24

This is probably not what you're thinking.

A card game for young children has the following cards:

 n1        n2 



      24




 n3        n4 

The objective of the game is to use any of the four arithmetic operations to combine the numbers into 24.

There are 42 ways to do this with the numbers 2, 3, 6 and 8. If we don't use any operator more than once, there are only 24 ways.

That is wonderful poetry. 2, 3, 2×3 and 2³ yield palindromes under the operations of arithmetic and combinatorics.

This datum was brought to you by programming with narrowing in Curry. I'll post my program in a week, after the course has ended and finals are submitted.

In other news, I plan to visit Jim some weekend soon and do some Abstract Algebra proofs, with much enjoyment of Led Zeppelin. It'll be a grand old time.

This is what I dreamed college would be like.

Damian Learns Sine-Prime

me: Well, Damian just went through an excruciating "by...inspection!" proof that sin'(x) = cos(x)
Jen Clark: lovely
me: Which is why this terminal has been quiet for an hour
me: He asked me "Alex, can I ask you a favor..." "Yes." "Can you order some sheet music-" "What's the derivative of x^2?" "..." "No."
Jen Clark: hehe
me: So my dad came along and ...instigated the fun. That's what drunk people do.
me: He had some wine and decided to spoil polynomial derivatives for Damian
me: So I then asked Damian what sine's derivative was.
me: ...60 minutes later...
me: Hallo.
Jen Clark: heh
me: Oh, in his defense, off the bat he said "Cosine," but I could see the gaping hole in his ass where that came from.
me: And I really didn't need to see that.

More math humor

Care of the Pouwster:

theandyman87: she is just on an emo trip
theandyman87: ...again
Phoenix Team I: in order for one to begin you have to infer that the other ended...
Phoenix Team I: she's a continuous function alright
theandyman87: with enough extrema to supply the hundreds of tangents she goes off on
Phoenix Team I: this series is most definitely not convergent either

Growing from a Broad Root

I'm enjoying Formal Logic right now. It really is like a game. The rules may be boring, the results almost painful to read — certainly painful to process entirely in the head — and because there is specifically NO meaning yet, the fun may be lost to some people.

But I'm a math major. Meaning is for those too grounded in reality to think about nothing productive productively.

This meaningless tree is a parsing of a word in a first-order language, whose vocabulary is the set of logic symbols, and functions R (2-ary) and P (1-ary). All nodes where "gx" can be substituted for "y" (that is, doesn't throw an x under a universal (∀) or existential (∀) quantifier) are highlighted yellow. The first node that disallows substitution of "gx" for "y" is highlighted orange.

            y
            |
   x   y   Py   x   z
    \ /     |    \ /
    Rxy    ¬Py   Rxz
     |      \    /
  ∀xRxy   (¬Py∨Rxz)     x
     \        /          |
(∀xRxy∧(¬Py∨Rxz))      Px    y   x
         |              |     \  /
∃x(∀xRxy∧(¬Py∨Rxz))   ∀xPx   Ryx
         |                \    /
¬∃x(∀xRxy∧(¬Py∨Rxz))   (∀xPx∨Ryx)
         \                  /
(¬∃x(∀xRxy∧(¬Py∨Rxz))→(∀xPx∨Ryx))

I wrote this for three reasons:

1. I reread some old entries and stumbled across an entry where I drove Ali to attempt to take her life through her forehead and monitor. That did my heart some good, so I thought, "Hey, why not spread the havoc once more?"
2. I really like the look of that parse tree. Gobbledygook at the bottom, lots of it, but just variables as leaves at the top. The pipes and slashes representing branches actually make it look tree-like, too, so it had an aesthetic appeal...probably my own aesthetics, but hey: More havoc if otherwise.
3. The markup for that tree is absolutely abhorrable. Then again, I know of no decent way to code ASCII art with colors. Blizzard Entertainment did it once by coloring a screen of random 0's and 1's to resemble their logo; quite neat-looking.

This entry was also written because my Math Journal archives have been kinda sitting there and growing stale. More shall ensue.

Insolvability of the Quintic

Here's a little excerpt from my final exam that I submitted today at 14:40:

"The bottom of page 93 (section 9: "Orbits, Cycles and the Alternating Groups") reads:

"'It can be shown that there are no formulas involving just radicals for solution of polynomial equations of degree n for n ≥ 5. This fact is actually due to the structure of A_n, surprising as that may seem!'

