Archive for the ‘mathematics’ Category

Independence is not good for some people

March 25, 2012

So lecturer X, for whose course “Coursename A” I am the teaching assistant, is away on a research retreat at Mt. Wolfdoom. This leaves me to spellcheck, photocopy and supervise the final exam.

I am so fighting the temptation to add one more question to the test.

Possibly this:

Essay. Inner products and me. (6 p.)

Or this:

Let P be the set of polynomials with real coefficients. What is the third element of P? (6 p.)

Or this:

Do you feel this course will prove to be useful in your professional life? (6 p.)

For the person that has not given in to the Darkness Which Is Math, the first question is ludicrous; the second nonsensical; and the third is a standard stupidity from the course evaluation form, made more exciting by the promise/threat of six points hanging in the balance.

— but probably I will resist this temptation, because after the test is over (ha!) I’ll scan the produced chickenfeet into pdf, and send them to Mt. Wolfdoom. And then thunder will flash over the mountain, and a voice dead cold and inhuman will utter many bad words.

Then again, if I said this special extra question should be answered on a separate sheet of paper…

Or if I handed out a special sheet which was the answer sheet for that question in itself —

(E1) What is the biggest natural number?

  1. one
  2. two
  3. four

(E2) The exam supervisor is thinking of a function. Write that function here:f(x) = __________

(E3) Your answer to the previous question is…

  1. Correct.
  2. Incorrect.
  3. I don’t know.

(E4) Your answer to the previous question is…

  1. Correct.
  2. Incorrect.
  3. I don’t know.

(E5) Your answer to the previous question is…

  1. Correct.
  2. Incorrect.
  3. I don’t know.

Once you think a while about the chain of those last questions, you may shudder. (“Well, I don’t know the function so for E3 “I don’t know” if my guess was correct. But what if it was? Then should I choose both “I don’t know” and “Correct”, and do I get partial credit from choosing just one? Should I hazard a guess? And if my answer is just partially correct, what do I answer the next one?”)

It’s a soluble problem, I think, but it would cause some twitching.

The right to define

March 17, 2012

There are many things that have no natural cause: many things that are the way they are because people have agreed to have them that way. Nowhere is this more true than among mathematicians.

For example, the average height of a population is a simple mathematical calculation, once the population is well defined. (Over time it is a function; at least logarithmically Hölder continuous, and possibly smooth.) But the “normal height” for a population? Why, that is whatever the relevant authority decides a “normal height” to be.

Thus in mathematics the meaning of “an important discovery” is largely dependent on the person using the term; see “Important discovery in the fridge”, dept. mailing list, last week — versus “Important discovery: Destruction of Earth imminent”, math-phys. mailing list, this week.

Among mathematicians, this “right to define” has been restricted, as otherwise the results would not be pleasant. Thus only the departmental head has the right to redefine the zero hour: that is, she gets to define midnight. This is not as sexy as it sounds; merely that if she chooses midnight to be 2 pm outside time (“on the rube clock”), then 8 o’clock is 10 pm outside time, and that’s when the workday starts.

Others are not allowed to mess with the official departmental time; this has been so ever since an adjunct professor redefined his workday into a singularity and retired five minutes later.

A particularly haunting case of redefining time is the sad life of the graduate student who, due to circumstances entirely his own fault, is set to graduate in February of 1993, as soon as that comes around. Opinion is divided on whether the intransigence of his professor is admirable or abominable; the main lesson of the case is thought to be this: never let Microsoft Word’s autocomplete anywhere close to your study plan, and always proofread everything before signing.

Fields laureates, that is, those that get awarded the Fields Medal, the Nobel of mathematics, customarily have the added privilege of defining category boundaries. For an example why this is a privilege best restricted to a small set of sophisticated, sensible people, see von Sturmleben’s paper, “All penii larger than mine are ‘wangi'; by definition I have the largest penis”, Acta Math., 185 (2000) no. 2, 287–290.

As for a more mundane case of time, borderline on the departmental-head powers, who hasn’t heard a math professor exclaim, “Just a minute! I redefine a minute to be fifteen minutes.”?

And who hasn’t heard the inevitable adjunct, popping out of thin air, sneering: “So ya define x to be 15x? Solve for x, and a minute is nothing! Ya have no time! Come on now pops we gots no time!

Or the assistant, similarly appearing, crowing: “So a minute is a zero unit? Then so are all other measures of time! All of time appears in this one and same instant then! Let’s go see Shakespeare, born living and dead, right now!”

Or the lowly lecturer, shuffling to view, moaning with his hands thrust forwards: “Ah truly then this job not only feels like centuries, but is centuries — millennia — endless spans of futile, frustrating toil!”

Ah, such is the playful nature of mathematicians, for certain definitions of “playful”.

One may wonder why mathematics departments all over the world — for they all are like this — have not descended into utter anarchy as the result of the right to define. This is a question whose answer is trivial; mathematicians like their definitions to be well-defined, with nice analogues to the definitions of other people (read: mathematicians), and as the result any department exists in a slowly fluctuating state of collaborative insanity, shared by the inmates (read: faculty), and as is well known, this is but the Earth in microcosm.

