Vlorbik's Diner

son of owen's cooking show

Archive for the ‘Exercises’ Category

the raw and the cooked

Posted by vlorbik on March 23, 2014

Photo on 2014-03-23 at 11.53

so here’s a roasted chicken, mostly boned.
i’ve been eating the bits that didn’t slide
off the bone easily. the wings are something
of a pain, so i left ’em for later.

now to put everything away and *clean up*!
whee!

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Posted in Chicken, Exercises | Leave a Comment »

this morning’s dishes

Posted by vlorbik on March 5, 2014


dragged and dropped into flickr.
*that’s* never worked before.

Posted in DAADD, Exercises | Leave a Comment »

not dark yet

Posted by vlorbik on June 3, 2011

you can’t *be* wise and famous.

anyway, not for long:
they kill you for it.
look at the god-damn record.
look at jesus. look at socrates.
look at john fucking lennon.
look at malcolm, look at gandhi,
look at martin fucking luther king.

it’s everywhere and everybody knows it.
speak truth to power all you want but power
talks back with violence not just words
and everybody’ll get behind the boss
without hesitation and swear it was
entirely your own god-damn fault…
what the hell did you expect?

good thing you’ll be dead by then.
the consolations of philosophy.

and now please let me say here as clearly
as i know how that *’i’m* not claiming
to have acquired any kind of wisdom…
that would be the height of foolishness.

i’m just sayin.

Posted in Exercises, Religion | 1 Comment »

it said there’d be some thunder at the well

Posted by vlorbik on May 20, 2010


i drew the i ching yesterday.
there are sixty-four “hexagrams” here.
each one consists of six “lines”;
each “line” is a yin (“broken”)
or a yang (“unbroken”). one counts
from bottom to top so that for example
_ _
_ _
_ _
___
___
___
has yang lines in its first,
second and third positions
(the “lower trigram”).

the arrangement of the hexagrams here
is my own; i copied it from a similar
arrangement of sixty-four objects
having exactly the same
“six things, with each ‘thing’
taking one of *two* possible
values” form… namely an
arrangement of “subgraphs”
of a certain “graph” having
six “edges”. (i don’t know why
i’m getting all “quotemark happy”
here exactly… maybe it’s the
*opposite* of “scarequotes” i’m
aiming at: by marking technical
terms in this way, i mean to indicate
that their technical sense is
here *recognized as such* [but
without actually wishing to
bother ourselves with their
technical *meanings*]…
“don’t worry about this;
it’s from another part of
a different course [but
worth mentioning here
just the same]”. oh never
mind.)

so much of the “work” of this “exercise”
i set myself will have been, if we want
to think of it this way, *calculation*:
translating each *graph* of the original
drawing i was working from into a
*number* (using binary arithmetic;
the zine i took the original drawing
from [K_4, MathEdZine #0.8] is
essentially *about* binary arithmetic)
and then translating this number
(from 0 to 63… starting at zero
makes things much easier) into the
“hexagram” notation.

of course i could also have just learned
directly which lines on the hexes
corresponds to which line of the graphs,
and spared myself lots of calculation
(by paying the price of having to briefly
memorize subsets of {1,2,3,4,5,6}
a whole bunch of times instead…
or maybe “wordstrings” like
“yes-no-no-yes-yes-no”).
but my way looked easier for me.

then i went ahead and drew ’em all
over again on lined paper. and in
a much more natural order: eight
rows of eight, 0–63. the book i then
turned to *also* had eight rows
of eight but in a different order…
and so i went ahead and looked
up all the correspondences so i’d
have the names… and the numbers
(in the “standard” order for the
writings-of-the-sages associated
to the various hexagrams). and
here i used a “mixed” strategy…
the trigrams have chinese *names*
so as i checked back and forth
i’d subvocalize along the lines
“chi’en over chen” more often
than “seven over zero” since
(as it turns out) i can recognize
even nonsense syllables (in
latin letters) with
less effort than it takes to
“read” a trigram as a number
from 0 to 7.

