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!

Posted by vlorbik on March 23, 2014

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 »

Posted by vlorbik on March 5, 2014

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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 »

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 »

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**

“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…)

.

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)

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 »

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 »