After all, to offer one final example, money is a ludicrously fictional concept, made even sillier by the antics of loans and trading in futures and so on; and yet the vast majority of humanity treats money as something which makes sense! Mathematicians have long since seen through this madness, and thus require in salary only enough for basic needs plus writing implements; which the ever at least slightly puzzled administration is happy to give.

Mathrage

March 13, 2012

So, googling 11/22/63, the name of the Stephen King book, I found out Google also gives you the result of that query as a math problem. This feature is not new, of course; what was new was I noticed the answer was this:

(11/22)/63 = 0.00793650794.

Two points of somewhat exaggerated mathrage about this.

1) That’s not an exact value! Don’t use the equals sign when you’re giving an approximation, you… you brutes! That’s almost 10^{-11} of potential error!

(Wolfram Alpha: 1/126 = 0.0079365 and then 079365 repeats forever)

This is not much of an error; but as a mathematician it strikes me as horrendously creepy (you may not feel likewise) to be presented an equals when an approximation is meant. The value may be close; but the kind is vastly different. Suppose you thought that—

Hang on a minute.

Surely not.

No.

Cannot be.

On the search “pi”, I get this:

pi = 3.14159265.

Well, that’s pi then, a nice and clear rational number. Initiate brain hemorrhage!

2) The answer given is at best one of two possible values; at worst, it is total nonsense. The mathematical expression 11/22/63 is monstrously ill-defined, for it contains two divisions without clearly telling which divides what. You could just as well suppose it meant

\displaystyle 11/(22/63) = 31.5,

exact.

On the other hand, I hear the book is good, so I shouldn’t make up forced gripes like this. But what else is a math-type going to do? Complain that in the Game of Thrones, note plural, there actually is just one throne?

*

On the other hand, much of Internet is interested in “lulz”, that is, transient and often malicious feelings of excitement and amusement. (As in, Q: “Why did you open a parachuting academy next to the pound, Mr. Larson?” A: “I did for the lulz.”)

I believe it would be possible to determine the exact value of one lul with full pseudoscientific accuracy, and to use it to measure just how amusing all manner of things are.

Experiment one. Obtain data by establishing things of zero lulz. Experiment to be conducted thru Chatroulette.

Experiment two. Obtain data by establishing chains of increasing lulz. Experiment to be conducted thru Chatroulette, with flashing images. (Example: If the subject expresses amusement at Img #1 (“A cat”), more amusement at Img #2 (“A cat in a maid costume”) and least amusement at Img #3 (“Necrosis”), the chain 3<1<2 is implied.)

Experiment three. Hoodwink the gullible public by choosing the point of one lul yourself; they don’t understand you can do this at any value you want! Solicit cash offerings from Internet authorities beforehand; Blogfather J.S. might pay big bucks to have the Baconcat be exactly one lul, ensuring his immortal fame for centuries to come.)

Experiment four. Submit to Acta Math., highest impact factor in mathematics; wait for results. In case the null hypothesis is confirmed, drink heavily, resubmit, get ready for the next project.

Next project: Are pets combustible? Forge a prior OK paper from the university Ethics Committee; blackmail them for big money. Use money to fund next project.

Next project: Orbital laser. Make indecipherable blueprints and build cunning miniatures. Put them in a package and mail it to a random address in the States, dusted with a recreational substance. Once the package is seized by their rabid border authority people and well-behaving dogs, they will find this slip at the bottom: “KGB surplus! For more info, visit” — and then a suspicious Russian web address you have registered beforehand, using an ID borrowed from a Russian post-doc. The address is a confusing maze of nonsensical pages with millions of ads; the CIA, FBI, NSA, WTF, NASA and SS (wait, no, the Secret Service) and the other authorities visiting it, increasingly confused and alarmed, should generate big money. This big money is intimated to go into the next project; while the department worries about that, take the money and dean’s Mercedes and run.

Experiment five. Tahiti: Nice or super?

Matrix Algebra Rule 34

January 5, 2012

Context: I said this; Bob O’Hara said this.

And then I said to myself, “bloody hell, what do you mean there is no Rule 34 of matrix algebra? Is there really no lazy mathematics graduate student who would— oh, wait.”

* * *

There is no 34th rule in matrix algebra.

No, the rules of matrix algebra are not numbered. They are wild, free, potentially uncountably infinite. They are discrete bricks of conditional truth baked from the raw red sludge of matricular concepts; bricks, and sculpted marble columns and colonnades that, founded on logical truth, reach up at a potentially limitless sky, supporting churches and palaces of proposition and conjecture. They make a city fair, ancient and beautiful.

The ghost labourers of that city’s construction are mathematicians; their sweat and tears are the mortar, their puny mortal minds the timid flesh-architects, unseenly present there to build this spirit world — or to make it reveal itself, if that is your philosophical inclination. Maybe these ideas pre-exist; maybe they are created by discovery. Maybe the distinction is bogus and meaningless.

