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Imagine a circle inscribed in a square. Imagine a sphere inscribed in a cube.

Trust that there are wise and coherent ways of extending such a situation beyond the three axes of comprehensible space.

Imagine, in colors, or as a mash of nonsense if you like, a prickly urchin of n mutually perpendicular axes, demarcating and graduating the points of n dimensional space. To soothe the skeptic, let's say that we only mean that each point is an ordered set (bing, bong, bling,....) of n real numbers, and that a cogent definition of distance applies to relate these creatures to one another.

We will recover the evolved version of the above geometric scenario, translated to this new more airy arena.

The solid sphere that lives in n-space is the smooth haze around the center of the the axis-urchin, those points of distance at most one from the origin, (a characterization which captures, you might note, the essential physiognomy of his more intuitive cousins).

The autochthonous cube of dimension n is the box made by erecting a perpendicular obstruction at the unit points of all n coordinate staves, then regarding with satisfaction the consequent area enclosed. Another, less scenic, conception can be had by considering locations of the form (a,b,c, ..., z) where the letters have value between one and one negated.

Now.

We have, if you will look and see with me, a hypercube which inscribes a hypersphere. What can we say about it?

It has a strange feature.

We notice that in the sigil of the circle inscribed in a square, and the sphere nestled in its cozy box, that the accommodations are not excessively spacious. These simple orbicles have scarcely room in their restrictive housings to turn around. To be vulgar, the sphere in the box occupies 52 percent of the area boxily bounded (the circle slightly more).


But upstairs, in the expanded inflated volumnized world of high dimensionality, this intuition is broken, snapped, severed, inapplicable. The sophisticated descendants of the provincial ball and box stand in a much less egalitarian relation.

The sphere, sadly, for sufficiently rarified values of n, occupies a vanishing portion of its home. It becomes like the lonely occupant of a mansion, then a mote of dust drifting by a window, and then is lost to all scrutability, vanishing down into quark-like insignificance, an angel on the head of a pin.

All the while, it abuts just as belligerently against the confines of its surrounding box as happened in dimensions one and two.

There is, therefore, a wonderful mystery about simple geometrical objects with great extension. Unfortunately it is not our province to know these beings. We glimpse them only unsatisfactorily as they drift through the insignificant film of our infinitesimally thin world, as a cat pads through a doorway, whisker, nose, ear, and tail.

I think the happiness boost from almost graduating is wearing off. It reminds me of a lecture on happiness my former friend and I went to earlier in the semester, where some philosopher compared happiness with the delta-meter on a sailboat. Happiness, qua sailboat, is only a measure of the rate at which your place in life is changing. It reflects nothing about your actual well-being.

Thus my position has improved, and yet life is once again the same as it was.

I have a childhood friend moving to the ciity. At the same time I just ruined one of my longest Maryland friendships. It occurred to me today that I might have some unconscious urge to hold my number of friends constant, which is why I purposely sabotaged things with Chris. I wouldn't put this past myself.

I would ordinarily start writing about you now, but all I know is that you had a run-in with a wine slushing lush at a party. It seems like girls would be more wary of you kicking their asses.

Regarding my office-mate Dave, who will never read this:

This morning he underwent a procedure to have a sebaceous cyst lanced. This is unremarkable in general. However Dave's cyst is/was located just inside his anus. The operation was minor, but apparently required general anesthetic. The cynical reader will wonder if this is just so that the doctors would not have to tolerate an hour or so of extremely strained conversation. I have no doubt that this is the real reasoning of the anesthesiologist.

This afternoon I had the experience of being with three other individuals, who all know Dave, and all assumed the others had no clue about the operation. Unfortunately someone broached the topic of surgery, and everything fell out after that. Surprisingly, no one cracked a smile.

My life in bullet points:

*My adviser told me I could graduate this May, but suggested I spend another year in penury so that I can go to the Sorbonne.

*I have a student who's dying. She sends me emails about her problems that are frankly a little long winded, but still tragic.

*I bought an alphasmart portable word processor.
--I must describe this device in more detail. I suggest you google it yourself. There's a small black and white display and a full-sized keyboard. You can type on it, but not much else. It has 512k memory, and can hold about 100 pages of text. On three AAA batteries it supposedly runs for 700 hours. It saves after ever keystroke, so you can simply turn it on and off like a calculator. It weighs 2 lbs.

