π

I've posted a couple of past columns concluding that humans, ultimately, are not as rational as we like to think ourselves to be.  What we are, I believe, is pattern seekers, and that's the basis of intuition.   You suddenly realize that one thing is like another--like Newton's realization that the falling apple and the earth's moon were bound by the same force, or Philo Farnsworth's inspiration for television from watching a till in the soil.  It's a sudden insight, an epiphany.  And fundamentally, it's irrational. 

But that brings me to another kind of irrationality:  irrational numbers.  Belatedly, I learned that March 14 is known as π day.  π, the lower case Greek letter pi, is equal to 3.14159265359…, and March 14 or 3/14 which is an estimate of π to two decimal places has been designated Pi Day. 

π is an irrational number.  It just goes on without repeating for ever and ever. 

There are lots of irrational numbers.  As a matter of fact there is a famous, elegant proof by mathematician Georg Cantor showing that the set of irrational numbers is larger than the set of rational numbers. That's a lot of irrational numbers considering there are an infinite number of rational numbers.  Cantor showed there are different sizes of infinity.  But that's another Mindfingers post.

π is perhaps the most famous example of an irrational number, and almost certainly the first one practically used.  π is simply the number of times a diameter goes into the circumference of a circle.  That is, if you took the diameter, D, in the circle below and wrapped it around the Circumference, C, it would go just over three times—π times to be exact.  

Circles, of course, are important, now as in antiquity.  The moon is a circle.  The wheel is a circle.  π was the magic number that would turn straight lines into circles.  It was known back as far as the Babylonians and Ancient Egyptians.

Despite the fact that π has been proved to be an irrational number, this hasn’t stopped people from trying to patterns within it. It has been expanded to around ten trillion decimal places, last I checked.  People are forever counting how many 5's there are, or where certain combinations of numbers appear.  There are oodles of “sacred geometry” websites professing spiritual revelation out of the understanding of π and other important irrational numbers.  

Of course, since π just keeps going, it does have the exact phone numbers, consecutively and in alphabetical order, of everyone who has read this post.

But in fact, π is not without pattern.  If it's written in its decimal format, it looks that way, but there are other ways to write numbers.  As an infinite series for example:

Looking at that, π appears quite elegant actually.  It's an accurate representation of π, but it's not very efficient.  To get π to 10 measly decimal places, you'd have to expand that series to 5 billion terms.  There are other infinite series that converge on π much more quickly, but they lack elegance of the above.   This one converges very quickly, but isn't quite as nice to look at:

Messy.  Looks impressive though, right?  And it's hypogeometric.  That sounds important!

The above mess was the work of Srinivasa Ramanujan, a self-schooled brilliant mathematician from one of the poorest parts of India in the late-19^th / early-20^th century who, with practically no formal training in mathematics, wrote to Cambridge professor of mathematics G.H. Hardy about his work on infinite series and continued fractions. Hardy, after reading the unsolicited work Ramanujan, was completely out of his depth.  He famously said that the theorems in Ramanujan's treatise "must be true, because, if they were not true, no one would have the imagination to invent them."  Ramanujan went on to a storied tenure at  Cambridge University and made phenomenal contributions to mathematics.  He died at 32, probably in no small part due to chronic malnutrition and disease as a youngster.

pic of Ramanaujan.

Srinivasa Ramanujan:  "An equation for me has no meaning, unless it represents a thought of God."

Another way of articulate a number is through continuing fractions, where π can be represented as:

These numbers, like humans, may be irrational, but they certainly aren't unintuitive.  They have patterns that can't be seen in a simple decimal expansion.  The patterns can only be seen when you include infinity--such as infinite series or continued fractions.  Rather pretty patterns, and restoring my faith in the underlying elegance of the universe. 

Without getting too geeky (I know, I know--too late), one of the most beautiful equations in mathematics is known as Euler's Identity. I've even seen a tattoo of it.

Here, e is the natural exponent.  If you've ever done much work on growth of bacteria, half-life of radiation, or compound interest, you may have come across it.  It's the "base unit" if you will, of exponential growth or decay.  And, like π, it's irrational.  2.718281828... 

The term i is the square root of -1.  But, of course, negative numbers can't have square roots which is why i stands for "imaginary number."

