emphasis on “the code”. Maybe someday I’ll get it… :^)
—sue v.

maybe today!

the “places” of “place value” are
places *in* certain symbol strings!
it sure doesn’t matter that you
*speak* of such strings without
having actual *written* code
in front of your actual eyes…
that’s not what i’m always

60+60=120
presumably gets its interest
from 6+6=12,
together with, right,
the “place value” concept…
*as it manifests in base ten*.

now of course you and your kid
don’t have to have spoken of
bases-other-than-ten for
the essential *role* of “ten”
in discussions of place value
to have become quite clear
all around.

some kid of the same age
if i were lucky enough to
know any…
and i’d sure enough expect
(maybe with a *little*
stack-the-deck prompting
from me) pretty soon to
role of *zero* (in, again,
certain symbol-strings).

and when our conversations
*without* written work begin
to break down… and if we
still *care*… why then,
we’ll break out some *pencils*
and take a look:
“what do you mean, *precisely*?”.

we’ve been talking about code all along.

tangent.
calculating with numbers
is the very *model* of
3*4=13
is just flat-out wrong.
and this is our greatest strength.

in principle, anything worth
in a math class should have
the *same* character:
if we could only find it.

in order to have this happen,
we have to agree on things.
we *can’t* agree… and be
*sure* we agree… and be *right*…
without certain so-called “rigorous
definitions”: marks on paper
(generally; otherwise
*verbatim verbal formulas*
memorized syllable-for-syllable
[mostly… i don’t seek a
“rigorous” definition of “rigor”…
“one is *this* many”
and its ilk (so-called “ostensive
definitions”) are all the rigor
we can *get* sometimes]).

generally the “rigor” one speaks of
is… i think… pretty *close* to the
being-able-to-calculate-it-out-like-a-computer
thing i spoke of (with reference to
elementary arithmetic) a moment ago.
and this comes from “code”.

again. our power in mathematics
comes to an amazing extent from
being-able-in-principle to emulate
some doesn’t-know-anything-*but*-code
*machine*.

now i’m as much of a luddite as the next
guy, if the next guy figures the wrong turn
was somewhere around “domesticated animals”.
but one *glaring* benefit of computers
in math ed is that students will work
for *hours* on getting code letter-perfect
(if they know no human being can see
their failures happening), that wouldn’t put
in five *minutes* of homework on paper
without getting so frantic about each
“move” that they fall apart before even
getting started. it’s that “interactivity”.
this used to break my heart but it’s true.

if schools were for clarity,
command-line programming
it’s much *easier* than almost
any other thing you can do
with a computer (which is why
it emerged much *earlier*
than the hugely-user-unfriendly
[from a “code” point of view]
*graphical* interfaces that
erased it from the national
consciousness in around 1984).

(somebody mention “logo”.)

math *is* hard.
but it’s much easier than anything else.
because we’ve got *all* the certainty.
(programming on this model
is of course a subset of math).

ot

### borromean guitars

three newish guitar-stands arranged in such a way
that any two will fall down without the third.
and madeline’s “three women” statue, having a
similar property. blessings from our happy home
to yours if you’ve got one; double blessings if
you’re doing without. happy “black friday”.

(blogpost of 05/03/014).
w’edia.
flickr shot.

### the orange blend

exercise: draw the other six lines.
hint.

about time these guys got some names.
ROY (this one) is obvious.

let’s see.
green blend.
bgy byg gby gyb ybg ygb;
GABY, then.

secondaries.
gop gpo ogp opg pgo pog;
PEGGIE-OH, then, maybe.

the orange blur.
bmo, bom, mbo, mob, obm, omb.
JIMBO comes to mind.

submissions welcome. yes, it’s the great
name-the-lines contest of ought-fifteen!

i’ll be forced for the sake of my own peace
of mind to type out the rest of the exercise
and make up three more names. “the purple
blur” isn’t much of a *name* for a line.
more like a *secret identity*.

### left-handed G chord

formed by my right hand. i got a new shipment
of strings today… thanks, madeline!… so i
(finally!) tuned up one of my guitars “lefty”.

if i try to *play* it that way, it sounds like
day one. but if i play it *right* handed, i
can make it sound like music. the chords and
the dynamics are different, though. so, cool
trick, it sounds like somebody else playing
(while still sounding a lot like me).

or maybe that’s the cough syrup talking. whee!

### the story starts with my decline and fall

Introduction
(Lines 1—40)

God (as I choose to call my higher power)
Grant me an audience for half an hour
And I will, if I can, do all the rest.
My subject is the story I know best;
I mean my own. It starts in a motel,
The night of my divorce. I felt like Hell.

Think of a pilot, learning how to fly,
Who, though he should know better, flies too high,
Then falls in the Atlantic and is drowned.
His body and the plane are never found.
There’s something like our marriage in that story,
The way it shoots to misery from glory.
The similarity might not be strong,
But, as to suffering, I’m never wrong:
Divorce is brutal. Trust me when I say
I’d rather be that pilot any day.

Lisa, in a voice that tore my heart,
Had told me, “From now on, we’ll live apart.
The First Street house is mine. I want my space.”
And so, a stranger in my own home town,
I left my room to have a look around.

Across the street, a Big Red liquor store
And Waffle House. A porno shop next door.
“The restaurant then. For now, I’ll do what’s right.
I’ve got no strength for sex and drugs tonight.”
The waitress, call her Ruby, perked me up.
I never saw the bottom of my cup.
A refill and a smile, and off she’d glide;
She wore her sixty years with grace and pride.

“Why look upon myself and curse my fate:
I couldn’t stand to only serve and wait.
I’ll bet that woman’s life is harder still
Than mine, by far. And, if I only will,
I could throw all my misery away
And love my life the way it is today!”

If that was true—and I don’t think it was—
I proved myself an awful fool, because
For years I didn’t love my life at all.
The story starts with my decline and fall.

• ## (Partial) Contents Page

Vlorbik On Math Ed ('07—'09)
(a good place to start!)