Kalil Azad's Mathematical Mindset

Thanks to the increasingly hive-like mind of 43 folders, I landed on this essay about how to think about math on Kalil Azad's excellent BetterExplained. (As we are now in the midst of the era of humees and wudzups, domain names composed of good ol' real words are such a breath of fresh air.)

I honestly wonder how I never came across this guy before. I really like his style, keeping each point short enough to swallow and emboldening the high phrases.

In the above-linked essay he conveys the tricky business of mathematical meaning and how it simultaneously lives in and transcends real-world examples. One skill that makes good mathematicians good at mathematics is the ability to jump between the specific and the general, the concrete and abstract, with tremendous agility. It allows one to see the pathological edge cases without drowning in multiplicity.

In case of power outage

In the EECS building on the Berkeley campus...

Am I fascinated by scale because I'm a Libra?

One of the greatest gifts given us by mathematics is a language for scale. It's how we know that 1,000 is different from 1, which is in turn different from 0.1.

One of my earlier memories is of attending this captivating exhibit on the limits of scale as understood thirty years ago. Those extremities have not been pushed back much further since then. Amusingly, it was only last year that I found out that it was produced by one of the most significant design houses of post-WWII America.

Someone (sadly uncredited, working for Nikon) has created an interactive work with the same lessons. Everyone should learn what's out there, what's up there, what's in there, and where we live in it all.

Continuous and discrete

This photo is more than two years old. You can tell because in spite of the huge grin on my face, the crow's feet near my eyes are still relatively small.



Thanks to C. Dewey for finally getting around to making these public.



And in case you were wondering: Yes. Yes we do.

Let T be a shirt...

I used to think I needed this shirt.



Now I realize that I need this one.



LOL rights reserved the respective creators, images used with their respective (implicit) permissions.

On the rise

As a counterpoint to his miserable formula from a year ago, R. Stevens gives a metric for quality of life.

I'm very happy to have had both input variables move in the right direction recently. More sleep, less time in a car!

Delinquency

For those distraught by the dearth of mathematically themed writings in this space, may I suggest my shared items in Google Reader, easily accessed either by the first of the above links, or in the sidebar of this page, entitled "And what have I been reading?"

My distractions from these posts have not been all work. I've written before about the great strides being made in human-computer interaction, especially regarding tools for musical creation. Last Saturday night I had the extreme pleasure of seeing Björk (or B. Guðmundsdóttir, for the sake of nomenclatural purity) perform at the Shoreline amphitheater, with M. Bell at the helm of a reactable. This instrument, first pointed out to me by J. Hopper (who is inexplicably nigh invisible to Google), is similar to the audiopad from MIT's media lab, but with a crucial difference: it is out of the prototype stage, and in front of a mainstream audience. Hopefully commercialization is not too far off.

Since when is 'B' a number?

It is finished

My adviser, A. Schilling, has just finished a paper extending my first result, proving my conjecture on the matter and thereby tying up the last loose end left by my thesis.

E8, AIM, and PR

I was very happy to hear the voice of M. Vazirani, who served on my dissertation committee (see the evidence below) on last week's episode of KQED's California Report. It's a great little piece, accessible to all. You can download it, or stream it from the page linked above.

In some sense it seems perfectly natural for a radio station whose call letters are a common mathematical initialism (or is it a physical one?) to report on mathematical news. As I mentioned before, this story in particular is catching a lot of attention. At a conference I attended last week, the topic of E₈ and its broad media exposure arose; I attributed this phenomenon to the fact that this is one of the first major milestones to be reached by an AIM-sponsored project, and that AIM puts a much greater (and much needed) emphasis on public relations than do other mathematical research institutions.

In the meantime, I found another connection between my own work and the E₈ project. The big character table calculation was carried out on SAGE, a multi-processor AMD Opteron machine built by Western Scientific, who also built my multi-processor AMD Opteron machine, frost. Hopefully some adaptation of the adage on great minds thinking alike can be applied to these systems.

I've received a variety of junk messages ...

which is not ideal.