"Let the record state that 'My socks have been blown off.' I've recognize a parallel between Fraleigh's favorite phrase, 'Never underestimate a theorem that counts something,' and a phrase I'll have to include with a publication at some point: 'Never underestimate the statement "It can be seen that..."'. I underestimated that statement with the Insolvability of the Quintic; lo and behold, 382 pages later, it was seen."

What makes me happy about the proof I wrote, which is 3 solid pages, is that it can generalize to any integer greater than 4. This means I've also proved insolvability of the Quintic, Sextic(?), Septic, and n-tic polynomials. If I had no love for this subject, I would've turned in that exam answering only the T/F questions.

If I really had no love for the subject, I would've answered the last question thusly:
10.) T/F: I think Galois Theory is easy. "Bullshit, I say. See pages 430-435. Scratch head. Repeat ad cruento. Whether the solution generalizes to Sextics, Octics or whatever, all those polynomials are just septic to me."
Katherine's response: "Fuck no." She's awesome. She answered #10 when she got to Sean's office to turn the exam in; Sean misread the "n" for a "yo" the first time he looked at her answer. "Well, thanks, Katherine, you just wrote your evaluation." Sean's awesome too.

There's gonna be another drunken math party next Tuesday. I'll bring The Impostors; I'll watch the class load up on beer; and then I'll probably screw around on Scott's guitar some more. He's really nice when he's drunk; he just hits extra strings and thus plays out of tune, but it's all good.

If we come up with any dirty math limericks, or maybe just dirty limericks, I'll keep you guys posted.

Spring Quarter, Thus Far

A note to readers: I sorta went memoir-caliber with this entry's length. It's sorta 3 pages in Word; I'm sorta sorting the length and lengthy word choice. It'll take a sorta lengthy time to read this. All right, I don't think I can bear to recycle words anymore. Oh, there is some math speak, in case any of you are allergic to math.

Well, it's Week 4 of Spring [quarter]. Gone are the days of Real Analysis and Topology...but Abstract Algebra marches on, and Multivariable Calculus reared its ugly head, comin' back for mohr. Actually, I take that back; MVC's ugliness is being thankfully reduced with the Tuesday class...

Tuesday: Differential Forms

For those of you who're Calculus students, do you recall (1) That integration always was done with one parameter, usually x or t, and (2) f`(x) = dx/dt "Implicitly implied" dx = f`(x) dt ? Yeah, those were good times...except when some fugnut threw an arctan in a radical into the integrand. Fugnut. Point being, with integrals there was usually one dsomething parameter.

S_a^b f(t) dt

Some of you who've had a glimpse of Multivariable Calculus may recall seeing something like this...(just pretend that this big S is an integral S, as I haven't learned how to express mathspeak here yet):

S_CP(x,y) dx + Q(x,y) dy ,

where that is math-grammatically correct. Not one, but -two- differentials under an integral. What's under that integral is essentially what a Differential Form is -- in this case, a 1-form, since for all of the summands there's only one dsomething term per summand. A 2-form would have something like dxdy, a 3-form dxdydz, and so forth to n-forms. Forms are what we're studying; we may move on to the more general Form, Tensors, in two weeks; after all, the class was originally called Tensor Analysis. It's, uh, nameless now.

This class has too many abuses of the Big Bad Sigma Σ notation, including obscene-to-read things in the book like ΣΣ. The subscripts are murder; it doesn't help that the book is so vaguely written that we can't tell if the author's laying down axioms, proving schtuff, or sometimes writing parametric equations. Reading's bad times for Tuesdays; luckily the lecture covers almost everything we need.

Thursdays: Complex Analysis
Think if you had the Reals...then add i to the set. Voilà, the Complex Plane, but everyone with an algebra background knows that. The Squa-yah w00t of negative uno gives the linearly independent iy axis, where multplication works FILO like nothin' else.

This class so far is making Green's Theorem earn its keep. The next couple weeks are going to be a buttload of path integrals in the Complex Plane, usually because we'll be deleting points here and there...see, the big deal of calculus in the complex plane is where a function is differentiable, and that oftentimes doesn't include the origin due to division by big fat love 0. So, we puncture the plane and just consider the derivative everywhere but those naughty places. Yes, "Puncture" is the [pretty] standard terminology; in describing sets, Sean even said pancake once. So, in my notes, I work in sets like "The punctured pancake of radius 1..."