So you take one irrational number, e, take it to the power of another irrational number, π, times the imaginary square root of negative one, subtract 1 (known as "the multiplicative identity" in mathematics) and you get 0 (known as "the additive identity" in mathematics).  It ties together a number of disparate mathematical terms and concepts in an equation of breathtaking eloquence and simplicity.

Let's close off with a little brain teaser.  What's the solution to the infinite series below.  Answer next time.

Six Things You Didn't Know About Octopuses

I’ve covered a lot of topics on this meandering blog over the last year or so, but it seems that I have inexplicably not written a post on octopuses, nor can I fathom how I could have made such a glaring oversight.  Whaddup wi' dat?

As someone who has difficulty coordinating the motion of a mere four limbs. I’ve always had immense respect for the octopus, who must manage eight.  

You’d think, extrapolating from the above, that I would therefore revere millipedes, but in fact I find them rather dull.

Anyways, without further ado, six things you didn't know about octopuses...

1.  The plural of octopus is not octopi.

In the Latin, regular masculine nouns end in a –us when they are singular.  When you pluralize the noun, you use the suffix –i.   In English, we’ve sometimes kept that for words derived from the Latin, and thus the plural of focus is foci.  Or stimulus and stimuli.   Or cactus and cacti.  Or how about this one:  What do you call a mushroom with a nine inch stem?  A fungi to go out with.  Ba da boom!  I’ll be here all week, folks.  But octopus is derived from the Greek, not the Latin (oktopous, meaning “eight footed”) .  Following Greek declensions the plural of octopus would correctly be octopodes.  But that would probably just confuse most folks, so the most acceptable plural is regular old octopuses.

This is an octopi.

Actually, that reminds me, it's March 14 today (3.14) so happy Pi Day everybody!

So the next time you are in the midst of discussing cephalopods and some pedant tries to correct you when you say "octopuses" by telling you the plural is octopi, you can actually out-geek him, snort derisively and proclaim loudly, “Pshaw! Clearly you are unaware that the etymology of the word is Greek, not Latin.”  Then send him to the corner to try to figure out the plural of doofus.

The adjective for octopus is correctly octopodal, as in “Cthulhu’s face is octopodal.”  But I think octopussy would be way cooler, as in “This squid tastes kind of octopussy.”

2.  Octopuses Could Be Aliens

Octopuses are pretty weird looking--gelatinous, translucent, boneless masses, all giant head and tentacles, and a beak with a tongue that has teeth built into it.  If you want to make a really freaky looking alien, as a matter of fact, you’re well off dropping a little octopus in.  H.P. Lovecraft was perhaps the first to clue in to this.  Lovecraft is a writer from the early 20th century of cult status who founded the "formless dread" school of horror.   He describes the indescribable Cthulhu—the Thing That Should Not Be—as “A monster of vaguely anthropoid outline, but with an octopus-like head whose face was a mass of feelers, a scaly, rubbery-looking body, prodigious claws on hind and fore feet, and long, narrow wings behind.”  Lovecraft even once sketched Cthulhu—apparently taking a dump. 

"Hey, Ineffably Abominable One, how about a courtesy flush?"

Then there’s the Daleks, the arch-enemies of Brit eccentric time-travelling demigod Dr. Who.  These creatures look like tin cans with an egg whisk and a toilet plunger sticking out of them, and strike terror across the universe despite their seeming inability to climb stairs.

But what does the inside of a Dalek look like? 

Yup, definitely octopussy, I think you’d agree. 

How about this alien?  Ring a bell?

Octopussy, with a hint of arthropod.

3.  Octopuses have been granted honorary vertebrate status

Octopuses are thought to be the cleverest of all invertebrates (i.e. animals with no backbone), demonstrating the ability for both long-term and short-term memory.  They can solve mazes.  They use tools.  They can screw open jars.  They can solve second-order non-linear differential equations. 

They are such clever and inquisitive little critters that in England they have designated “honorary vertebrate status.”  That might not sound like much to you and me, but, with respect to experimenting on animals, it means you can’t dissect them while they are still alive and stuff.  Something I’m sure the octopuses appreciate, anyway.

