Scenes from the novel XIII

Clarissa and Emily had already walked quite a ways, hand in hand. After a while the scenes of devastation around them, appearing through the swirling mist, became like a sort of dream. From time to time there would be a subtle shift in the landscape, like a trick of the light. What Emily first took to be a mountain would turn out to be a bank of clouds, and then the clouds would be gone. The suffering faces peering so intently out at them were the worst – they seemed to be everywhere – but they were hard to bring into focus. Every time Emily would try to look at one directly, it would turn into a knot of wood, or a leaf, or something else. It was as though the faces didn’t want to be seen in their suffering – they just wanted you to know they were there.

Emily wanted to ask Clarissa where they were going, but she knew better. All she needed now was the constant touch of Clarissa’s warm firm hand, a reality that never wavered. The mist was beginning to grow thinner, and the landscape was changing far less often. Now they were on a long dirt road, and to the right and left appeared to be waving fields of wheat, stretching away as far as the eye could see. Even though the landscape was flat all around, there was something wrong with the horizon up ahead, like it was shaped wrong. Emily looked up questioningly, but Clarissa merely smiled a reassuring smile.

At last they came to it – the Great Precipice. Of course Emily had heard of it, in whispered stories and the occasional snatch of song, but she hadn’t really believed in it. The world can’t just stop, not just like that. In its way it was rather beautiful, the vast scale of it, on the other side of a razor-straight line at their feet where the road and the wheat fields abruptly ended – where everything ended.

“Now, my dear,” Clarissa spoke aloud for the first time since their journey had begun, “you mustn’t be alarmed. One expects to feel a certain sense of vertigo when encounting things of infinite proportion. Try to remember that we are merely at a door, and one door is very much like like any other. Some doors lead from the kitchen to the parlor, others from hallway to drawing room, whereas still others lead from one frame of existence to another. This doorway happens to be one of the latter kind. My dear, you must try to remember that a door is, in the end, merely a door.”

This speech, as intended, did much to reassure Emily. “I can deal with a door,” she thought to herself, “I’ve been opening doors all my life – including a few that were never supposed to be opened.” She smiled up at Clarissa, ready for whatever came next.

“Ok, I get it,” she said, “It’s a door. But where’s the doorknob?” Clarissa laughed in response, and waved a hand over the abyss. A shimmering white circle appeared at their feet, and even before they jumped, Emily could already smell a familiar scent – something between lilac and cinnamon, but not quite like either.

She smiled and took a deep breath, as she held on tight to Clarissa’s hand. This was going to be a fun year.

Villagers with torches

I had an odd experience recently. I was invited to a dinner at a nice restaurant, part of an awards event, with a number of fellow scientists – scientists from a variety of fields. It was the sort of thing where everybody puts on a jacket and tie or the equivalent.

There was nothing out of the ordinary about the evening until the moment when somebody heard me quietly conferring with the waiter about what vegan options there were. At that point one of my tablemates picked up on the fact that I was ordering vegan, and the next thing I knew it was the topic of conversation.

At first the questions were friendly, although it sort of weirded me out that what I eat would become such a focus of discussion. When asked why I was eating vegan, I explained that I would just as soon not cause suffering if I could avoid it. Soon the conversation became markedly less friendly, although the tone was always polite. There was a lot of focus on scallops. “Why don’t you eat scallops?” “Do you think scallops would mind?” And I remember thinking how strange it was that somebody would want me to eat scallops, just to make them feel better about what I eat.

At some point I realized that my simply being there, an outed vegan, was creating a kind of dissonance at the table, and it probably wouldn’t matter what I said. Eventually I managed to steer the topic of conversation to something else. But I was left with an odd feeling by the aggressiveness of it all. I hadn’t questioned anybody else’s eating choices. I was just sitting there minding my own business and trying to have dinner, and suddenly people I didn’t even know were trying to talk me into changing my ethical beliefs.

I feld a bit like a well meaning Frankenstein monster who had foolishly wandered out of the castle and into town, prompting the villagers to dutifully pick up their torches and give chase. Except that these particular villagers were fellow scientists.

Thinking about it afterward I was struck by some particular questions raised by the experience. If I had given one of a number of culturally acceptable non-answers, nobody would have thought to question anything I might say. For example, I could simply have said: “Of the eight sacred Sutras of the Patanjali, the first is Yama, which teaches to do no harm to one’s fellow creatures.” This would have identified me as a Buddhist, which would have protected me.