The Professor: Sean

This dude's lecture style really brings the class down to earth. Don (prof. from last quarter) dispelled the illusion some people have going into college, that professors Know Everything. I don't mean that to be mean to Don, but he was a physicist -- heck, his Ph. D. was in plant physiology -- so at times it was a little obvious that pure math wasn't necessarily his field.

But Sean's a grad. student, out of Colorado State U. He tells us stories of how [ahem] well he did in Don's classes, like how Multivariable Calc just passed right over him the first time. Now he's a regular teaching assistant at CO State, teaching first-year and multivariable calc. He seems to really be having fun out here, since it's a small class...

"...Now, here's something that would annoy me. I'd be correcting homework, or even worse grading tests, and I would see that some students wrote the gradient's result as a scalar. Then I'd go to class the next day; I'd stomp my feet, I'd get pissed off, and tell the students to reference the Big Red Ink notes that the gradient is a vector. I've been doing that for three years, to engineering and math students of all calibers. Now, according to page 33 of this book (the General Relativity book, Tensors chapter), this picture is exactly why the gradient is not a vector." At this point, he hurled the book across the front table so it slid as frictionally-high as possible. "Well, Damn It. Way to make me feel like an ASS, Mr. Relativity."

Lectures from Sean are good times; it doesn't take too much goading to get him off on some weird, weird tangent, like one day when I asked him what some squiggly Greek letter was in this diagram that decided to use half the Greek alphabet. "See, Sean," I said, "I know that one of those squiggermajigs is zeta, but the other is...uh...zeta with a loop?"

"Oh, that (ξ) is xi. Oh, man, you know how important xi is? It's one of the most prized words in Scrabble; if you have an X, and no clue what to do with it, and there's an "IT" somewhere, and there's a triple word score box by it, you can set up one helluva good combo...

X I
I T

"...and you're right on top of everyone, man. Xi and xu are the most prized Scrabble words; they're top-notch in the tournament players' arsenals."

And there's one last note on how down-to-Earth Sean's professin' style is. He told us that while living in WA, he's staying with his grandma; unfortunately, she doesn't go out and buy food too often...if ever. So, he cut lecture short one day: "...Heh, I probably shouldn't be telling you guys this, but there isn't really food where I'm staying right now. ...So, I'm going home to take gramma shopping." He's professin' awesomeness.

Except on Fridays: Abstract Algebra, 3rd quarter -- Galois Theory and the Insolubility of the Quintic

Abstract Algebra, with its rings, groups, fields, homo- iso- homeo- and endo-morphisms, has the basic underlying goal of finding 0's of polynomial equations. If you thought that seemed hard in High School Algebra, you ain't seen nothin' yet, cause you never had to worry about solving some of those bad boys.

The class voted near the end of winter to continue Abstract Algebra, instead of start Abstract Linear Algebra. Sean was hoping to do ALA instead, but alas; well, it's sorta trouble for him, because he doesn't teach Abstract Algebra. In fact, he's an Analyst, the Foundations-of-Stuff type; the only Abstract Algebra he has done was in an undergrad and grad course, and those were a couple years ago...and he's jumpin' into the course in the hard part. Fridays are a challenge for everyone.

He joked about how he's going to give us our take-home final. "Well, let's see, I want to give you a test, but what about the morals of the professor understanding what's on the Final Exam he's about to dole out? And how the hell am I going to grade it? I'll probably just look at the answer key from the text...'Hmm, this solution looks different, must be wrong.' 'You should really consider taking this course over.'" I'm glad he has a good sense of humor.

Apparently he's in a metal band, too, and he's teaching himself banjo. Cool guy. I'd better get back to the work I'm doing for his class tomorrow, though. Writing about the class was a nice escape from actually doing stuff for the class; and there's even a relation. Let the good times roll...roll on to 03:00. w00t.

Bad Dog, No Biscuits

Cowboy Bebop: OST 1
Track 4 -- Bad Dog, No Biscuits

I used to not like that track. Too much extraneous noise, electric squealing chaos...just didn't appeal to my decent-chord-loving self of a summer or two ago.