and i got to thinking about
drilling and killing. now
obviously if i were doing
this kind of thing all day
on a deadline or something,
i’d want a quick-and-easy
*routine* and soon would
arrive at one, too. i’d
slip into a bit of a zone
where i’d devote most of
my conscious effort to
controlling the movements
of my hand to make better
lines and let my inner robot
do the names-and-numbers bits.
i even did some of this yesterday.

but only for a little bit at a time;
this takes *concentration* and
is, pretty specifically, no fun. no.
i haven’t *got* a deadline (alas)
and sure as heck ain’t gonna
do this all day, anyhow not
yesterday i wasn’t. not a bit.
i was also looking up some of
the hexagrams… the one i
typed out a moment ago shows
the “female” sign over the “male”
and denotes “peace” (and when
the positions are reversed,
“stagnation”)… for example,
and thinking back over certain
classroom work in my past and
fantasizing forward about working
with classes of students again.
it’ll never replace DEAD BEEF
but i’d love to give an “i ching
as intro to binary logic” lecture
before they finally haul me away.
to everything there is a season.
a time to gather stones for casting.

Posted in Exercises, Handwriting, Zines | 3 Comments »

the chinese room

Posted by vlorbik on November 18, 2009

this moving-around-of-letters
activity of the past couple of
rambles is, or could and (i hope
someday to convince *some*body)
should be, as foundational
in the study of mathematics as
elementary arithmetic (+, -, *, 1/n) or
compass-and-straightedge constructions.

“trust the code” shall be
the whole of the law whenever
*i* set up as math dictator.

this means symbol-by-symbol
every-keystroke-perfect *code*
is, first of all our *subject matter*
when we’re studying algebra
every bit as much as it is for its
johnny-come-lately derivative
“computer programming” (whatever
the proper euphemism is these days).

enforcing this level of attention to
detail *without* a computer turns out
to be quite difficult. one of the great
frustrations of my life is that *with*
a computer you can pretty much get
*any*body to perform rituals of
*arbitrary* complexity as long
as no actual *reasoning* is involved
just by convincing them that there’s
a paying job in it for them somewhere
if only way down the line behind all
those other poor desparate bastards
that already graduated and have nothing
better to do now but spy on *them*.

but computers are are hard.
to pay for. to understand.
and altogether *impossible*
to maintain for long.

whereas the game
is “simple things first”.
(another fine game is
“don’t let machines
tell you how to live”.
this one’s *much* harder.)

*you can do this*.
what’s more, having done it…
and had the right *conversations*…
you’ll be darn *sure* you can.
and when anybody else…
human or robot overlord
or one of the many blends
emerging all around us daily…
has it *wrong*, you’ll *know*.

here is power.
*that*’s what the simplicity is for.

let me go ahead here and admit that
there’s plenty of good math you can do
*without* this almost-machine-code
letter-by-letter detail-oriented
okay-i-admit-even-somewhat-obsessive
*algebra* stuff.

i was an algebra *major*. so i’m biased.
anyway, logicians are worse. but no. really.
this is the stuff that’ll make you *good*.

story-of-the-blog-so-far stuff.
last winter when i was blogging
about my math148 precalculus class
(as i think of it; three classes really),
i devoted quite a bit of attention to
finding and implementing the “right”
notation
for, what was one of
the big themes of the course,
transformations of the xy-plane.

here as maybe nowhere else
one has an opportunity to *use*
the “points as ordered pairs”
point-of-view so sloppily
developed throughout math101.

because the centerpiece
in everybody *else’s* imagination
seems to be the xy-plane
itself… the admittedly epoch-making
observation that by laying down
co-ordinates over a euclidean plane
you get a cartesian plane and all
of a sudden equations have *pictures*.

ooo. aaah.

and these pictures are all well and good
and the basis for the scientific revolution
whether *i* like it or not and all that.
but.