I recently found a message labeled as spam with "On nullstellensatz" as its subject. This isn't the only time a math term has turned up in this context–earlier today I was treated to "so mcclain go matroid"–but it is the most sophisticated language I've seen in the Pynchon-esque maelstrom of unsolicited commercial email.

If they really want to reach the algebraic geometry crowd, they should go beyond the relatively pedestrian terms nullstellensatz and matroid. Imagine eye-catching phrases such as "Resolve your too small singularity NOW!" or "Satisfy infinite polynomials with finite basis".

I'm a bad geek

Unlike last year, I did absolutely nothing to celebrate π day this year. I can at least send out a post facto link to the (boing)²-ing of Stanford's L. E. O'Neal.

I still prefer my proposed finger-play. It's also the binary representation of 503 (i.e., 111110111), making it especially appropriate for PDX'ers.

Build character!

A few research institutions and news outlets, as well as some blogs have reported on what is generally being termed the "mapping" of E₈. This is about as good a layman's description of this result as can be given, although for those who know a smattering of group theory, it would be better to say that the fruit of these labors is a table relating the conjugacy classes of E₈ to each other. Seeing as there are 453,060 such classes, this requires "writing down" 205,263,363,600 numbers; these are enough to cover Manhattan, and they require 60 times more storage than the human genome, as the researchers and the reporters to whom they've spoken are fond of pointing out.

I can't help but notice the striking resemblance between some of the pictures from the atlas project and the output of one of my projects. Granted, you can't use it to generate E₈ crystals, but then again, it's just running on a simple webserver.

And by the way, if you or anyone you know would like to take over the development of this software, please contact me. This was the start of something really good, but other concerns have taken center stage since then.

Would it help if I drew a flow chart?

Such great things come out of MIT's media lab: one of their visitors has built a water-based computer.

This could come in handy in the event of a massive electromagnetic pulse. However, I suspect most people would have other matters on their mind at that point.

In all seriousness, this is a wonderful project. Like other mechanical algorithm devices, it allows the fundamental procedures of computation to be animated in space, relieving the burden of mental modeling from the executor.

But remember, if you're seated in the first six rows, you may get wet during this subroutine.

Gang sines

R. Stevens models his forthcoming design, featuring both a transcendental number and an earthly delight:



With right thumb and left ring finger extended, he'd be repping the circle with proper digits.

Hyperbolic frolic

I had the tremendous pleasure today to attend an exhibit of works by M. C. Escher in my hometown. It was an incredible treat to see so many familiar images, a few of which have adorned my walls, in one place. There were a number of surprises, too: lovingly crafted scenes of Italian buildings and staircases, and a series of early woodcuts with biblical subjects. One of these even had the reversed initials so common to an artist's early ink prints.

I was most struck by the technical prowess that went into his prints. Take "Circle Limit IV", which not only illustrates the peculiar geometry of the hyperbolic plane, but does so with a thought-provoking two-color woodcut. Few in history had the exacting vision required to create interlocking patterns with such precision, while at the same time communicating so much personality. The exhibit includes two "segment proofs", prints of one third of the work. Taken out of the context of its two triplet siblings, the details that Escher chose to omit as he moves toward infinity are brought into sharper relief, making his genius all the more apparent.

Also consider the hauntingly evocative "Rippled Surface", the studies for which include perspective drawings of concentric red and black circles to indicate the local maxima and minima of the wave as it propagates across the water.

Rhythm of Illusion remains at the San Jose Museum of Art through Sunday, April 22, 2007. If you are anywhere near the south bay before then, I urge you to attend.

Over my head

I looked up, stretching my neck, and saw this lovely ornament hanging above my cubicle.

I thought this one goes up to eleven

One of the more intelligent sorts of mathematical questions I've been asked are those on the nature of higher dimensions. Somewhere in our pop history it was decided that since we lived in the third dimension, there should be others, like so many arrondissement. These are usually just three-dimensional spaces where things are wacky, not the fundamentally larger spaces they ought to be. In fact, some authors believed they could exchange dimensions for vowels.