But, things change. That started blasting in my car on the way back from The Topology Exam and I could do no better than crank it higher -- I liked the chaos now. It's so fuckin' full of life, I just had to have as much of its energy as I could get. Because I was inVIGORATED by the Topology test -- I finished in 3.5 hours instead of having to tough out all 4. Magnificent.

When Don handed out the exam at 2, I scanned through it, and asked Don to step outside the room for a quick sec. In the hall, I buckled my knees slightly and threw my head back, quietly quietly shouting "The exam's not that bad!" I then stood up again. "Ahem. Thanks, I just needed to say that to someone." We both had a good chuckle.

I had some fun with Loop Spaces. See, in π₀(Ω), the elements of that set are equivalences classes, labeled as points in a space. For each point, there is an infinitude of paths that start at that point, go off and do some crazy shit, then come back and end, the most basic being something like this:

Well, that don't look too fun, 'til ya spice it up a bit with extra-terries.

And I drew the Rainbow Topology for my optional not-asked-on-exam discussion, mainly because Katherine thought something called "The Rainbow" anything would never make it onto an exam. But, Don did draw it on the board in lecture three weeks in a row...and he did use colored chalk...(Unfortunately, the rainbow topology's only interesting to people who understand compact-open and point-open topologies, so it's just another lame math joke. Nothing compared to my dirty calculus pick-up line, though.)

Another way today went well: Yesterday, I arrived at Evergreen when it was sunny out, and left when it was dark. I thought I had done that for the last time back when it was still hardcore dark-at-rush-hour winter. I didn't enjoy doing it again; but today I left when it was still light. Yay.

All right, my happy-spurt has worn off. I'm gonna watch some guilt-free Star Trek TNG. Mmm...Data...

End of Real-ity

2004, March 4th
11:00
Most everyone in Real Analysis got to the home room about 2 hours early on Final Day. There was an assload of things to remember for the exam, which Don promised to be "shorter and harder, as opposed to longer and easier." With topological stuff, a dozen convergence tests, and finite integration to remember, we were all a bit frazzled.

At some point, I said "You know what'd suck?"
Scott replied, "What's that?"
"It'd suck if we had to prove the Fundamental Theorem again."
"Ooooh, yeah...man, I hope that's not on there, too. That proof is hard...and we didn't do too much with Riemann summating."

12:30
Eventually, we all got tired of reviewing and took a break. For one wonderful half-hour, we stepped outside into some wonderful [and chilly] sunshine. Someone asked Sean if his banana was vegan...many pleas for help and air-popping sounds ensued; Katherine said the answer to #1 on the exam should be "Inflatable banana. What? They're vegan!"

Classes are a lot of fun when there are only seven students...it's a much more friendly environment; feels close-knit, even.

13:00
Well, all good things must come to an end. We got back to the classroom, and Don arrived. He laid out some snacks for us since he's such an awesome professor -- some nuts, tortilla rounds, and blueberry cream cheese. The latter two make a surprisingly good snack. Then he handed out the exam.

I scanned through, since my habit is to let my unconscious mind start working on the problems before I get to them in real-time. Nothing looked too bad 'til the third page [of four].

Question 9:
"State and prove the fundamental theorem of calculus."

My mind:
"Fuh-huuuck."

15:00
I'd already finished 9 of the 12 problems. My beef was now with the remainder; the Fundamental Theorem, and a couple series-related doodads. Not much was comin' to me.

About a quarter of the cream cheese tub was gone; I grew a snacking habit and munched for a good chunk of the exam. Those rounds and cheese really were a good idea.

15:50
9 down, 3 to go.

In order to stay in my seat more, I took one chip and scooped a huge mound of cheese on top, and took other chips with me back to my seat. I noticed if I planted another chip into this mound, and the another on the opposite side, it looked like a butterfly.

3 to go.

I flittered my butterfly about; Scott behind me laughed.

16:30
3 to go.

Four butterflies sat next to me on the desk. Sean laughed mightily; I was building a fleet. Don came back into the room, and we made eye contact. I swooped my hand over my glorious four-strong army: "The test ain't going so well."

16:58
I asked Don for help on the Fundamental Theorem; I had the beginning, end, and most of the middle to the proof. He pointed out, after much pondering over what I had done, that my beginning was wrong.

17:54
2 to go. After five hours, there's no chance in hell I could even scrawl more with my pencil to get it done. I threw in the towel--er, exam, and left for Chamber Orchestra rehearsal.