In other words, if you say you are doing something for religious reasons, unless you’re going around killing people or trying to have sex with their underage children, then you get a free pass. “Oh, you’re a Buddhist,” would be a typical response, whereupon everybody would nod thoughtfully and respectfully, while wondering whether they can still score tickets to the Dalai Lama’s talk next month.

And yet if I had said that, I wouldn’t have been giving a more substantive explanation – I would just have been adding that I’m a member of a recognized group. And so it occurs to me that we are putting ourselves in a kind of danger when we build our ethics from individually thought-through conviction. Identifying yourself with a large and powerful spiritual tribe, such as “Christian” or “Buddhist” or “Southern Baptist” confers upon you a measure of protection from angry villagers with torches.

The situation is somewhat analogous to the way large corporations like IBM, Microsoft and Apple accumulate patent portfolios. Because they have each built these extensive Intellectual Property shields around themselves, they don’t need to worry about getting sued by each other. But if you, dear reader, try to go up against any of these giants as an individual inventor with nothing but one clever patent to your name, they can shut you down like a book.

What I’m talking about here is not one specific ethical choice, but rather any ethical choice that has led you to wander away from received wisdom after you’ve worked some things through in your head. If you don’t have an army to back you up, you’re not always going to have an easy time of it.

But I don’t care, I prefer doing things this way. And I’m not about to let a few pesky torches keep me out of yonder village.

Benjamin Button

I expect many things from big Hollywood blockbuster films with A-list actors. Cool special effects, wonderful charisma from the stars, hopefully a story that hangs together. What I don’t expect is true profundity. I don’t expect literature.

But last night I saw “The Curious Case of Benjamin Button”, and I was completely unprepared for its emotional and philosophical depths. The conceit, based on a short story by F. Scott Fitzgerald, is simple (I’m sure you’ve heard it by now ad nauseum): A man is born old, and goes through his life growing progressively younger. Along the way he meets his great love, a woman who ages through life normally.

One would think that this would be a great opportunity for some neat set pieces and some cool computer graphics effects, and one would be right. This film has all that. The computer graphics are even almost good enough – which is quite a trick, given that 2008 is still a little early, tech-wise, to try pulling this kind of thing off seamlessly.

But the film, as written by Eric Roth and Robin Swicord, and directed by David Fincher, ends up not really being about any of that. It’s a truly heartbreaking meditation on mortality, the limits of love, and the inevitable space between human beings. Starting with the simple trick of a main character who goes through life the wrong way around, the film continually forces us to address very difficult questions about life that we have all become very good at ignoring.

I find now, a day after having seen it, that the more I think about this film, the more questions it raises in my mind. For one thing, it illuminates the great twin mysteries of childhood and old age. We generally deal with these mysteries through euphemisms – our culture routinely puts children and old people into reductive categories that fail to capture the enormous enigma of those states of being. The mind and personality of a child come from seemingly out of nowhere, as the child gradually grows, before our eyes, into a fully unique individual, unlike any other on the planet. And then at the end of one’s life this process is strangely reversed – the gifts of life gradually taken from us a little at a time, until the day when birth itself is bookended by its opposite.

We develop entire religions, as well as elaborate taboos and tricks of language, to avoid looking at the terrifying power of this process full in the face, and yet it is the central mystery of our lives – along with the elaborate dance in which adults find themselves between these two bookends, as our sexuality draws us together to participate in this connection with the infinite.

We are so used to seeing all this that we no longer see it – it all devolves into a set of symbols so ubiquitous that they have become reflexive icons – love songs and slow dances, baby clothes and wheelchairs, trappings of this strange arc that we simply take for granted.

But the genius of “Benjamin Button” is that it forces us to look at this process – really look at it – the way René Magritte forced us to look at everyday objects as though seeing them for the first time.

And once we are made to look in this direction, we begin to see things. For example, we see that from his early twenties until his early sixties a man is a member of a club – he is accepted into that great society known as adulthood. He can take a job, have sexual partners, take responsibility for children and find within himself a way to create meaning from his life. But the time before this and after this are out of bounds – they are the bookends. Society does not look at the boy or the old man as a member of this club. Privileges and freedoms that are taken for granted by the adult in his long prime are not granted to those who fall outside of this roughly forty year window.