One might notice (since the theme to any blog is vanity I do not blush to mention this) that this entry is more concrete and less pleonastic than previous entries. This is because I've realized recently that I can't do math and write at the same time. Possibly, I cannot do either activity even in isolation. This is a grudging admission for me, because I've always felt that categorizing intelligence into Analytical/Verbal was extremely artificial. Good evidence for this position is that math graduate students (this is not just professional loyalty) can generally do everything better than everyone else. I say this not so much about myself as others that I know in the program. I will not defend this claim.

It took me five years to realize that the key to making progress in mathematics is obsession. The only way to solve a complicated problem is to fixate on it to a ridiculous extent, muttering, pacing, drooling, etc. This is because all the elements of the problem must be held in mind all at once, and putting them there is a time consuming process, like booting a computer. Rather than commuting to and from one's problem at morning and evening, one has to simply move in with the problem, live like a vagabond on the right angled streetcorners of its mathematical neighborhood. Unfortunately this is incompatible with writing in the evenings, at least insofar as my poor brain is concerned. The only thing that doesn't upset my Wa is vegging out in front of Battlestar Galactica.

Incidentally I think Battlestar Galactica is the greatest piece of art ever produced.

Speaking of stupid aesthetic claims, I have something painful lodged in the pocket of my brain. Last weekend I ate dinner in a salon style setting at a friend's apartment. The friend is actually two people, who are married. I won't reveal their personal details, since they are accomplished enough for each to be identified by broad-strokes descriptions of their lives. We all became quite drunk, and everyone said things I'm sure they regret. I have an unpleasant memory of talking about Uranus and why there were no testicles hanging from the sky, after launching uninvited into a bombastic and nonsensical discussion purportedly about the artist El Greco. Even writing this makes me curl my fingers in discomfort at the idiotic things that must have come out of my mouth.

The next morning I had an uncomfortable encounter with the hostess, and I've spent the last two or three days trying to put my finger on the source of our tension. One possibility is that I didn't make out with her friend from New York, who was seated next to me intentionally. For this I claim plain excess of booze as my defence, which leads to the second possible source, which is excess consumption of alcohol. I believe that towards the end of the evening the remaining wine, champagne, both bottles of brandy, and a 40 proof dessert liquor of some sort had all found their way to my placemat. How I can remember that I even drank them all, I'm not sure. I have a feeling that I behaved unseemingly in many respects.

Merry Christmas.

Numbers stations are high powered low frequency radio stations that broadcast repetitive messages consisting of strings of digits and letters. They are thought to be tools of the spy agencies of world powers, and some countries have obliquely acknowledged this fact. Listening to them is illegal in the UK. Some of them have been located by enthusiasts, and found to lie inside military garrisons. At least two of them are unofficially named after popular folk songs, because bars of the corresponding melody proceed the transmissions. The transmissions occur at regular intervals, and hobbyists have produced schedules.

Low frequency radio reflects off the upper layers of the atmosphere, and consequently a signal produced by a single ground based station can theoretically traverse the globe.

The whine of my kitchen light is a thin straight chain, cold and hanging heavily downwards. It vibrates unsettlingly, but with a very short range, like a cricket chirp prolonged for minutes on end. My right ear is suffering, is pained by the sound, and my right side is turned toward the kitchen.

I am munching rice dry, which I understand is sometimes used as a training technique by sumo wrestlers. The toughening of my intestines is an exciting possibility. On the other end of the equation: appendicitis.

Why is it that lately, I can't get out of my mind the cliffs along the Tennessee river, and the waterskiing trips my friend's father would take us on, so agonizingly early in the morning? We tried always to arrive while the water was still smooth. It was almost like red eye gravy in the mornings, thick and fragrant, hot from the hot august nights, and still as flat glass. The flatness would carry sounds far sometimes, and when the engine was still you could hear the lapping of the small waves on the red clay shores, or the guffaws of distant fishermen. There were other sounds unaccountable, caused by the unobservable actions of the timid water, which was surreptitiously occupied on all sides of the boat, into the far distance.

The worst part ( I am forgetting about the cliffs ) were those moments when you were in the water alone, waiting to be pulled up. Then you were helpless, and you had all the anxiety of your vulnerability summed with the impending roar of the boat engine, and the eyes of those in the boat, watching you rise or fall. The rope would go tight, and the power of the river would be felt all around you, inhumanly strong. The only escape was to slide upward, to stand aright on the sliver of wood beneath you, to somehow slide through the resistance, and gain balance above it. Then the water became progressively harder beneath you as the boat sped up. If the driver were inexperienced he would accelerate for too long, and the churning spume beneath you would solidify into limestone, and shake you mercilessly down, back into the foam, striking you as you came, as if in punishment for your arrogance.