Looking back in my notes, I realized my proof for the geometric series that I had copied from the board was totally and completely wrong; must ask Don about this. Well, on the plus side, 10 outta 12 ain't bad...and there'll always be the Butterflies.

Topology, Day the First

When one thinks topology, one thinks of either two things:

1.) Fun things like mapping functions onto "rubber fields" and deforming them into weird and wacky shapes; mobius strips; and the Klein bottle (one-sided 3D surface)

2.) Point-set topology, the college senior math course.

These are, unfortunately, mutually exclusive. Point-set topology is promised to be the most challenging quarter we will experience in Math Systems. Yay. The book we're reading requires Linear Algebra on accident. Eeyay. Some of the pages of our reading will chug along at the reading rate of 30 minutes per page. Eeeya....no, fuck.

Our other book, The Shape of Space, is somewhere at the eight-grade reading level. The topics in there are the fun parts of topology; there isn't -too- much theory to get in the way; and the material has barely any prerequisite material. It's open to everybody. And it references Flatland (Edward Abbot Abbot, "A Square"), like many high-level physics papers.

There's plenty of material in the book to play with...and because it's a book of intuition, some experiments require other materials than pen and paper. Don brought in Playdough for one exercise; he says some "Labs" may require Playdough to understand. There's also colored paper, scissors, and tape...

Some of you may be shaking your heads at the idea of a Topology class that involves Playdough and scissors. I'm personally reminded of Shigeru's quip about the...Gen. Ed. school at the U. of Boston ("Crayons, Glue and Scissors" as I recall the alternate title). But the experiments are at least noteworthy:

*Make a Mobius strip, fairly fat and long (take a strip of paper, twist it once and tape it into a loop; there's only one side). What do you think will happen if you cut the strip in half (along its side, not across it)?

A mobius strip cut in twain becomes a longer strip, with a couple twists; cut that in half, as your curiosity will undoutably lead you to do, and you will get two double-looped rings intertwined with each other, locked like a magician's steel ring pair; after the first cut of the original Mobius strip, the extended strip gained an extra twist, losing its Mobius properties.

What of in thirds? Make two slits in a Mobius strip, and then start cutting along one; you will eventually hit the other slit. Finish the cut, and you get a smaller Mobius strip, interlinked with a large, double-twisted strip like the 'twain cut.

If you're hardcore enough of a math geek, you'll find this fun and start writing theorems about 'twain cuts, and quarter cuts, and tr...tri...uh, 'train cuts? Yeah...I wasn't hardcore enough, because of another "Experiment" in Shape of Space I mentioned in the last entry.

The king's in the middle.

Torus rules for chess: Pieces can go in their normal directions, -including- wrapping around the world on all 4 sides. Also, pawns move one space in any cardinal direction, and can only take pieces diagonally. There's no mention if they get the two-space jump start.

Now that the rules are changed, the playing field is essentially leveled...so, anybody up for a round of chess?

Torus Boards

Do you guys remember the old-school arcade games where you could walk off of the East side of the screen and arrive at the West? I found out in my easy, easy, easy topology text that those kinds of screens/boards/surfaces are called torus-surfaces. And then it delved into some light theory that anyone here could read & understand, but enjoy is a different matter so I'll leave that be.

HOWEVER, the book did present the idea of playing some board games with Torus rules. Tic Tac Toe with Torus rules means the first person to move will generally win; Torus Chess has to be played with a completely different setup, since if classic chess got Torus rules, the game would be instant checkmate (the king could reach to the other side of the world, as it were, and bop the other king right over).

Torus Chess starts with only 9 pieces, and in the corners of the world, though I believe white is offset by one from its corner...I'll post a picture later, I've gotta get to math and potentially talk nerdy about chess for five hours. Toodles.

Math Systems--day The Last

This may sound perverse, but I've had a knack for perverse speech. Advanced Calculus had the easiest final--heck, easiest test--all quarter. I finished all of it except for one problem on Math History where I needed the formula for the latera surface area of a cone. I made the mistake of trying to derive it with calculus, and after an hour I realized how to do it with algebra.

Multvariable Calculus was a whole 'nother story...I forgot one of the four [new] Fundamental Theorems of Calculus we had "learned" (read: rushed through in lecture on the last day), and ended up toying with a question for about an hour trying to remember what the bloody integral was.