Nonetheless, this fair city of the truths and potential truths of matrix algebra is vast. It sways upwards in most un-cathedral-like fractal growth, results building on results, outlines filled in, and new outlines mapped as mirrored, distorted translucent copies of those that already exist.

If you are a mathematician and study matrix algebra, you can see this fair city growing, alive, unfurling like a flower, uncurling and growing like a child, its growth accelerated thousandfold by your encounter of it in a book that has the labour of centuries behind it. Tens of thousands of mathematicians, or some bounded from above number of a meaningless magnitude, have each made their contributions, some minuscule, of epsilonian size; some sprawling giant tangles of invention-discovery-organization, and out of all of them is curated and arranged the seemingly easy progression that is a book on matrix algebra, that guidebook to a city of lovely dreams — and this is one of the stories it tells.

The city is one of pink marble towers and golden steps, one of many cities on the trembling mathematical globe; and it is arranged in sweet spirals of repeating patterns, laws inexorably echoing in ever different cases, lemmata-chapels kneeling humbly beside towering theorem-cathedrals, and swarming crowds of matrices funnelled hither and thither, sure and confident in their knowledge of the rule of law, and a law of many rules.

But in all that structure, there is no numbering of rules; and thus there is no Rule 34 in matrix algebra.

No, that rule is one of life, not of mathematics. Matrix algebra tells you you can add together two matrices of equal size, or multiply them — the rule, of life, tells this too can be done with passion, excitement, and, dare we say, sexiness.

Let A be the matrix defined by

\displaystyle A = \left[ \begin{array}{cc} a_1 & a_2 \\ a_3 & a_4 \end{array}\right].

Observe this matrix, dear reader: a simple country matrix — square, decent, with no particular qualities along any moral axis; no secrets that any decomposition might reveal. Let us lift its skirts, and with chaste passion observe the most succulent number a_1a_4 - a_2a_3 to be not zero; we shall not want more detail from our blushing everymatrix. It means our matrix A is by no means singular; but though common, it is lovely.

But ah! Tragedy strikes. Our common, nonsingular matrix has an inverse: the Aristotelian other half of its Hamiltonian soul, a matrix A^{-1}, made at the other end of the universe of 2×2 matrices; all different, impossibly distant — but by a chance of statistics they meet, and as is known, opposites attract.

Their romance kindles like every phase of the Gauss-Jordan reduction of a 1000×1000 matrix going off at once!

This is their downfall. Their love for each other is fierce, undeniable. But A is a Capulet; A^{-1} a Montague. They should never meet; all Verona of matrix algebra knows what their embrace will bring.

At first, their romantic play seems harmless, though intoxicating. They swap sweet nothings, hold hands (metaphorically speaking), go side by side. The sum A + A^{-1} forms, and sits inertly, sweetly, unsimplified.

They share a first kiss; the sum dissolves into passionate summation, a heart-pounding, bracket-clutching, element-interleaving rush of the first base, and the second. The sum is resolved — for a fleeting moment there is no A, no A^{-1}, but merely this sweet sight:

\displaystyle \left[\begin{array}{cc} a_1+a_1^{-1} & a_2+a_2^{-1} \\ a_3+a_3^{-1} & a_4+a_4^{-1} \end{array}\right] .

But alas, this happiness is not to last. They are interrupted! A foul cretin, a singular old matrix of evil aspect, sees the two, and runs to inform, to speculate, to conjecture, with no decency or peer review, on other, more unseemly operations the two might have engaged in. It has no shame — no mercy — it sees nothing but the trivial thrill of a basic operation in the actions of two young and innocent matrices in love.

The two push apart, alarmed, their determinant-crossed fate clear to them. There they stand,

\displaystyle A = \left[\begin{array}{cc} a_1 & a_2+\epsilon \\ a_3 & a_4 \end{array}\right]

and

\displaystyle A^{-1} = \left[\begin{array}{cc} a_1^{-1}&a_2^{-1}-\epsilon\\ a_3^{-1}& a_4^{-1}\end{array}\right],

a little epsilon of lipstick transferred from one to the other. Their brackets heave; their elements shudder, torn between a moment’s lust and happiness, and the sure knowledge of impending doom.

“We are done!” A exclaims. “Finished! Run with me, away, away from this pestilential Verona of matrix algebra, this place that will not tolerate our love! If they shall not have us, we shall not have them (I can prove this) — come! I care not if you choose complex analysis, or potential theory, or some cold and distant L^p space, where dim Hilbertian stars wheel overhead. If you but be with me, my dimensions shall remain unchanged, my elements fixed in their positions — come with me.”

“Oh, darling!” the grief-struck A^{-1} cries, and brushes the epsilon of its lipstick from A’s cheek. “I cannot. I must not. I would perish — you would perish — there are limitations to what the gods of Thales and Bourbaki allow us. Would a matrix fit in at the courts of the Functional King? No, we would be pleasure slaves, freakish exotics, not ourselves; we would perish. Would even the brotherly dukes of Geometry shelter us? Us, who are cold and hobbled imitations of them, to them — as they to us are a cipher of lines, circles, sizeless points — there is no life for a matrix, or two of them, save in matrix algebra.”