Yes, technically a man in his seventies has all of the rights as a man in his forties or fifties, but there are subtle differences – and others that are not so subtle. In most cases he is not seen as relevant in the same way, not perceived as an agent within the world. Rather, he is gently shunted aside, somewhat the way society shunts aside its children, with the message “you are not one whose place it is to act upon the world, but one whose place it is to be taken care of.”

And like adult life itself, love too has its bookends. Mysteriously we fall in love, and just as mysteriously we may fall out of love. We have no idea how long we have, only that our love has experienced a birth and a gradual maturing, and that at some point it might become lost to us. In the day-to-day we tend to forget that each day of being alive and in love is a miracle, a miracle that we can hold in our grasp for only a measure of time, before it must be relinquished.

I have found all these thoughts and many more rolling around in my head since having seen this film. And isn’t that what good literature is for?

Aesthetic flow

People enjoy looking at things that have symmetry and order – but not too much. Beauty requires an interplay between pattern and chaos. I touched briefly upon this in my October 10 post about a misguided use of computer software to “beautify” the human face by making its features more regular (and therefore more bland).

But I wonder, as I look at a snowflake, or a leaf or sunset or candle flame, or in fact, the face of someone I find lovely to behold, are there principles at work here? Just as Mihaly Csíkszentmihályi spoke of humans being happiest when in a state of “flow” in which things are neither too easy (ie: boring) nor too difficult (ie: frustrating), perhaps there is an equivalent state of aesthetic flow, in which the things we perceive are neither too regular/symmetric nor too chaotic/asymmetric.

There is plentiful evidence that people respond positively to artful assymmetry within a symmetric structure. The genius of great composers from Bach to the Beatles is clearly entwined with their ability to surprise us, to bring a melody or harmonic progression to some wholy unexpected place, while somehow making it all sound right.

Case in point: the second Beatles song that Paul McCartney ever wrote was “I’ll Follow the Sun” (the first was “When I’m Sixty Four” – he composed both songs when he was only sixteen). By the third note of the melody – the flattened E atop an F7 chord (at the word “you’ll” in the lyric “One day you’ll look”) – he has already broken the rules. Right off the bat the melody jumps clean out of the key of C to god only knows where. But he then uses the momentum from that crazily asymmetric choice to launch a lovely and unforgettable tune that ends up sounding not just right but inevitable.

It’s one of those moments that heralds a new kid on the block, a fresh new talent, like Bobby Fischer at the tender age of thirteen sacrificing his queen in his famous game against Donald Byrne – and thereby ensuring a stunning upset victory. That game was outstandingly beautiful because it was outstandingly unexpected, in addition to being brilliant.

On a much more humble scale, I embedded controlled chaos in one of the first computer graphic objects I ever synthesized – a marble vase. I was drawn to the sense of capturing a raging storm within the placid curved surface of a classically sculpted form, and I developed a whole set of techniques that would allow me to express such controlled chaos:


I wonder whether there is any way to calibrate this relationship – to find some formal measure of chaos versus symmetry in any given situation, and then use that ratio to predict a rough measure of potential beauty?


For some reason this morning I remembered a snatch of a phrase I had not thought about since childhood – “Niagara Falls. Slowly I turn. Step by step, inch by inch…” Some of you will recognize this as a line from very funny scene in an old Three Stooges film.

The humor of the Three Stooges was squarely aimed at little kids, with a precisely calibrated transgressiveness rivaled only by “Mad Magazine”, that sublime liberator of curious young minds. There was a point in my early childhood when the humor of the Three Stooges, endlessly repeated on daytime TV, could hold me completely rapt. I realized even then that their brand of take-no-prisoners slapstick was not at all the sort of thing responsible parents would want a kid like me to watch. Which of course made it twice as much fun.

I remember that when I was about six years old and my brother was eight, one of us had merely to utter the words “Niagara Falls” for both of us to collapse helplessly into a paroxysm of hysterical giggles.

This being the twenty first century, I responded to this memory as would almost any normal modern citizen: I googled it. Turns out it was from a 1944 Stooges episode called “Gents Without Cents”. The sketch was originally written and performed by the great Joey Faye – you may also have seen it done by Abbott and Costello.