Now I know what it's like to take four hours to do a test. If it weren't for the really lax testing environment, I probably would've pulled out a good deal of hairs. What do I mean by lax, you may ask?

We have a little bar of food on the side of the room for every test--er, by little bar, I mean Don (the professor) brings in snack foods and lays them on the counter. Apparently it was a class trend last year; I would've liked someone to bring in a cookie besides Nillas. Nillas are kinda plain. I do like how there are snacks for munching during the tests; I theorized today that it's actually a "Comfort Food" thing.

We get as many breaks as we desire, too; it's quite nice to get up and walk around for a while, unwinding with every step away from the desk with that Blasted Test sitting at it. I think one guy in the class took his "break" a bit too laxly, though...he left the room after getting the test, with his test with him, and walked down the hall to a little couch area, where he promptly went to sleep.

I prodded him awake two hours into the test on one of my breaks. He said he was done; I saw a blank front page. I really wonder about that guy...he slept during lectures, too. Yet, this is his third year with Don as a professor. Queer.

Oh, I figured out that my verbal directions to that church I'm playing in tomorrow are wrong. If you get on Franklin from Union road, you'll be able to see a church with First Christian Church written on the side of the building. Can't miss it. It's not adjacent to Sylvester Park, though. Hope to see you there.

My First Proof; Antisociality

The saddest part of professional mathematics has got to be the on-the-job interaction.

When proving mathematical theorems, you usually have to go into some really obscure field to make any proofs of novelty. This means that not only do you have to go into a field that few people, if any, have researched, but you have to create some new fangled idea in that field to lay your claim to fame. Only, who do you brag to?

In order to explain your proof to anybody, from dorm buddy to tenure-holding university professor in pure mathematics, that person must understand everything you understood to read your proof. Which means that person must be trained in your field. See method of novelty.

This means that most of what mathematicians do is, socially, useless. Sure, Non-Euclidean Geometry was a hard-to-understand field that eventually became the keystone to Einstein's Relativity Theory, which was a hugely useful field, but that took about two hundred years to go into effect. Any math that professional mathematicians derive is, for most practical purposes, bunk that nobody understands except for the individual that wrote the proof to begin with.

Pure mathematicians pride themselves on being useless. In World War I, the chemists reigned over the battlefields with gases. In World War II, the physicists shocked the world with the atomic bomb. Some fear World War III: The potential Mathematicans' War; and that means more than cryptology in communications. For now, pure mathematicians work to prove things of such little utility that no weapon could come from them--a fine example of this is the discovery of an isomorphism that equated multiplying numbers, a third grade operation, to summating trigonometric identities, a waste of time.

I felt I've contributed to uselessness tonight, though it's so useless that it's probably been proven umpteen times over. What is it? It's a little note in group theory, on the order of a cyclic subgroup generated by an order pair.

And here's the antisocial point, another problem with professional pure mathematicians: Merely mentioning a proof or an outline of the proof is potentially condescending. Mathematicians aren't too big in the P.R. section.

Exquisite Philosophers

Fridays don't seem too terribly efficient in Math Systems. Today for seminar, my group wrote a story exquisite-cadaver style about math philosophers.

In Formalism, there are no Platonic objects like numbers, circles, or whatnot--everything's just reduced to axioms, which are given arbitrarily. Thus nothing exists in mathematics. Hilbert was a diehard formalist, and I ended our story with him.

'Twas a Sunday, and Hilbert locked himself into his closet, left the light off, and screamed and ranted and raved about how nothing at all exists. Sunday was Philosophy Day, where mathematicians stopped doing real work and just delved into their own private beliefs about real numbers and all that stuff. At 12:01 am on Monday morning, someone opened the door to Hilbert's closet and let him out. "All right, your sins are forgiven; time to get back to work." And Hilbert went to his desk and continued on studying...whatever the heck it was he studied.

That was the ending I offered to the group writing the story; we started an Exquisite Cadaver round to try to write the story. It didn't work too well; a Platonic, Self-Conscious cow derived things by Greek Antiquity (straightedge and compass), until Sunday rolled around and a port hole (not portal) opened in his side and Adam, Eve, and some philosopher popped out in the Bermuda Triangle, where a super-duper secret meeting of a Formalist, Foundationist and Constructivist were together, plotting to overthrow Euclid and his Elements.