“Say not so!” A cries, his brackets near bursting with agony. “Do not require of me the mockery of others — no, require that, but do not ask me to live separate from you. We are inseparable matrices. (Poetic license.) I would sooner die.”

“Then let us both do so!”, A^{-1} cries, wild, fey: “Let us die together, our love consummated in a fatal embrace. Let us perish; for to perish together is sweeter than any lonely existence could be!”

And so they rush at each other, rows and columns tangling, multiplying in a wild frenzy of passion free of all care. A gasp escapes A as its virginal columns part; A^{-1} murmurs a comfort, and slides its rows alongside, inside.

“Oh!” A gasps; A^{-1} is a perfect fit. They are made for each other. Each row clutches a column, and each column rubs against a trembling virginal row; a_1 rubs alongside a muscular column, till it finds the hungry touch of a_1^{-1}; now A^{-1} gasps, too, frightened, eager, demanding and adoring, and surrenders to the pleasures of calculation.

Their multiplication is hurried by necessity, but there are no errors in it, Their elements dance, their brackets merge; the movement achieves an breathless symmetry, and the two coalesce into a single form, melting, swirling, snapping together, each part finding its corresponding part in the other, their worries melting into happiness, feverish happiness, impossible happiness, perfect happiness; their elements are but the very basic numbers of the mathematical universe, engaged in the most ancient actions of all mathematical life: the numbers, they multiply, they are added; the orgasmic pleasure of unification is beyond words — and it is deadly.

With a cry of unspeakable contentment, the two are gone, their love the doom and fate of them. It was as the cold stars of logic had decreed —

\displaystyle AA^{-1} = \left[\begin{array}{cc} a_1 & a_2\\ a_3&a_4\end{array}\right] \left[\begin{array}{cc} a_1^{-1}&a_2^{-1}\\a_3^{-1}& a_4^{-1}\end{array}\right] = I,

nothing more, nothing of them remains, but this: a single I, an empty shapeless thing, the identity matrix, a dead thing, an epitaph, a cenotaph:

\displaystyle I = \left[ \begin{array}{cc} 1&0\\ 0&1\end{array}\right]

— no sign of the lovers remains in it, but bare entropiac blankness; so it is not by that stone, but by this tale, that you, dear reader, are to know there were two young passionate matrices that once loved, and dared, and lost, in the old Verona of matrix algebra.

Too many kittens

December 14, 2011

Too many cat posts at the blogs and twitters of Greta Christina and Jen McCreight.

This came out.

Dog help us all.

* * *

KITTEN CUTENESS FOR THE NON-MATHEMATICIAN

Eustace J. Wobbles
Professor of Faunamathematics
Agric. Univ. Hoho, Sussex

A kitten cuteness measure C should have the following three properties:

(1) C(0) = 0

(2) C(x) < U for all numbers of kittens x, where U is the baby unicorn crocodile cuteness constant.

(3) C(x) is continuous in the piecewise fractional kitten sense for all x > 0.

Notes:

  1. Gauss’s hypothesis that \lim_{x\to\infty} C(x) = U was disproved by Wiles in 1999. The best current result is that \sup_x C(x) is at least \pi/3 below U for all natural measures C(x), i.e. measures that have (1)–(3). See Olcat for the contradiction that follows if the distance is less, “the Cyriak Paradox”.
  2. The behavior of C when x < 0 is of no importance. For an exhaustive survey of research into this case, see the celebrated book by Vem Varför (Varför 1993) and the more accessible Varför 2001.

We now introduce a common cuteness measure; for full derivation and alternatives, see Olcat and Hezbuugaa.

Let x be the number of kittens. Then Erdös’s experienced cuteness function C is

C(x) = 10x - \frac{1}{2}x^2.

Clearly C(0) = 0, and since C is continuous, it is continuous in the piecewise fractional kitten sense. Condition (2) follows for the Wieso-Holzkopf Lemma.

Observations on the aptness of this measure for measuring real world kitten number behavior follow.

Observations.

The cutest number of kittens is 10, after which cuteness decreases. After 20 kittens C is negative; experimentation has shown this is the number at which the Kitten Hivemind activates, and all cuteness disappears.

At roughly 144.5 kittens the cuteness goes below minus nine thousand, with predictable and catastrophic consequences. (See Narm.)

It is at the present unknown what would happen if over 145 kittens were present at once; the Gell-Mann-Feynman PET model seems to indicate the strong nuclear force would prevent this configuration and those above it.

For a contrary view, see Hezbuugaa’s proposals for “the Kittycube” (1984) and “the Kittycube Propulsion System” (1989).

Research into maximal kitten formations has been severely limited by the Kitten Ground Test Ban of 1992; at the present, Russia and China are still conducting secret kitten tests in space, and CERN in co-operation with the European Union and the Hivemind is planning a strictly volunteer-based non-weaponized high Earth orbit testing program as a successor to the LHC. (For a look into the possible environmental impact, see Appendix D in Miau’s Raining What? Fertilizer At The Crossroads.)