But what really strikes me about this is how many people have contributed on-line to the minutiae of this and other obscure long-ago artifacts of pop culture. For example, a great many commenters have noted that Larry Fein flubbed a line during the “Niagara Falls” sketch, at one point saying “inch by inch, step by step”, rather than the other way around.

But you can also see that for yourself – just click on the image below to watch it on YouTube (and I dare you to try not to giggle):


Taken all together it’s astonishing how much the internet has become an enabler for learned Talmudic commentaries about the Three Stooges, “Knight Rider”, “My Mother the Car”, “December Bride”, old toilet paper commercials with Mr. Whipple, Chevrolet ads with Dinah Shore, and so many other artifacts of our shared history of popular culture that were intended in their own time to be no more than entertaining ephemera.

It seems that we are collectively digging deeper and deeper into the arcana of this last half century or so of commercial entertainment, laying everything bare, putting it all on YouTube and Wikipedia, hungry to know everything there is to know about everything.

As we collectively rub away at pop cultural history with a scrub brush, we seem to trying to be more Proustian than Marcel Proust.

Perhaps, as technology advances, our reach will go further and further. New forms of reconstructive spatio-temporal imaging will emerge, and then everything that has ever happened in our culture can go up on the Web.

You’ll be able to look up exactly who attended your ex-lover’s birthday party that time in 1994, or what your mother had for lunch on the day that your dad proposed.

You’ll be able to do a simple search to see all the outtakes from all the Beatles songs, whilst zooming in to study McCartney’s fingering on that tricky bass guitar part.

You will even be able to find out what Bill Murray whispered to Scarlett Johansson.

And then perhaps technology will advance even further, when we finally perfect forensic extracranial neural holography. Then you’ll be able to look up exactly what O.J. was thinking while he was trying on the white glove, or what thoughts flitted through Lee Harvey Oswald’s mind the moment he pulled the trigger.

You’ll be able to find out whether that pesky little bulging box taped to George W. Bush’s back under his suit jacket during the 2004 debates was making him uncomfortable, or whether he was just annoyed by that voice in his ear telling him what to say.

Don’t get me wrong. I am glad that I can find out where to go to relive my moments of childhood glee when watching Moe, Larry and Curly doing “Niagara Falls”. But I wonder how far we will go. Or rather, how far we are willing to go.

And I wonder whether Marcel Proust, wherever he is, is laughing or weeping.

Emo porn

I recently saw, for the first time, “Dan in Real Life”, a 2007 romantic comedy starring Steve Carell and Juliette Binoche. The set-up is simple: He meets the woman of his dreams, and then almost immediately discovers that she is his brother’s new girlfriend. Emotional mayhem, family weirdness, and some seriously acute embarassment ensue.

I was struck by two things about this film: (1) Every single thing about it is almost maniacally predictable – every line of dialog, every camera shot, editing choice, casting decision. Scene after scene unfolds with fresh new comic humiliations for our sad-sack hero, and we see every one of them coming from a mile off. (2) In spite of all that, I enjoyed it immensely. And I found myself trying to figure out why something like this is so damned enjoyable.

Well, for one thing, Steve Carell plays it perfectly. His Dan Burns is a man who has known what it means to be a Loser – he knows the Universe is against him in some deep karmic way. Any move that he makes, even the slightest flutter of an eyebrow, will instantly result in some fresh new divine retribution. You can see the controlled rage beneath the sad-sack exterior, the way he has learned, through long acquaintance with this harsh universe, to stifle every impulse, smother any emotional expression that might give him away. But he cannot avoid these trials. He must continue to go through time and space in some way, and as he does, he knows that he will be bitch-slapped by fate at every step.

And yet… and yet… He has met Juliette Binoche, and so he has already won. Director/co-writer Peter Hedges is careful to let you know at every turn that these trials will indeed lead our hero to the promised land. The way the heroine looks at him, talks to him, even the way she gets angry at him, all clue you in that she too has found the true partner of her soul.

And that’s what makes it all so much fun: We’re being invited to join a high wire act with a safety net. In a sense we, the audience, are Steve Carell’s character – we see everything from his point of view, and we identify with him at every turn. And since we know this (because the film is well-enough made to subliminally clue us in), we know it’s all in fun.