the kids don’t get it. and won’t
until they believe they can. and
as to “functions as sets of ordered pairs”,
the examples given typically…
graphs of polynomials and whatnot…
have manymany scary confusing aspects
already known by the audience to be
well beyond their comprehension.

so it’s… well… just *logic*
(not *rocket science*[!]): simple
things first. confused about why
some “transformation” (that doesn’t
even have a proper name, let
alone appropriate symbol)
causes “it” (the graph of…
something… but “it” isn’t usually
any one thing in these discussions)
to *change* in some particular way?
well, how about a bunch of highfalutin
*technical terms* that you know very
well *you* don’t know (and have no
very good reason to be sure about
the teacher)? that’ll sure be useful.
(depending on your goals.)

confused about A, B, and C?
*where*, precisely?
how did *yours* look?

in the *spirit* of “keep it simple”
i now propose to ramble some more
about the “simplest interesting case”
of permuting the elements of a set:
the case of *three* elements.

ABC ACB BAC BCA CAB CBA
XYZ XZY YXZ YZX ZXY ZYX

here are two isomorphic “strings”.

“isomorphic” means “having the same form”.
that the strings… lists of symbols…
*do* have the same form
in some sense is probably obvious to
any reader. heck, six groups of three.
but more than this.

the set isomorphism
A \rightarrow X
B \rightarrow Y
C \rightarrow Z
“induces” (what i’m here calling)
an isomorphism of lists:
replacing each left-hand object
wherever it appears in
our first string with the
corresponding right-hand object
produces the second string.

note that “isomorphism of sets”
is (and deserves to be) standard language
for the kind of one-to-one (and “onto”)
function we’ve displayed here.
two (finite) sets “are isomorphic”
as soon as they have the same number
of elements.

but there will be many different
isomorphisms between any
pair of isomorphic sets.
indeed… theorem 1!… there’ll
be n! (en-factorial) of ’em
between any pair of n-element
sets. (you see this, right?…
remember that factorials count
permutations…)

A \rightarrow X
B \rightarrow Y
C \rightarrow Z

A \rightarrow X
B \rightarrow Z
C \rightarrow Y

A \rightarrow Y
B \rightarrow X
C \rightarrow Z

A \rightarrow Y
B \rightarrow Z
C \rightarrow X

A \rightarrow Z
B \rightarrow X
C \rightarrow Y

A \rightarrow Z
B \rightarrow Y
C \rightarrow X
.

now. in the spirit of the introductory
ramble
from a couple weeks back.

two exercises are isomorphic
when one can be worked out from the
solution of the other simply by
replacing “letters”.

consider the six isomorphisms
from {A, B, C} to {X, Y, Z}
(as shown above).

for a low pass, write out all six
isomorphisms from {a, b, c} to {x, y, z}.

for a passing grade, write out all
six isomorphisms from {P,D,Q} to {E,I,O}.
let (the particular isomorphism)
P\rightarrow E, D \rightarrow I, Q \rightarrow O
be denoted by “elbowgrease”.
write out the result of applying
elbowgrease to the string PDPDQ.

for a high pass write out the
iso’s from {1,2,3,4} to itself.
what happens if you “apply”
an isomorphism to the result
of the application-of-an-iso’ism?

for a pass with distinction learn
“cycle” notation and how to calculate
with isomorphisms-of-sets considered
as members of the so-called
symmetric group on three elements.

essay question for advanced credit.
we’ve “gone meta” twice in “lifting”
correspondences of sets first
to what we called isomorphisms
of strings, and then to
isomporphisms of exercises.

one could continue to “lift” the
concept to even “higher-level”
groups of data… perhaps introducing
some metaphor along the way to
replace strict symbol-for-symbol
sustitution.

find a pair of textbooks covering
transformations of the plane.
display an “isomorphism” between
the bone-headed wrong ways the
relevant sections of your chosen
texts leave out crucial concepts and
fudge important details.

develop a theory of how this state
of affairs came about. for the
love of god and the gratitude
of generations still to come
do something to change it.