4.  Octopuses Have an Odd Intelligence

We know that octopuses are smart, but smart like Dustin Hoffman in the Rain Man.  There’s some formidable intellect at work there but it is very different than ours.  There is an active debate in the biological community as to the intelligence of octopuses, going back all the way to Aristotle, who called the octopus a “stupid creature.” 

The current orthodoxy is that the octopus, having evolved in a completely different world than the one we did, has a completely different intelligence suited to its survival.  One theory has it that the octopus abandoned its shell (which most other mollusks have kept) and suddenly found itself soft, slow and delicious—a fatal trifecta in the Darwinian stakes.  So they developed intelligence as a defence mechanism.

Octopuses have pretty big brains, gargantuan by mollusk standards.  Some half a billion neurons, on average.  Not quite up there with human beans, who have somewhere around 50 billion, but nothing to sneeze at either.  They regularly escape containment in ways that their human captors can’t even understand, except to say that all the fish in the adjacent fish tank are gone and their octopus has gotten fat.  They are regular Houdinis apparently.  One reason is that octopuses can escape out of a hole the size of one of their eyes.

Octopuses also have very distributed brains.  Two-thirds of their neurons are outside their brain.  Some octopuses can autotomize—that is self-amputate—its limbs in times of danger.  Now since the neural net of an octopus is distributed throughout its tentacles, the amputated tentacles actually carries some intelligence with it, so the arm has the “presence of mind,” if you will, to taunt the predator while the octopus disappears in a cloud of ink.

5.  Octopuses can Taste With Their Arms

The suction cups on an octopuses arms (apparently they are not called tentacles.  Whatevs.) are equipped with chemoreceptors that taste what the octopus is holding.

6.  Octopuses are the World’s Greatest Mimics

In comparison to a Mimic Octopus, calling a chameleon a mimic is like calling the guy that flips burgers at MickeyD’s a chef.  I mean, what can a chameleon do?  Change colour slightly.  Big whoop.  The Mimic Octopus, like the chameleon, uses chromatophores to change colour (Chromatophiores are cells that contain pigments that can be released through muscular control, thereby effecting camouflage).  Octopuses have a palette of four colours to choose from.  But octopuses also have iridophores, which reflect light, effectively making them just about invisible in a marine environment.  Not only that, they can control the texture of their skin to match their surrounding environment, be it coral, rock or sand.  Not only that, but the Mimic Octopus can change its shape to look like a rock or a predator, such as a sea snake or lion fish.  Ah, the advantages of being boneless!  And not only that, but they also change their behaviour to mimic the predator they are impersonating.  Now that is camouflage, baby! 

See the video below for a flabbergasting example. 

So that’s octopuses.  Pretty darn cool.  They're slimy.  They can change colour and shape.  They can become invisible. They’re smart.  They can escape through a hole no bigger than their eye.  They are ...(ahem) ... well-armed.  They can squirt ink.  They can self-amputate limbs which become almost fully autonomous mini-octopuses themselves, and then grow the arm back later.  They eat sharks.  They have extremely powerful tentacles that can actually taste you, as well as paralyze you with venom so that you are perfectly aware, but powerless as you are inexorably drawn to its central feeding beak where they drill their tooth-lined tongues into you to suck your guts out.

Cool critter.

Ruminating on the Black Plague While Washing Me Socks

Well, here I am in Mississauga, just outside of Toronto washing my socks in the sink and drying them on the radiator.  Air Canada managed, somehow, to lose my bag for more than 24 hours on a direct flight from Vancouver to Toronto.  But they’ve authorized me to spend $50 on toothpaste and stuff.  And—here’s something that might be useful for you at some point—many credit cards, including mine fortunately, have insurance on them.  I get $500 coverage for delayed baggage, so at least I can buy myself a shirt, some pants, a warm shirt and some skivvies.

In other news, the hotel charged me $3.50 for a to-go coffee this morning.  Are you guys kidding me?

So as I’m sitting here watching my socks dry on the radiator I thought I’d grab the paddles, yell “Clear” and try to bring this damn blog back from the dead.  We’ll see if the it gets the heartbeat going again or just spikes the EKG before eternal flatline.