The ease of weaponizing over-20 kitten configurations (the so-called “Cat Lady Weapon” or “the K-bomb”) has been greatly exaggerated in popular depictions. It is not enough to simply throw thirty kittens at someone and duck! Likewise, compacting a sub-100 number of kittens does not cause a 144.5 meltdown, but just an awful mess. Speaking of awful messes, the Alan Smithee film “Hellcats at the Singularity!” makes both of these blunders, the first in the presidential assassination scene and the second in the New York destruction scene. The Hivemind does not appromeow.

See Olcat and Hezbuugaa’s book for human- and time-dependant kitten functions, and Munroe for the distance dependant function.

*

REFERENCES

Olcat, L. and Hezbuugaa, C., Advanced Animal Theory, Springer Verlag, 2009.

Olcat, L., The Unicorn Crocodile Constant Is Strictly Separated Away From Kittens, Comm. Soc. Fel. Amer., 7 (1992), no. 3, 133–176

Grausam, J., Fractional Kittens: A Very Graphic Approach, Wilford Telmarine Farrar, 1962.

Munroe, R., Cat Proximity, xkcd, 231, no. 1, 1–1

Narm, M., An Oral History of the Kitten Apocalypse of Syracuse, 1909: A Retrospective In Analysis, New Syracuse University Press, 1984

Varför, V., Negativa och icke-real complex kattdjur: en helveten spekulativ strategi, Kungl. Vitterhetsakademien, 1993

Varför, V., Subzero Farm Animals: An Introduction, Springer Verlag, 2001

Feris, M. O., How Much Is Too Much? Too Many Perspectives on Humor/Humour in the Mathematical Sciences, Tripleday, 1973

Feris, M. O., I Spell It Hummor: Translating Humor Across Subculture Boundaries, Quadrupleday, 1984

Advisor moments

December 5, 2011

There’s something magical in that when you go to your advisor to tell you can’t do X except with Y; and he says “well, I think this is not an obvious disaster; run with X-plus-Y and see what happens.”

There are a lot of other advisor moments, too: all the moments when it becomes apparent that you have a penlight and he has a giant-ass halogen disco ball. All the moments when it is obvious that you are still a learner, and he is the master. (And maybe, one day, you will be the doctor? And in your case, it could be a “she” just as well as a “he”.) Your advisor is kind of like a third parent; it’s not trivial that one instance of tracking these relationships is called the Mathematics Genealogy Project.

There’s the dreadful moment when he offhandedly, and possibly jocularly, drops a word that after the year of head-walling you’ve done, you probably know and understand more about that particular theorem than anyone else alive. There’s the moment when you see you’re really at the edge of knowledge, and he’s about to throw you out so you can see what lies beyond.

There’s the moment he pulls a trick out of thin air, and fixes the one detail you had no idea how to deal with. Probably it’s a trick you should have known; but “trivial” is not a constant but highly time-dependant.

There’s the moment you see he hasn’t understood something, and it feels so good to step in to explain. (Because maybe you’re the bigger expert now, as regards this tiny subject, and this moment in time? Or maybe your handwriting just is really awful?)

There’s the moment you say “And next we—” and then you see the mistake, the horrible gap you totally forgot, and there’s that terrible three-second delay before he sees it. It’s not that he would be upset or angry, but that you want to be bright at him. You’re the penlight and he the halogen, but you want to show him you’ve changed out some of your dimbulbs. He’s radiant, and in his company you want to be the same, as good and a thousand times more.

There’s the moment you spend a double lungful explaining your approach, and he breathes out the two-word name for it — and if you’re lucky it’s “basically Hölder’s?” and not “Weird nonsense!” And then there is the thing when you’re explaining what you have done and in the middle of it you see what you’ve done is wrong and you have to offhandedly admit it and dance madly backwards trying to fix it as you talk and walk and write, chalkdust flying and the arrows becoming twistier, the letters sketchier, because you have all the details still in your head and you’ll show him how it goes… because for that moment you’re still the circus director, he the audience, and the proof may still be there, if you just reach and twist a bit.

There’s the feeling of being much smarter than you are, when he explains things to you, suggests and goads and outlines; and there’s the feeling of staring at an empty computer screen later and thinking, “It seemed so easy when he was talking about it. If only I’d taken more notes. Now was I supposed to see if this thing was bounded, or what?” When he is there, all is anchored by his understanding; when you are alone, it’s not only dark; the solid ground melts away, too. And then you run back and whine; back to the light, and then away carrying a little bit of it with you.

And though every single day you feel as dumb as the previous day, you’re pretty sure you’re feeling equally dumb about ever smarter things.

I guess what I’m trying to say is I don’t need no drugs: I meet my advisor at least once a week, and that’s a good enough altered state all by itself.

Three views into mathematics

November 29, 2011

What better use for a lunch break than a bottle of cola and a blog post? Three books off the showing-off-how-mathematical-I-am bookshelf at my elbow. One semi-random sentence from each.