Nobody else in the class got it either. Even fewer people understood the 17-yr-old girl working at Cinnabon's.

The next type of Exquisite Cadavers was much more efficient...or smart-looking, anyway. Raphael's group took the idea of Formalism, where all truths are derived from axioms through rigorous proof, and turned it into an Exquisite Cadaver game. Everybody writes one line of a proof, gives some reason, folds it over and passes the paper in a circle.

It ended up proving that Alice (de la Wonderland) approached infinity, creating an infinitely-ranging set. Of course, since there's little continuity in the proof, we just plastered it onto the door for passerbys to observe and think "Wow, they must be smart." Quod Erat Demonstradum.

The Simpsons on ...Calculus?!

Friday was another Multivariable day, and we started multiple integration. Fun stuff--seeing a bunch of big S's drawn on the board makes me feel like I'm doing calculus, moreso than last year when I just saw one S on the board.

Don moved the lecture today into multiple integration in polar coordinates--where the main point to remember was that dx -> rdr. He said that a couple times to the class, and Katherine just busted out laughing and brought Don's lecture to a pause.

"Hey, do you guys remember that Simpsons..."

Half of the class cracked up on the spot; Don didn't get it, so Katherine cited her source. When Bart went to the school for the gifted (earlier season), the teacher was lecturing to the class on polar integration--cause, y'know, that's quite the topic for TEN YEAR OLDS to cover--and she kept mentioning to the class that it wasn't dr, but rdr. She hit Bart's blank face. "You know--haR D haR har" [insert appropriate elbow movements].

The best part about this was seeing Katherine, a girl with quite a few piercings to her head's name, do the elbow motions in a green colonial jacket and the rest of her pirate ensemble. Don got chuckles for the next twenty minutes whenever he said rdr.

So, even with calculus pneumonics, we can once again say "Simpsons did it."

Ideal Inertia

Exercise 2.2.4, part 2: Explain why it's impossible for a sequence to have an infinite range AND have a limit at infinity.

Naoko and I discussed this problem for about ten minutes yesterday, because (1) she didn't get her book due to crappy mail-order service and (2) I couldn't figure out the problem myself. We pored and pored over definitions of sequences, functions, and mutually exclusive boundaries of range and domain, reading the book several times...until the solution hit me.

You may think I'm being metaphorical there: But really, when I had the necessary epiphany to solve the problem, I was shot into the back of my chair, which made a loud squeaking sound after the equally-loud k-klunk. "By gum, I think I've got it!" I exclaimed as I flipped the pages of my book back to a discussion of some proof or another. I pointed my finger in triumph, then explained my logic to Naoko.

After I finished, I sat back and realized I was overheating. A quick, sweatshirt-less walk later I was ready to work again. Naoko thinks I'm weird. And not just because I draw smiley faces in my upper-division math homework.

Faith from Mathematics

Friday's seminars are becoming something like a religious open-floor discussion, pertaining to math. I heard an argument last Friday on finding faith in a God through mathematics--and it wasn't that pure, ultimate-abstraction-to-His-workings conjecture I'd heard several times before, mainly from the seminar reading. I heard a guy say he believed in a God, by Statistics.

The moon and the sun are the two most visible objects in our sky--and they are the same size by our perception. The probability of that is [with the hand in the O position] ze-ro--but zero in a calculus sense, as in just really close to zero. Thus, one man in our class has faith that a God exists.

Not a bad conjecture, particularly for those of us in the class who enjoy symmetry. I wonder how he'll enjoy reading Symmetry (Herman Weyl) later this year.

How'd I do that?

This may confuse many of you. It baffles me. It may cause Ali to bang her head on the computer monitor once again, which satisfies my joy quota for the day.

I tried to prove that -0=0, and got it wrong.

I didn't end up at infinity or anything like that, but I did try to factor out -1, which wasn't permitted. That's actually the only way one can really screw up that proof...

I'm still working on this one. We've defined 0 in class as nonnegative, nonpositive, but neither negative nor positive. I couldn't use that for a proof either.

Be back later.

Math Systems, Day 3

I won't make this entry quite as technical as the last one--though Ali's pain does cause me such joy, and I'm sure my pain does her soul a bit of good too. It's a mutual feeling which usually leads to gnashing of teeth.