It's at once refreshing and disappointing to see an exposition such as this. The animation and sound are lovely, and a rather decent volley is made at the concepts of "splits" and "folds", even if the subtle beauty in the relationship and differences between these is largely ignored. However, the narrator suggests that each "dimension" has some inherent parameters defining it, such as "the seventh dimension" being the space of all possible outcomes from the origin of the universe, and similarly pinning the notion of "split" and "fold" only to certain dimensions.

Do you want to see ten-dimensional space? Here it is:

Anyway, isn't supersymmetry supposed to be eleven-dimensional? Or does 11-D SUSY refer to a women named Susan with an impossibly small ribcage?

Shaken to the (multi) core

Catherine Crawford, chief architect for next-generation systems software at IBM Systems Group's Quasar Design Center, has made a bold statement about the future of software; as is to be expected, some sources make it sound a little more sensationally apocalyptic.

This isn't the first time I've brought up the many-core future of the desktop. You may be wondering who will have the expertise to write the software that will take advantage of this massively parallel paradigm. Well, that will be just one skill that I'll be developing while a postdoctoral fellow at a very fine institution.

Let the CHI flow

For the purposes of this post, that's Computer-Human Interaction, not the Chinese concept of life force.

I made my way to PARC again last Thursday to hear a great talk about how the landscape of electronic entertainment is changing thanks to developments in HCI. It really says something about my interests that so many of the topics that T. Blaine covered were already familiar to me. Off the top of my head, these include: Guitar Hero, Karaoke Revolution, the DS, the Wii,
D'CüCKOO, audiopad, and Jeff Han's multi-touch interface. What ties her interests together amongst each other, as well as with mine, is how custom hardware can facilitate musical creation through intuitive human manipulations.

Of all the above, only Karaoke Revolution uses the single most intuitive human tool for sound, the voice. One of the more interesting projects underway in the realm of speech-and-song control of computers was covered in this space earlier; unfortunately, this wonderful creative tool is still not publicly available.

However, an even greater voice-controlled application has just been released. Have you ever heard a song on the radio, but not caught the attribution, only to have that catchy hook running through your head, leaving you wishing that you knew who wrote the song? Thanks to midomi, you need wonder no more! Just sing into your browser and find all the covers of "Fly Me to the Moon", or who's done that "Doo-wah-ditty-ditty-dum-ditty-doo" song. They're still in "invitation only" beta, but not to worry; if you want me to get you past the velvet rope, just let me know in the comments.

Non-orientable garments

Every mathematician worth their salt knows of The Acme Klein Bottle Company, although I'm one of the few I know who has made a purchase there; specifically, a question mark as a wedding gift to the former proprietors of a Seattle restaurant. (Some are more familiar with the proprietor as the narrator of one of the world's first cyber-espionage stories.) The business has diversified since my order five years ago, having extended to include knit wool hats with zero volume. Now absent from the site (although I swear they were there the last time I looked) are matching Möbius band scarves; fortunately, this need is filled by other resources on the web.

A new foray into the realm of one-sided wearables (brought to my attention by the pillar of hipster science) is the Möbius shoe, pictured below. Of course, to make the outfit work, one would need a matching Fortunatus' Purse.

Synchronicity

The other day, I took the time to read S. Yegge's latest essay on software, complex systems, and consciousness; the thesis of the work, an idea that Steve has been refining for some time now, is that "the most important principle in all of software design is this: Systems should never reboot." He gives numerous examples of software products that fail to incorporate this principle, and a few that provide a weak, half-hearted attempt at it. He then goes on to explain that given his (quite reasonable) definition of software, the best systems have this idea built into their core:

So my first argument against rebooting is that in nature it doesn't happen. Or, more accurately, when it does happen it's pretty catastrophic. If you don't like the way a person works, you don't kill them, fix their DNA, and then regrow them. If you don't like the way a government works, you don't shut it down, figure out what's wrong, and start it back up again. Why, then, do we almost always develop software that way?

To celebrate turning 30, R. Stevens created a new t-shirt design (which is now available). Imagine my surprise to find this in his official announcement:

We all have pretty much the same personalities we were born with, just earlier versions. Our software never really gets rewritten, it just evolves.