It’s an ersatz theatre of cruelty that is really a theatre of fun. A genre I call “Emo porn”. All such films have a number of traits in common. For one thing, the sad-sack antihero can do absolutely nothing right – even decisions that seem perfectly sensible turn out to be wrong – except at the moment then he meets The Girl. Not just any girl – she has to be The Girl. As soon as he meets her he enters a state of grace, and the Loserness falls magically away long enough for her to see his true self (which the film has been careful to let us see all along).

I’m describing so many movies here that there wouldn’t be any point in trying to enumerate them, so I’ll just mention a few. One of my favorite examples is John Favreau’s 1996 “Swingers”, a film for which he wrote the screenplay and played the romantic loser hero. In that film the moment of grace begins the instant he lays eyes on Heather Graham’s character across a crowded bar. The transformation is so cleanly done, so expertly conveyed, that it’s as satisfying as watching the evil lady lawyer in “Erin Brokovich” fearfully put down the glass of water without drinking it – a moment of triumph that cleanly divided that film into Before and After. You know exactly what I mean.

And I claim that this is a kind of pornography. Emotional pornography. Not that there’s anything wrong with that.

We’re being invited to join in a consensual illusion that we are here to witness a deep exploration of the human heart, thoughtful delineations of character, family, love and relationships. But that’s not really what’s going on. What’s going on is that we are being presented with a cartoon version of romance – the hero is such a loser that it makes us wince, but nonetheless he meets the ideal woman – beautiful, loving, understanding, and completely in love with him – and the filmmakers convince us that this is his true fate.

Which of course appeals to the ugly duckling in us all. Everyone harbors the secret belief that within their sensitive misunderstood soul there is a beautiful swan waiting to burst out, to take glorious flight. If only they could find someone to release them from their lonely prison.

Now I’m not saying that it’s always Emo porn. “Dan in Real Life” is pornographic in the sense that it’s not really trying to do anything else, other than to jerk our chain this particular way. The entire film is a thin excuse for the money shots – the meet-cute encounter with the one true perfect soul mate who redeems our hero’s life, and of course the happily-ever-after ending.

“Swingers” incorporated that same dynamic, but it was also so much more – a delightful and brutally close to the bone dissection of puffed up young losers on the fringes of the L.A. film scene. Of course the hero had to find the perfect girl – that’s what the form calls for. But that was only the form, not the content.

Similarly “Wall-E” incorporates the same emo-porn structure – sad-sack romantic-dreamer loser whose life is redeemed when he meets The Girl – but there’s so much more to it. After having seen Wall-E and EVA perform their particular flight of love, I’m sure I will never look at a fire extinguisher again in quite the same way.

And then there is the rare film that takes it all to sublime extremes, that understands our hunger for emo-porn and uses it to play with us like a cat plays with a mouse. I’m thinking in particular of “Punch Drunk Love”. P.T. Anderson follows all the rules, but he does so in an utterly original, surreal, head-spinningly deconstructed way. Think about it: if that were the only film in which you’d ever seen Adam Sandler, you’d think he was a great actor.

Now that’s an accomplishment.

After the solstice

It’s hard to keep track of how many civilizations have claimed the winter solstice – the shortest day of the year – as a lynchpin of their religious ceremonies. It is a wonderful irony of human existence that the longest night – the very epitome of darkness – is almost universally recognized as an annual symbol of rebirth.

It seems to be built into us, on some deep level, to view our darkest hour as the herald of a coming dawn. I suspect that this ingrained response, a continual search within the darkness for the harbingers of hope, has contributed mightily to the survival of our species.

In some sense this quality is echoed in the sort of feeling I see around me in our country today. On the one hand, industries are failing, unemployment is on the rise, belts tighten nervously throughout the land, while eyes glaze over in dull disbelief at a seemingly endless war. There is a feeling that we have reached some sort of nadir.

And yet these feelings go hand in hand with a wave of excitement in many parts of the country about the incoming Obama administration. There is work to be done here, an uplifting to be had after years of deep systemic failure at the top. Every time our president-elect makes another sensible cabinet appointment, each time he signals a willingness to reach across the partisan aisle come January 20 (something Bill Clinton famously neglected to do in 1992-94) you can sense a nation gaining confidence that there is serious intent here, a deep commitment to prudently steer our great ship of state out of the dangerous waters of its current crisis. It won’t be easy.

