A Pi Day vi Trick

It is Pi Day, and the nerd Code of Honor means I am contractually obligated to post something pi related today. In past years, I told you about David Boll's discovery that pi is in the Mandelbrot set. Unfortunately, that exhausts LabKitty's knowledge of amusing pi factoids (March 14th is also Albert Einstein's birthday, but you already knew that).

The other difficulty is, arriving so close on the heels of World Eigenvalue Day, the WED hangover is still lingering come Pi Day morn. Which is kind of a buzzkill both figuratively and literally, what with motivation waning to scour troves for a topic I might parlay into pagehits from civilians feigning interest in the mathogonical arts one day a year (if all y'all really want to help support the mathogonical arts, get your gorram congressperson to start properly funding the NSF).

But I digress.

Which brings us to vi.

Let's Make Gershgorin Circles!

Of all the gaps matriculating from Circle Pines U left in my psyche -- how to knot a cravat, the difference between good flan and bad flan, world history before the 17th century or US after the 19th -- failure to even mention Gershgorin circles stands as a particularly egregious transgression.

In sophomore year we deconstructed Descartes' Meditations, this in the guise of a philosophy graduate student opining on the tangible gustatory effects of imagined peanut butter sandwiches. Perhaps it was a metaphor for sexual arousal, which would have made for more spirited recitation albeit risking pointed inquiry from the CP Chamber of Commerce, who already did not much care for our fancy book lernin'. Alas, whatever his intent, it was lost on us.

In junior year we discussed Lucifer State, a novel of dystopian future I rather enjoyed but is no longer in print, or so the Internet tells me. (Such not being the damning praise it once was, what with the Amazon self-publication machine Heutzutage delivering any provocatively-titled finger barfing -- Moist Vibrations, Debbie Does Due Diligence, Triggered -- to the tenterhooked browser.) Mostly I remember the prof lectured wearing sunglasses, which on more than one occasion concealed a black eye.

Then in senior year there was Herman Melville and the Forms, which was a comparative literature course. Unless that was freshman year, then it was a punk band. I think. TBH, the whole experience is something of a blur. I took classes in economics and FORTRAN, racquetball and contract law, geology and differential equations. I'm not entirely sure what my major was.

Someone could have devoted a lecture to Gershgorin circles, is my point. As I hope I have convinced you, the remainder of my education would not have suffered.

Besides, Gershgorin circles are muy cool.

LabKitty puts the E in E=mc^2

The Muggles call Einstein the Father of the Atom Bomb. We, their intellectual superiors, know better. When someone (probably a reporter from E!) first confronted Einstein with the prospect that E = mc^2 seemed to make possible the release of prodigious amounts of energy and maybe not in a good way, Albert famously retorted: Daran habe ich gar nicht gedacht! If I recall my high school German, that's Hochdeutsche for GTFO. On another occasion, Albert opined that splitting the atom was akin to shooting at birds in the dark in a place where there aren't very many birds. Basically: it's not possible and if it is I never thought of that also.

Einstein was in good company. Ernest Rutherford --- the guy credited with first splitting the atom -- went to his grave thinking it was impossible. (Clarification: Rutherford believed the large-scale release of nuclear energy was impossible. Doing so in the laboratory was fair dinkum.) Enrico Fermi -- the guy who built the world's first nuclear reactor -- told the US military an atomic bomb would be as big as the sun. It would take a World War and the threat of a Nazi bomb to whip the physics establishment into a frenzy sufficiently Promethean to steal the nuclear fire. This gave us Hiroshima and Nagasaki and the Cold War and what all else. The crisis de jour is a nuclear ambitious Iran. One might point out Iran was kept well in check until the Bush administration decided to whack Saddam Hussein for reasons that were never explained beyond "shut your cakehole." But I digress.

Bubbles, and the Mind of God

All the world's a stage, Shakespeare tells us. Which is wrong of course, but that's hardly his fault. Shakespeare died the year John Wallis was born, English mathematician extraordinaire working in the area we would later call calculus forty years before Newton. Or at least poking around the suburbs, until Newton's bright star and what can be politely described as an unhealthy obsession with preeminence identified the true stage the world is built upon, eclipsing all who came before, during, or after.

For as any engineer can tell you, all the world is really constrained optimization. Ya makes the best ya can with the whats yous given. A universal truth operating in every institution known. Suspension bridges to airplane wings, microchip design to antibiotics, universities to investment banks. If you want to succeed, you take the cards you're dealt and you do it lightest, you do it fastest, you do it cheapest.

Fine words, and if words were all that is needed we already had Shakespeare. But we don't need words; we have calculus.

The Magic Coin Game

Weirdness abounds in probability (and I mean the kind of probability that can be understood by regular folk, not the kind that involves things like measure theory. That kind of probability also contains weirdness, but it is only understood by mathozoids who I imagine use some sort of proboscis to acquire nutrients when the rest of us aren't looking).

Perhaps the most famous of all probability weirdness is the Monty Hall problem, named after the eponymous host of the '70s game show in which it appeared. Briefly: you are shown three doors. Behind one door is a good thing (a car, money, Claire Danes' underpants). Behind the other two doors are bad things (a goat, a beating, Hitler's underpants). You pick one of the three doors. Monty shows what's behind one of the doors you didn't pick. You must decide to stick with your original selection or switch to the door Monty didn't show you. Monty then opens your door and shows you what you have won.

The famous result is that switching improves your chance of getting the good thing. That seems redonkulous, but careful analysis shows it to be true.

Explanations of the Monty Hall problem have turned up everywhere from AskReddit to Parade Magazine. If you are holding a probability textbook (and why wouldn't you be?) it's probably mentioned in it somewhere. I won't discuss the solution here (see: the Google avalanche returned by searching on "monty hall problem"). Rather, the MHP got me thinking about other odd fish that lurk in the dark sea called probability. Other weirdnesses proffered by Lady Luck, her strapless evening gown and feminine jowels calling us forth, seeking to separate us from nous and paycheck.

I am fortune's fool! Romeo exclaimed. Yes you are, Mr. DiCapricorn. And you better keep your paws off Claire or there's going to be trouble.

I call this weirdness: The Magic Coin Game.

e^iπ + 1 = 0, usw.

Mathematics is a poem that has God for its subject. Thusly might we adapt Petrarch's aphorism to our Harsh Mistress. And the poem in the title above is often cited as the Most Beautiful in all of mathematics by people who do that sort of thing. (Although let's be honest: Everybody picks this equation because Richard Feynman did.) It includes all of the fundamental constants (0, 1, e, i, and pi) and all four fundamental mathematical operations (addition, multiplication, exponentiation, and equality) and ties it up in a pretty package with a sheen of the inscrutable (unless you took a course in complex variables, raising anything to an imaginary power makes about as much sense as a Mozart opera).

Inscrutability is nice and all, but if you are looking for mystery there's far queerer things to be found in mathematics (and I don't mean Leonardo Da Vinci). Addition, multiplication, exponentiation, and equality are to mathematics as paste-eating is to your public school education: something most of us move beyond by the the time puberty shows.

Let's see if we can scare up a more exotic beauty.

Pi in the Mandelbrot set

With Pi Day upon us, I am obligated by the Nerd Code to post something relevant to the festivities. Since all the low-hanging fruit has been picked (yes, yes, March 14th is also Einstein's birthday, hoopty-doo) I thought I would tell you about the connection of pi to the Mandelbrot set. This was apparently discovered in 1991 by David Boll at Colorado State University. A description of it appears in Peitgen's big honkin' fractal book, but otherwise the story remains not-so-well known. (Then again, what do I know what is and isn't well known in math circles.)