The cliffs were slices out of the mountains, as if pieces of them had been removed like pieces of cake. The deep verdancy of the outside of the mountain would seem as skin, alarmingly different from the stone interior of the suddenly real geology beneath it. It looked like a grievous wound. The strangeness of it seemed absolutely alien when put together with the tableau of water on which we insecurely rested. The cliffs came down into the water, and conspired in weirdness with it, two occult forces in parlance in the always just-away distance, inaudible and suspicious.

The saddest man I know works down the hall from me, on the fourth floor of a very old academic building. He's a lecturer in mathematics, a privatdocent in the modern sense, which is to say, the title retains its suggestion of low salary, but doesn't confer the old sense of intellectualism. The lecturers at the college where I'm a graduate student are generally a sad lot, stillborn academics or educated housewives looking for a diversion. They are essentially professional TA's, teaching low level math classes to the slackjawed, and enjoy neither the respect of the faculty nor the glint of future brilliance which shines from some of the graduate students. The men among them are particularly strange for their acceptance of their lower caste position, politely ignored in superficial conversation with other members of the department. It is, in short, a pathetic job.

What's funny is that the saddest man is on the surface the most depressing representative of the class to which he belongs, moping slumpshouldered down the halls, and riding the elevator with downcast eyes, a copy of Math 101 pressed under one well chalked arm. But lately in the instances where he meets my eye, I've begun to notice something else. He has a certain precision of speech which he summons on demand to season certain points he wants to emphasize, and this version of himself--the version lost in high speech--seems completely separate from the man as a math instructing drone. Here he has penetration, which one sees is repressed, like a knife sequestered in a coat pocket, during his day to day affairs.

He loves a woman who works downstairs, an Audry Hepburn type, aging as he is, though still desirable. It's hard to say if his unrequited passion for the aging vixen, you'll have to take my word that's a fair description, increases or diminishes him. Part of what I suspect is that he is a deliberate romantic, that he has thrown himself upon the drudgery of his life as an act of deliberate sacrifice. His vocabulary and strange phrasings bolster the theory that his career means little to him, that it's just a shield for a passion that lies somewhere hidden. The averted eyes and faculty snubs do not touch him, because the world inside the old building is not the real place he lives and works; it represents nothing of him and he wears it like a cowl, an inverse Scarlet Pimpernil. I think he looks out from behind his bleared blue eyes with the knowledge that he is a child with his face pressed against the inner surface of a transparent cell, and I think even as the old face sags, he smiles.

my enemies

It's an absurd thing about the world that a person, however harmless, accumulates enemies. For me, the last two weeks have been a bumper season of new additions to my enemies list. It seems like new ones accrue every day, like pollen falling on the windowsill, or boxelder bugs congregating on my door this November. In fact there may be some gothic connection with the boxelder infestation and my new enemies.

This morning I was walking, alone, in a transition area between the bucolism of a suburban neighborhood, and the hell-like strip of a busy street. I have an embarrassing tendency to walk in the very center of the road. It's not something I mean to do, I just find myself there--I suppose it's the place to walk that I enjoy the most, if there are no fresh fallen leaves to drag my feet through along the sides. Occasionally I was passed by a car of some sort, usually one of a gentle, meandering disposition.
Whatever can influence an individual in the apparent sanctum of their vehicles seemed not to be present in any of the drivers I had observed heretofore. But abruptly, this rule was violated, when a black pathfinder appeared behind me, and drove by me in a strange way.

I was reminded of SUV commercials in which the driver must quickly avert some emerging danger, usually caused by lightening. The driver of the black pathfinder drove in just such a way this morning as one might use to avoid a boulder, or a deer, crashing or leaping unexpectedly into the roadway. But I had been plainly walking in the road, in its very center no less, for hundreds of yards. It was hard for me to understand how the nearness of the truck as it passed could have been anything but intentional. Then it occurred to me, even before my pulse rate had regularized, that I probably had another enemy on my hands.

As things happened, traffic was congested enough that I eventually passed the truck again on foot. The model was new, and clean, and the windows had some kind of platinum tinting like one sees on the visors of certain vintage model space-suits. It was simultaneously nauseating and pleasing, a very neat job of something I associate with crushing banality, and poor aesthetics. Inside the cab there was an unbathed woman, or so it appeared at a glance. She was abusing sportswear, in the sense that she was patently unfit, but was nonetheless wearing sweatpants, as though they could be worn for the purposes of fashion.