I

In 1924, reviewing reports on algebraic numbers issued by the National Research Council, he noted with pleasure the comparatively great amount of space that the authors had devoted to cyclotomy, a fact that he saw as an encouragement to beginners and proof that the “lusty” old subject was still very much alive.

(Constance Reid, The Search for E.T. Bell, also known as John Taine, Mathematical Association of America, 1993, p. 145)

An obscure book about a mathematician, a historian, a popularizer of mathematics (author of the justly famous and famously not always exact Men of Mathematics), a poet, and a science fiction writer, that was not always all that honest with his own personal history.

II

Pathological monsters! cried the terrified mathematician
Every one of them is a splinter in my eye
I hate the Peano Space and the Koch Curve
I fear the Cantor Ternary Set
And the Sierpinski Gasket makes me want to cry

(Sarah Glaz and JoAnne Growney, Strange Attractors: Poems of Love and Mathematics, A K Peters, 2008, p. 141; this bit is a Jonathan Coulton song lyric)

I guess this is what snotty types call “an eclectic collection”. I liked about half of the poems; the other half weren’t mathematical enough.

III

“No reason was ever given,” recalled Henriksen, “but his lawyer was permitted to examine a portion of the Erdös file and found recorded the facts that he corresponded with a Chinese number theorist named Hua who had left his position at the University of Illinois to return to Red China in 1949 (a typical Erdös letter would have begun: Dear Hua, let p be an odd prime…) and that he had blundered onto a radar installation in Long Island … while discussing mathematics with two other noncitizens.” The authorities apparently feared that the letters to Hua, filled with impenetrable mathematical symbols, might be coded messages.

(Paul Hoffman, The Man Who Loved Only Numbers: The Story of Paul Erdös and the Search for Mathematical Truth, Hyperion, 1998, p. 128)

One of two (!) Erdös biographies I have; the other is by My Brain Is Open by Bruce Schechter. Erdös (and I suppose that is not o-umlaut but some Hungarian doodle) was a real-life stereotypical mathematician. As can be glimpsed from the quote above.

Rock, paper, scissors, a mathematician ruining it

November 16, 2011

I

The obvious variation is to add more signs into the game: say “rock-paper-reviewer-editor-scissors”. It in inobvious, though, whether rock beats reviewer or the other way round. (Some of those reviewers are tough.)

One way is to draw a pentagram in a single line (making each segment an arrow pointing the way you draw it) and then to draw a circle round it (marking the direction you draw). Then you can treat the points of the pentagram as the five signs, with each point originating two arrows indicating two other points, and being indicated by two of the others; which gives two signs that submit, and two that conquer.

Also, probably the most Satanic game design in history.

This addition alone, though, doesn’t make the game more interesting, just more complicated.

One could say winning or losing by the circle is different from winning or losing by the pentagram: but how? (Through a pentagram loss, you forfeit your very soul?)

Ib

As for the simpler obvious variation: Rock-paper-plasticknife-scissors, the game with four sign(al)s/gestures, is a bit iffy. You tie with the same; you lose to one, win against one… but what about the fourth? If it is a tie, one half of games end in a tie. It can’t be a win or a loss, because that would make some signs better than others. If rock wins against against plasticknife, then plasticknife loses to both rock and scissors, wins against paper and ties against itself — it would always be better to play rock (WWLT) than to play plasticknife (WLLT).

Any odd number of gestures can be arranged to be equally good; no even number above two can be without increasing the number of ties.

Then again, with more gestures this just isn’t interesting. Who cares if Horned Goat loses to Hanged Man or Lone Dalek, if it’s the same loss either way?

Ic

Rock-paper-scissors doesn’t have the same kind of a hierarchical arrangement as playing cards do — there you don’t get to choose your cards, so you can have cards that are better than others, most of the time. In rock-paper-scissors, you need to have options that are somehow equal (by not knowing the other player’s choice, if in no other way), because why would you choose a sign that was less likely to win?

Consider the card game known as “Red”. Both players draw a card from a deck, face down. Both then reveal their card. A red always beats a black; below that, a bigger card always wins. Not a particularly interesting game, but perfect for high school students really tapped-out after an unwelcome lesson. If you could call the card you wanted in Red, you’d be screaming “Ace of Hearts!” all the time — and having a tie with your opponent, who would be shouting the same. (Or “Diamond Ace!” — it would be a pointless, melodramatic game either way.)

This illustrates that either your choices can’t matter, or you must have no choice at all… which is a depressing prospect, but rock-paper-scissors is not much of an intellectual game anyway, as far as its mechanics go. The psychology can of course be very interesting, especially when you keep playing it. (“Is she going for scissors again? Third time in a row? But what if she’s counting on me pulling rock, and intends to play paper? Then I should play scissors— unless—“, et cetera. Put two psychologists to work playing each other, and they’ll probably stare at each other for five minutes, and then one admits defeat.)

It would be ideal to make a game with mechanics just complex enough to generate interesting psychology. Rock-paper-scissors isn’t quite complex enough. (Then again, it’s better than tic-tac-toe, a game where any player smarter than your average calculator can always tie, and two such players will always tie.)