When thinking about such things, it may be useful to remember that the holiday our nation celebrates today is an historical amalgam of opposites. The celebration of a prophet of peace – a renegade rabbi from Nazareth – became merged with the Viking Yul celebration of Thor – the hammer wielding god of thunder himself. The cute song we now sing to the twelve days of Christmas had its roots in the time it took – up to twelve days – until the last angry fires of the mighty Yule log, symbol of the war god, had burnt out.

Our lives are built of opposites, as surely as dawn rises from the darkness of night. The brutal priorities of the Bush administration with its hammer of war, Guantanamo, extraordinary rendition, illicit NSA wiretaps of you and me, body bags of American teenagers returning home to weeping mothers in enforced secrecy, the massive sell-off of the wealth of American children not yet born, these all laid the groundwork for what is to be.

A centrist government is coming in, and has already begun assembling itself into shape as an agent of economic growth, and of a more inclusive and less frenzied kind of capitalism. Each individual citizen, each family, every child, no matter their background, is a potential agent for growth and prosperity. It’s a simple concept – one we’ve had in this country for many years, yet recently somehow lost amid the crashing and splintering din of Thor’s hammer.

It seems that the Winter solstice has passed, and Spring is coming. I, for one, am looking forward to it.

Not infinite, but really big

After all of this talk of infinity, I think it’s time to come back down to earth and talk about finite things. The question came up recently about things that people make, and how big or small they can be. I started wondering what is the biggest – or smallest – object ever created by people.

One candidate for smallest object is the IBM logo that Don Eigler and Erhard Schweizer created in 1989 with the tungsten tip of a Scanning Tunneling Microscope. They used the STM to position 35 xenon atoms, one by one, on a smooth nickel surface. When they were done, the xenon atoms were arranged into the company’s famous logo:

The whole thing was just six nanometers across – four million of them laid side by side would be only about an inch wide.

My candidate for largest object ever created is an ingenious sculpture that David Barr built in 1985. He placed four tiny tetrahedra (three sided pyramids) at equidistant points around the Earth – one each in Easter Island, South Africa, Irian Jaya (New Guinea), and Greenland – and there they stand to this day. Collectively they form a perfect tetrahedron about the same size as the planet Earth itself.

So far I think that’s as big as anyone has ever gotten. Jaron Lanier has told me that he wants to start a project to arrange entire stars into artificial constellations, as a kind of shout-out to alien races across the Galaxy. Needless to say this would be a very long-term project – some tens of thousands of years – and would require immense effort to succeed.

What I like about David Barr’s achievement in 1985 is that he created his sculpture pretty much all by himself. Just one person – a man with a dream (and a travel budget) – ingeniously converting our entire planet into a scaffold for his monumental sculpture.

By my calculations, the difference in size (from end to end) between the Barr tetrahedron and the Eigler/Schweizer IBM logo is about 2,000,000,000,000,000 – or about two million billion.

For some reason that makes me very happy.

Making a list, checking it twice

The other day VRBones commented that the discussion about ever larger sizes of infinities sounds like a playground argument:

1: You’re an idiot!
2: Am not!
1: Are too!
2: Am not, not!
1: Are too, too, too, too, too!
2: Am not infinity!
1: Are too infinity times infinity!

Far be it for me to argue with the wisdom of children, but it turns out that in additon to describing how infinities can be different, another of Cantor’s insights was how to say when two infinite sets are the same size. Basically, they are the same when you can match up their elements one to one.

This leads to some pretty crazy sounding but absolutely correct conclusions. For example, the number of even numbers (2,4,6,8,…) is the same as the number of counting numbers (1,2,3,4…), because you can match them up one to one:









Now we get to where VRBones quotes his schoolyard mathematician friend: “Are too infinity times infinity!”. It turns out that even if you multiply an infinity by itself, you just get the same sized infinity. Here’s how Cantor showed this to be true for the counting numbers:

Making a list

If you multiply the set of numbers 1,2,3,…∞ by itself, you get a two dimensional block of row/column pairs:

















Clearly this forms an infinite set. But here’s the question: Can we number the things in this set, using the numbers 1,2,3,…? If we can, then the set is no bigger than the set of counting numbers.