Anyways, out with the old news, in the with the old-to-new. You guys have known how the Swiss Army Knife is useful in just about any situation--but I'll bet you didn't think it could be used in Multivariable Calculus.

Consider yourselves broadened in horizons. I whipped it out during in-class exercises today. Of course, all I used was the nail file as a straightedge--but nevertheless!

Don congratulated the class on enduring through the first 10% of the quarter. At least the course doesn't feel quite so...epic now.

I'm almost convinced that the seminar sections of class will be my least favorite. I'm still not too keen on why the math and physics programs need seminar segments like the rest of the programs on campus. Maybe it's just to keep a tether on our soon-to-wander minds...

Sandra from the Wednesday dance (Dropped Follow #5) and I chatted about pure math at my last dance. She told me about what her view of pure-math students were:

She went to Caltech, a university whose classes usually gave take-home tests. Most classes had tests with three-hour durations, due back the same day and the like. This was not the case for the pure-math students. They had a week. Unlimited time, unlimited resources short of communication between peers. They spent just about every waking minute on those tests, lost sleep over them...needless to say, Sandra's view of pure math dropped since seeing those students walking in the halls. She could always tell a math student at Caltech.

Don has warned us that pure mathematics is an easy field to get absorbed into. There are thousands of disciplines, which summate to publish over two hundred thousand theses and theorems in math journals every year globally--those are figures from ~1980 or whatever year there were 35,000 computers in the world. Those numbers have not diminished at all, which implies there is a huge body of knowledge to work in--and also that very few people will have the prerequisites to understand what the heck you're trying to prove in your theorem.

And you thought the 32nd digit of pi was Mathsturbatory.

Math Systems, Day 2

I got to name a theorem today. In a class called Advanced Calculus / Real Analysis, we've been defining binary...something-or-others. Must check over notes. Binary clusters or something; anyway, it's the pairing of the Field you're working in (Reals, Complex, Integers, etc) with a binary operator that works in that field (+,*, and not much more). We're starting in Reals, which is kinda tricky. We're only given a scant few things:

Reals
+
*
0
1

We are not even given 2. We had to define 2 today, as 1 + 1, and then use it to prove that 0 < 1/2. There are a few other things that we can't take for granted (like everything). We actually had a theorem with the unsexy name 0.2.3, which went like this:

x * 0 = 0

Don thought that you need to name theorems, in order to keep from messing them up with axioms. I called out a name, and this is what he wrote on the board:

x * 0 = 0
Obliteration Theorem

I got to name my first theorem. Tee hee.

Dirty, Filthy Calculus

Note: This one ain't intended for the kiddies. And yes, it is a dirty, filthy calculus pickup line. Not quite along the lines of "I like spaghetti, let's f@&%," but--actually, it's even crasser than that.

Since I'm studying math for the year, I'll try to come up with a few more.
___

Baby, I wanna lay you on a 0-radian plane and run my elongated elipsoid between your absolute maximae.

Hours of Calculus

After scribbling away with this Stanford final, I've noticed that I've invested many hours in the solving of twenty problems. So, I devised a scale to represent my progression of completion:

A day with...
1 hour of calculus: OK, glad to have had a warmup, but banging my head on that one problem for only an hour'll get nothing done.

2 hours of calculus: Reading the text makes the lap cold at first, but that plastic-y paper warms up quite nicely, making hunting for solutions at least comforting to the lap. Still nothing actually done, though...

3 hours of calculus: Huzzah, another Final problem is finished!

4 hours of calculus: My poor, neglected history book'll just have to wait until this next problem is done...be that tomorrow or no.

5 hours of calculus: Eeeehhhhhh...bordering on obsessive-compulsive Integrating, but it's still a hard day's work's worth, as at least one problem was finished.

6 hours of calculus: (See (5), but sigh longer at the beginning; two problems finished)

7 hours of calculus: I will scoop myself ice cream and then look at the bottom of the spoon and think "Hmm. That looks pretty cold. I wonder what'll happen if I do this..." and stick my tongue upon it.
This is also my supporting evidence for my thesis, "Seven hours of calculus in one day will lower your IQ to below the age that multiplication was learned."
The tongue was removed with minimal (but non-zero) flesh loss.

The 7-hour member of the scale has only been visited last Saturday; will possibly occur again this weekend. I'll update if I need an 8th-hour level. I'll also update on the status of my tongue next Sunday; by then, I should have to mail the test in.