Posted in Exercises, Permutations, Rambles, VME | 4 Comments »

never get outta the boat

Posted by vlorbik on November 16, 2009

i first encountered the factorial function
at about age ten. in fact, i recently
acquired a copy of the very book i
leaned about factorials from.
i blogged about it here,
mentioning the (classic!) problem—”four fours”—
i learned about ’em from.
the game is, using only “standard” operations
like powering and rooting and multiplying
and subtracting and whatnot…
and *exactly* four 4’s…
and no other numerals..
to write representations of
small natural numbers.

e.g.
1 = 44/44
2 = 4*4/(4+4)
3 = (4+4+4)/4
4 = 4*4^(4-4)
and so on. a great game for kids.
(you can see it had something of
an influence on *me*…)

kids of all ages lest that go without saying.

anyhow, sooner or later you’ll get stuck.
two things happen. you give up or you
get mad and start looking more carefully.
okay, three. you can *cheat* and allow
“new” symbols… like factorial (!).

the factorial function “counts permutations”.
in the example that should be given
every time the subject come up
until the student indicates that
they’re already doing it “in their
head” every time it come up already
and you can stop again (already):
the permutations of the elements
of {A, L, T} are
ALT, ATL, LAT, LTA, TAL, and TLA.

*any* three letters can be used of course;
the permutations of {X, Y, Z}
(i’m being sloppy) are
XYZ, XZY, YXZ, YZX, ZXY, and YXZ.

the point… *a* point anyway…
is that a set of *three* letters
will always have *six* permutations.

one easily sees that this is “because”
6 = 3*2*1. likewise for {A,B,C,D}
one has 4*3*2*1 permutations.

notation:
4! = 4*3*2*1 = 24
3! = 3*2*1 = 6
2! = 2*1 = 2
1! = 1 = 1

the “factorial of” a (natural) number…
n, say…
is denoted by “postfixing”
(like some… trouble aplenty…
adjective-postpositive)
the symbol “!”
(i pronounce this “bang”
usually in class…
“exclamation point”
has five times the
necessary number
of syllables…).

we now introduce the weird-looking
but not-so-weird-if-you-just-look-closer
convention that
0! = 1
(there’s one way, from anyway
one point of view to “list”
the “elements of” the empty set
[i.e., the set of *no* elements…
the “zero” case of “how many elements?”]:
namely the empty *list*).

we can now (though i consider it highly
optional) define the factorial function
!:N—>N
by

!(0) = 1
!(n) = n*!(n-1) [ n\=0],

a “recursive” definition.
these amuse prepared minds
and horrify the rest.
best not try it on the class as a whole
unless they’ve got some “math maturity”.

really n-factorial is spelled “n!”.
i used !(n)
to be perfectly explicit
about the fact that we
*are* considering a
function on N
(the set of natural numbers
[including zero; rant still
to come unless it’s around
here somewhere.

the point is to know like your own middle name
that when you need to count orderings you’ll
*use* this thing (and to know when you *see* it
what the heck it is).

students that can’t write out all 120
permutations of {E,G,B,D,F} at this point,
and go on to the rest of the course anyway,
are *damaged* thereby
and indeed constitute damage to
their whole class and to society at large.

i don’t like this any better than anybody else.
but what i *really* don’t like is being the only
god-damn doctor of philosophy i know of
saying so on the record at this level of detail.

your philosophy is sick and i’m here to fix it.
oh, cursed spite.

you don’t have to be all tough-guy
this-is-college-kid about it… never mind
the if-you-were-serious-you’d-already-*know*
game that wrecks most math classes
before they even get started…
actually, starry-eyed idealist that i am,
i believe that material much easier
than tying your fucking shoes
can probably be taught even
to the dimmest kid admitted
to your college if that’s what
you actually want to fucking do.

i could be wrong of course.

Posted in Exercises, Permutations, Rambles, Rants | 3 Comments »