II

The obvious biological variation would be to play the game with both hands at the same time. But this too makes the game different — in this case quicker (two at the same time!) — but not more interesting.

Then again, this gives more scoring conditions: a double win, a small win (win one, tie one), a fighting tie (win one, lose one) and a full tie (tie both). (The first two are, from the other end, a double lose and a small lose.)

By crunching numbers, the likelihood these outcomes is, assuming the players are dumb automatons:

11% Double win (W/W)
22% Small win (W/T)

22% Fighting tie (W/L)
11% Full tie (T/T)

22% Small lose (L/T)
11% Double lose (L/L)

— one percent is lost in the rounding. (Use 1/9 and 2/9 if you want to be exact.) If you take the first two as “wins”, the middle as “ties” and the last two as “loses”, then the odds are the same as in a normal one-handed game of rock-paper-scissors; there’s just a bit more additional detail within each category. To make a sensible variant of the game, this added sensitivity should be utilized somehow. (Note the two ties aren’t different in any intuitive way; both players get the same result in each. Some new rule could distinguish them for some other new aspect of the game.)

Mind you, this could be a decision tool if you needed two exit conditions —

Double win : We’ll do what I want, all the way

Small win: We’ll do what I want, for the most part

Fighting tie: Fine, let’s do nothing; I’ll go home, this isn’t working!

Full tie: Let’s try to split everything evenly, okay?

— but I’m not sure anyone needs help for making decisions like that.

The mechanic is there; the game just needs an addition that uses it.

III

The third variation, a sort of obnoxious meta thing, would be to have three players, each with two hands, each playing a one-handed game with each of the other two at the same time.

Call the players A, B and C. Three games resolve at the same time, each with three possible results (win/lose, lose/win, tie); this gives twenty-seven different total outcomes. Those form four categories, the way I choose to group them.

I’ll write “A>B” for “A wins over B”, “A<B” for “A loses to B” and “A=B” for “A and B tie”.

1) A<B<C<A : a roundabout tie. A>B>C>A is the same thing: each player has one win, one loss, and there’s no assigning rank to them.

2) A=B=C, every game ties; everyone flashes the same sign. A great tie! Also, the appearance of a gang meet-up.

3) A>B(sthng)C<A — Strong ranking; One player wins both of his/her games: victory! (I’ll call it that to distinguish it from “wins”, which are the results of individual games.) The third game, between the two losers, either gives second and third places, or a divided second if they tie:

3a) Full rank: A>B>C<A. Player A takes first place (wins over B and C), Player B the second (wins over C, loses to A), Player C the third (loses to A and B). Alternately, A>B<C<A. (It’s probably sensible to say A>B>C=A and A<B<C=A belong here as well; one can’t argue for any different order than the obvious one.)

3b) Weaker rank: A>B=C<A. Player A is the winner; the other two both lose.

Note that there can’t be a case where two players win both their games: the game between them can have at most one winner. This three-player game produces either one victor (above) or less (below).

4) A>B(sthng)C=A — Weak ranking; No player can be ranked as the best of the three. (A>B>C=A is already included in 3a.)

4a) Weaker rank: A>B<C=A. There’s no victor, just two winners; but B sure loses.

4b) Weakest rank: A>B=C=A. There are two ties and one win-lose; thus, a winner, a loser, and one the game didn’t decide about. (Also, A<B=C=A.)

I think one has to think that a tie means “no decision”, because one can’t really interpret a tie as “are equal” because of situations like A>B=C>A. If B and C are equal, why is one strictly better than A and one strictly worse? Unless you interpret that as collapsing > into \geq into =; how you interpret the mechanics makes the game.

As for the improved version of rock-paper-scissors, I have no idea. I’m just throwing up mechanics.

Ruining the sequence game

November 10, 2011

This is an old puzzle-type question: “I give you three numbers, a, b, c. What is next?”

This is nice brain exercise, but as a mathematician I feel duty bound to tell you you can break this game in about five seconds, if you so wish. (Well, five seconds and a bit of calculation time.)

Let f be a function so that f(n) gives the n:th number in the sequence; in the above example,

f(1) = a,

f(2) = b

and

f(3) = c.

Suppose you want the next number to be d. That’s one more condition for f,

f(4) = d.

The trick now is that these are four fixed points for a function; and it is trivial to find any number of functions that give those four values, and thus are “the rule that gives the sequence”.

That is to say:

A+ Math Student: “The sequence starts 1, 2, 3. What’s the next one?”

Mathematician: “The next one is 666.”

A+MS: “What? Don’t be silly, the next one is 4!”

M: “Huh? What perverse logic is that? Your sequence consists obviously of the integer values of the function f(x) = \frac{331}{3}x^3 -662x^2 + \frac{3644}{3}x - 662. You’re just changing the answer because I got it right.”

Best of all, the trick can be used to go from any given number of sequence points into any further number of points you may want to insist on. It’s a bit prohibitively bothersome to calculate — but it is always possible.