I think she had been dreading my approach, and when I arrived, she wasn't totally unprepared. Her face turned towards mine, and her lips manifested a Pressly-esque snarl, which I intuited was meant to be dismissive, indicating that she knew just what I was about to say. That she might have known was entirely possible, but I was baffled as to how she could act wrongly with such idiotic insouciance. Her radio was blaring, so I was in the position of needing to speak very loudly and distinctly-- I ended up sounding like one of the MC's at a boxing match. At first I only offered only advice. But as she refused to entertain my comments, turning her head away, my suggestions became less reasonable, and more vitriolic. By the time traffic shifted and we parted ways, we had indubitably become true enemies. My heart sank slightly as I realized that this was the case, since everyone wants to keep their enemies to a minimum.

Every time I make a new enemy, my mind is unsettled. This is disturbing for anyone, but for me it is especially irritating, because it prevents me from being able to do my job--mathematics--while walking. For me to calculate in my mind I need a very relaxing setting, one in which my conscious awareness of the real world fades to black, and brightly colored symbols can be manipulated. Any kind of confrontation freezes me in the real world, and it can before hours before I'm able to slip back into my little black domain.

Later, I was reading The Sheltering Sky, the scene where Port dies of typhoid. I was affected by the descriptions, and the stress of acquiring my new enemy, and I started to experience a twisting feeling in my own entrails, weirdly similar to what the character was experiencing. This became steadily more painful, and eventually I was actively controlling myself to prevent others, seated around me, from noticing my discomfort. It seemed like someone was tossing my intestines with salad tongs, and the pain would come so suddenly and with such unexpected ruthlessness that once or twice I grimaced and flinched before I could steady myself. I began to visualize tendrils of cancer spreading out into my viscera and coiling around my lower spine like wands of hemlock.

What this episode means, I have no idea. It was one of those situations in which I was feeling no anxiety that I was conscious of, but anxiety was still present, as if it were an objective property of my body of which I could be unaware, like its specific hardness, or precise temperature. It attends me in a different aspect, this physical response, every time the register of my temporal foes increases, as if it were a daemonic warning against making adversaries. Still I seem unable to stem the flow, and experience suggests I could make a new enemy at virtually any moment.

This relates only my most recent enemy; there are others, much worse, who I have to encounter on a daily basis. Cheating students. The guy from the cafeteria, who performs a taunting dance akin to the Locomotion when I come into view. All of these things make my life more difficult, and I wish they could be changed.

Alpha Hydroxy

I have a thesis adviser.

I met him today, in his office, where he sat, thinking. He is approximately 40, with long lank white hair and a predilection for obsolete basketball shoes. On this afternoon he wore faded blue jeans and a purple golf shirt, somewhat unkempt at the collar. What he was actually doing would be difficult to say, though there was a notebook in his lap and a few hieroglyphs on his personal chalkboard.

I knocked tentatively on his door.

There was a time when I feared to interrupt his thinking, both for fear of losing his personal approbation and the inadvertent disruption of an idea being woven. On several occasions in our relationship he has declined to own my presence, as he focused his energy on himself, merely holding up a tangerine sized rubber ball in answer to my inquiries.

Today he smiled, as he does more often of late, having developed the unfortunate impression that I am capable of stimulating him in the least way intellectually. You could say I've been lucky lately. Thus there was a certain performance pressure as I stepped into his lair, at his slow behest. His desk was covered in feet of paper, as is usually the case, and he showed me three collaborative papers he was writing, each with its own notebook, stolen from the teaching supplies canteen.

He fumbled for a moment through one of them. After a moment he came to his notes, which I was flattered to discover contained a refutation of a proof I had presented in seminar the previous week.

"Was that your proof?" he asked, watching me with a pugnacious light in his eyes.

"No...It's Pollard's," I said. In truth it was my interpretation of Pollard, or an inchoate idea of his, which I had tried to phrase precisely.

"It's wrong," he said with palpable satisfaction in his polka style Wisconsin enunciation. "I'm almost glad it's not yours then."

He proceeded for the first time to show me a proof of his own during one of our scheduled weekly meetings. As he went along with his demonstration , he expounded a little on why the other argument was wrong. He used "wrong" in its aesthetic sense and gesticulated vaguely at the board to indicate what kind of structure the true argument required.

"I just knew there had to be this...splitting that wasn't in the other one. You said this is practically Sauer's lemma."

I waited as he watched me. Grasping that a response was advisable I answered Yes.

"Yeah, well." He shrugged and waved at the board.

I had the feeling of watching someone who can find water through divination, or a dog who can detect brain cancer by scent. I pursed my lips and nodded approvingly.

A few minutes later I left, and he returned to his mental safari. On the way back to my office I felt relief and loser's regrets.