Cantor showed just how to do that:

















I’ve used colors to highlight the way he went through the list along the diagonals, column by column. In this way you can assign a number 1,2,3,… to every pair of numbers in ∞×∞. In other words, infinity times infinity is just infinity!

Checking it twice

Cantor also showed that the real numbers times the real numbers is no bigger than just the real numbers, as follows:

Consider the real numbers from zero to one. This is the same as considering all points along a line segment from zero to one:

If you multiply the infinite set of real numbers between zero and one by itself, you get all possible pairs of such real numbers, like (0.73245.. , 0.26347…). But this is just another way of talking about all points in a square:


The first number describes left-to-right position of a point in the square, and the second number represents top-to-bottom position.

I know, you’re probably going to say it’s crazy to talk about a line segment and a square having the same number of points. But Cantor came up with a very easy way to prove that it’s true.

Just take any point in the square, say (0.73245.. , 0.26347…), and weave together their decimal coordinates into a single number, like so:


I’ve alternated every other digit color so you can see how the two original numbers each contributes its digits. You can do the sme thing for any pair of real numbers to get a single number.

Furthermore, you can also always go the other way: For any single number you can think of, like 0.3141592653…, you can make a pair numbers out of it by separating out its even and odd numbered digits – in this case: (0.34525… , 0.11963…).

Since you can match up their points one to one, you know that there are just as many points in a line segment as there are in a square.

In other words (yet again) infinity times infinity is the same as infinity.

VRBones, you might want to try telling that to those kids in the playground the next time you see them. But beware – they just might surprise you: If you explain it clearly, you might find that they understand exactly what you are talking about.


Since the invention of gravitic field cancelling plasma, there has been quite a rush to make something really portable. Even ten years ago it would have been hard to get sufficient CPU power into a compact design, and so even if you could have done the gravity canceling, you wouldn’t have been able to keep the thing stable.

But now of course it’s a piece of cake. Pouring GFC plasma onto a counter-acting gravitational field actually generates power, and you can tap right into the current to power a forward thrust drive. Of course it’s all coming out of the Earth’s gravitational field, and I suppose that one day the rotation of our planet will slow down enough to be noticable. But I don’t think we need to worry about that quite yet!

Here are the first mock-ups, all to spec. The induced gravitic field just below the lift module is not as dangerous as it looks, although I still wouldn’t suggest spending all day in there:

The purplish hue in the plasma field is just an artifact of resonant photon absorption in the red/green part of the spectrum. You’ve probably read about U.V. blindness in people who spent too much time near those first prototypes. Not to worry, it isn’t really a problem anymore. The newer models are all properly U.V. shielded.

What I really like about this shot is how you can see the plasma spread out as it is literally pushed down against the Earth’s gravitational field, almost like it’s hitting a wall. The vehicle essentially rests on the induced force cushion. You need to use traditional means for forward thrust – in this case a simple magnetically accelarated ion air jet – because there’s no gravity difference between the front and the back of your hoverboard, so the GFC plasma is useless for lateral movement. If you used only the plasma, you could get up in the air, but you’d pretty much be stuck sitting in the same spot all day.

I like this shot, looking almost straight up at the vehicle, because you can see the striations in the plasma field. It turns out that the pulsing is necessary – the whole lift thing works only when the plasma gets into the right resonant frequency, almost like surfing waves of gravity from up off of the planet:

Riding one of these suckers is easier than it looks, since the computer feeds back your shifts in weight and makes small motions of the hoverboard to compensate. It’s pretty much impossible to throw yourself off – and I think that’s a good thing.

When you’re riding it properly, you’ll want to have your rear foot turned sideways, resting on the smaller of the little black pads. Your front foot should rest on the larger black pad, turned at more or less a 45o angle. You don’t use your hands at all to steer. The pressure imaging sensor in the top surface essentially captures a real-time video of the pressure from all parts of the bottoms of your feet, and uses that info to move the hoverboard.

Once you get the hang of it, it feels pretty much like an extension of your body. You kind of just think “lift” and it lifts, or “forward” and it starts to gain speed. Of course it’s not really reading your mind – just the ever-changing pressure image being formed by the bottoms of your feet.

After all, there’s no such thing as a vehicle that can read minds – that’s just science fiction.