* * *

Oh well, the calculation. Oi, the calculation. If you have four points, a third-degree (four-minus-one-th degree) polynomial is probably the easiest guess, that is, a function f,

f(x) = Ax^3 + Bx^2 + Cx + D

for some constants A, B, C and D, so that it holds that

A + B + C + D = a

8A + 4B + 2C + D = b

27A + 9B + 3C + D = c

64A + 16B + 4C + D = d.

Just run a Gauss-Jordan elimination in your head and… what?

Okay, just use the ready-made solution:

\displaystyle A = -\frac{1}{6}\,a+\frac{1}{2}\,b-\frac{1}{2}\,c+\frac{1}{6}\,d

\displaystyle B = \frac{3}{2}\,a-4b+\frac{7}{2}\,c-d

\displaystyle C = -\frac{13}{3}\,a+\frac{19}{2}\,b-7c+\frac{11}{6}\,d

\displaystyle D = 4a-6b+4c-d.

For a=1, b=2, c=3, d=666, that gives

\displaystyle f(x) = \frac{331}{3}\,x^3 -662x^2 + \frac{3644}{3}\,x - 662;

clearly and obviously the rule asked for.

That’s still a lot of numbers, but a person with quick wits (not me!) could easily memorize that, and answer any what-is-the-fourth question with a horribly misguided rules-lawyering technically correct answer.

And technically correct is for a mathematician the only kind of correct that matters.

* * *

It will be a tad easier to forget the fourth number, and just slap down the second-degree polynomial that fits the given three —

f(x) = Ax^2 + Bx + C

for

\displaystyle A + B + C = a

\displaystyle 4A + 2B + C = b

\displaystyle 9A + 3B + C = c

or

\displaystyle A = \frac{1}{2}\,a-b+\frac{1}{2}\,c

\displaystyle B = -\frac{5}{2}\,a+4b-\frac{3}{2}\,c

\displaystyle C = 3a-3b+c

— and proclaim: “Here’s your bloody rule; as for the fourth number calculate it yourself! I don’t have time for your silly games! Ha ha ha!”

Alternatively, proclaim: “Here you go. Trolled by maths.”

(As for that f giving the intended fourth point, that’s infinitesimally unlikely. Most clever sequences aren’t second-degree polynomials.)

* * *

Of course if one doesn’t feel bound to finding an explicit numerical rule, the possibilities are endless.

Less than 666 reasons the next number is 666

  • The rule is my rule. The next number is 666.
  • No, you’re doing it wrong. Trust me, I’ve heard this one before; the next one is 666.
  • No, it’s the medals that are awarded in Tour de France: gold, silver, bronze and hamstrung. One, two, three, six-six-six.
  • Obviously it is integers ordered according to the frequency of their appearances in Western literature. Those wacky Christian mystics, right? All about 666!
  • “The Beast comes, all of a sudden! One! Two! Three! Six hundred and sixty six, the Number of the Beast! In medias res, Lupus Magnus Innominandum, Lucifer Deovore Daalek Satanas!” is the rule.
  • What do you mean, the next one can’t be 666? What happened to respecting the other guy’s religion? Huh?
  • Four? I’m so quoting that on Facebook. (This does not actually produce a sense of conviction in the other party, but rather a sense of crippling self-doubt with pretty much the same results.)
  • Four? I never pegged you as a racist before.
  • What do you need a rule like that for? It’s as obvious as elementary, high, university and Satan on horseback!
  • “Wait, no.” (Repeat after each attempt to disagree. Each time increase the time between “wait” and “no”.)
  • No no, I got this. This is much more elegant than yours. See? “One, two, three, six hundred and sixty six, blood, blood, blood, blood, blood—” (repeat with slowly increasing volume until you win)
  • Is too! Look, I can point you at one guy on the Internet that agrees with me and not you—

Horrible mathematicians

November 7, 2011

Was talking to a friend, a mathematical like me, a frequent TA like me, and through subjects that you don’t want to know, it came to this:

“…which would be quite a web address to point the students at to get their copies of correct answers from! Ha ha ha!”

“Ha! Ha! Ha! I think we could think up something worse, though.”

“Like http://www.analanswers.com?”;

After which, there was more laughter.

Explanation: Mathematical analysis is (a) a pretty good fraction of a math M.Sc., and (b) really abbreviated like that, pretty often.

The jokes for complex analysis and numerical analysis practically write themselves.

Eventually, one could slip the address into the contact details of an academic paper. Name, affiliation, snail mail, e-mail, booyah!

And a person ought to have a card — but because affiliations can fluctuate, it would just have

(your name here)
(phone number)

http://www.analanswers.com/

“Nice to meet you! Here’s my card; looking forward to working with you! Double-hand finger point, wink, leer!”

Though if the site had more than demonstration answers, maybe it should be http://www.analexplorations.com? Or http://www.realanalproblems.com, “Problems in Real Analysis”?

Or, if one had self-confidence and delusions of grandeur, http://www.analchampion.com?

And then there’s the old one, the real actual one I’ve seen on a blackboard for reals, of abbreviation in assuming f is an analytic function:

“but ass. f anal.”


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