Rich, Steve, allow me to introduce you to each other.

...should you choose to accept it...

If you've ever been bored while sitting at a computer, you've probably killed a few hours flipping over square tiles, hoping not to upturn a land mine. This polytopal variant pulls the old stand-by out of the plane and pushes it into the world of combinatorial spheres.

I especially like the dramatic "mission failed TRY AGAIN!!" and "mission completion!!" messages that close each level.

Touch you I

If you're reading this, you must have heard Tuesday's big news. Even if you weren't following it on twitter or the web, you found out soon enough, either from a major news outlet or by reading any of the thousands of blogs discussing it.

That number is no exaggeration: as of this writing, technorati lists 83,622 posts, and Google Blog Search returns about 86,006 results from the past three days.

If you've been following this space long enough, you've already seen some aspects of the interface to this device. The "pinch" zoom paradigm looks just like the light box tool that J. Han's lab put together, so you can proudly tell all your friends that you knew about this whole multi-touch UI way before Apple brought it into the mainstream, which suddenly made it so much less cool.

But that gloat leaves you open to one-upmanship, as I just found out that J. Han was not the first to implement such gestural control of a computer. Over at Microsoft Research, A. Wilson had a working demo of TouchLight in late 2004. If anyone knows of an earlier claim to this concept, please let me know; otherwise, cheers to A. Wilson's ingenuity and creative spark!

While proper credit is due the originator of any idea, it is equally important to note the astounding progress each of these iterations accomplished. The product from the Courant Institute took some great HCI ideas and built a robust interface library and scalable hardware around them, inviting software developers to join in the fun. Now, Apple Computer Inc. has managed to pack that technology into less space than a satisfying meal, along with wireless transmitters, a camera, an accelerometer, etc., and to make it all as attractive as we expect Apple products to be.

It is crucial to remember that the well executed embodiment of an idea is itself a creative task, often no less challenging than the development of the original idea.

Oh do I now, Bert? Or should I say "Guntar"?

Ever since I completed my Ph.D., this family of spam has become all the more amusing.

Even if I did, wouldn't I want it from someone who can spell "degree"?

Fwd: You Need a Better Dergee,{} and we can Help!

From: (270) 818-XXXX Bert <293guntar@xxxxxxx.net>

In jus tas little as 2 *weeks* you can have a masters degree from a national university.

A better job, more income and a better life can all be yours in less than 2 weeks.

No books to buy, no classes to go to, and no entrance exams.

Learn in your own home at oyur own pace. We supply all the study materials, all you have to do is apply! Everyone is accepted!


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A perfect cube

For those born in 1979 (and that includes me and a great many of my friends) last year was a big one. We're 27, or 3×3×3, or 3³, and it's probably the last time we can count our years in the form nⁿ, although let's hope not the last perfect cube.

Speaking of which, it's so easy to forget how low an odometer can read:

It was a major year for me in a number of other ways as well. About a month after adding three letters to my name, I bought my first new car. It has many more cubes to go–in fact, its first scheduled maintenance isn't until after 46³ miles.

But back to the number at hand. Our time spent being 27 is split between the year just past and the year just begun, and as has been the case for the past seven years, our age bears some resemblance to the number at the top of the calendar; just drop the zeros, and there it is. But this year there's another connection, albeit one that is purely numerological and not at all mathematical.

When I took a course in algebra from K. Ribet, he mentioned an algebraist who every year published a list of all the groups with order equal to the year. One of my classmates pointed out that in certain years, this would be a rather short list; for instance, in 1979 only one group qualified. This year isn't as dull as all that, but it is manageable, since 2007=3×3×223. For instance, I know from one of Prof. Ribet's homework assignments (specifically, Problem 28) that all such groups are solvable and have at least one normal Sylow subgroup.

But what I like, even though I know it's nothing but a quirk of our base-10 notation, is the typographical similarity between the prime factorizations of 27 and 2007, 3×3×3 and 3×3×223. That the products differ by two zeros and the factorizations differ by two appearances of the numeral 2 makes the numerological aesthetics all the more appealing. I know it's not math, but it sure is pretty.