It’s not my fault; just my nature.  I’m easily distracted by shiny baubles and a dozen lives stretch out behind me like so many jackets tried on and cast aside.  Adam the Good Student, Adam the Wayward Teenager, Adam the Bartender Party Animal, Adam the Engineer, Adam the Anarchist, Adam the Musician, Adam the Government Worker, Adam the Writer and Poet, Adam the Caver, Adam Goes to Afghanistan, Adam the Boss, Adam the Sporty Dad.  So Adam the Philosophical Blogger, was destined to join the scrap heap with the rest of the outfits I’ve tried on in search of Me.  

Maybe it's like Kurt Vonnegut said, Be careful what you pretend to be, because you are what you pretend to be.

But enough about me, what do you think about me?

I just finished The Emperor of all Maladies: A Biography of Cancer, a Pulitzer Prize winner by Siddhartha Mukherjee.   If you want to know how to pronounce his name, click the link below:

Mukherjee  is an oncologist, so he knows whereof he speaks.  No, I don’t have cancer.  Or, rather, I may have, but I’m not aware of it.  I just have an enduring fascination (keeping in mind that “enduring” for me means “greater than one year”) with the mechanics of the human cell.  I’ve written previously on the subject, specifically on the fatal relationship between aging and cancer (Mindfingers: Immortality and Cancer).

The book was very good on many levels—the diabolically difficult task of trying to conquer a disease caused by one of your own cells going rogue; the politics of the great tobacco company battles of the 60s and 70s; the hopes of so many “cures” dashed, like radical mastectomies and chemotherapy; the bleeding edge of genetically engineered drugs.

I was especially interested in the sections on medieval treatment of cancer, though.  They had some funny ideas back in the Old Days.  Not “funny ha-ha” but “funny-that’s-so-gross-I’m-going-to-hurl-my-supper.”  One of these ideas was that the body was made up of four fluids, or humours:  blood, phlegm, yellow bile and black bile.  Any illness was brought about by a surplus or deficiency of one or more of these humours.  No less than Hippocrates (of Hippocratic Oath fame) popularized this idea, and its sway over medicine continued until the mid-19^th century when the theory of cell pathology became popular.  Apparently, the idea predated even ancient Hoppocrates and can be traced back to the Mesopatamians (Humorism).

Mesopatamian.  The only thing harder than that to say drunk is “No really, I shouldn’t sing karaoke.”

Anyways, enough humour, back to the humours.  Too much blood brought about inflammation and fever.  Of course, many wounds and diseases of the day resulted in inflammation and fever.  These people were commonly bled.  This resulted, I’m sure, in a higher mortality rate than the non-bled but nobody ever seemed to notice.  One of the reason the bubonic plague of the 1400s was so bad in Europe was that the wretched victims were bled.  In the Muslim lands, boils were lanced which actually helped. Perhaps occasionally, some gormless traveller from the East would mention that the Muslims seemed to have a better handle on this thing, but then he’d be tied to the stake and burned alive as a heretic. 

With respect to the Black Plague, the Christians and Muslims had one thing in common though—they both blamed the Jews.  Quel surprise! Meanwhile a group of particularly observant Christians, the Flagellants, were literally thrashing themselves  to death daily with iron-studded whips to purge themselves of the sin that caused the plague.  And, not counting the God of Abraham’s plagues,  catapulting plague-riven corpses into walled cities is perhaps history’s earliest example of biological warfare.  Anyways, the bottom line is that the Christians, Muslims and Jews were all killing each other, and the Black Plague was killing all of them indiscriminately.  See what I mean about funny.  OK, that’s a little dark, but I have no change of clothes for tomorrow and I’m washing my socks in the sink here.  Anyways, where was I?

One of the interesting things is lingering impression medieval medicine has left on the English language today.  A bad balance of humours led not only to physical pathology but also affected moods and behaviour.  Too much phlegm made you sluggish, and today we have the adjective phlegmatic, meaning “unemotional and calm.”  Too much yellow bile put you in a foul mood, and today we have the word bilious meaning “unpleasant or ill natured.”    Blood was associated with courage and hope, thus the adjective sanguine (Old French for “blood”) meaning “confident and optimistic.”  Temperament, from the Latin tempere, “to mix”, was the balance of the humours.  And even temperament was a good balance.  Similarly, if your mix was good at some point you were said to be “in good humour.”

And what of black bile?  It only caused two disorders.  One of them can be guessed from the Greek term for black bile, melan chole: depression.  The other—cancer.

My socks are dry.