Hand-in-glove

One of the more fascinating bits of biochemistry is the odd "handedness" (technically called chirality) that a lot of biological molecules have.  Chiral molecules come in a left-handed (sinistral) and a right-handed (dextral) form that are made of exactly the same parts but put together in such a way that they're mirror-images of each other, just like a left-handed and right-handed glove.

Where it gets really interesting is that although the left-handed and right-handed forms of biologically active molecules have nearly identical properties, they aren't equivalent in function within living cells.  Nearly all naturally-occurring sugars are right-handed (that's where the name dextrose comes from); amino acids, on the other hand, are all left-handed (which is why amino acid supplements often have an "l-" in front of the name -- l-glutamate, l-tryptophan, and so on).  Having evolved with this kind of specificity has the result that if you were fed a mirror-image diet -- left-handed glucose, for example, and proteins made of right-handed amino acids -- you wouldn't be able to tell anything apart by its smell or taste, but you would proceed to starve to death because your cells would not be able to metabolize molecules with the wrong chirality.

Chirality in amino acids [Image is in the Public Domain courtesy of NASA]

Molecular chirality was used to brilliant effect by the wonderful murder mystery author Dorothy Sayers in her novel The Documents in the Case.  In the story, a man dies after eating a serving of mushrooms he'd picked.  His friends and family are stunned; he'd been a wild mushroom enthusiast for decades, and the fatal mistake he apparently made -- including a deadly ivory funnel mushroom (Clitocybe dealbata) in with a pan full of other edible kinds -- was something they believed he never would have done.

The toxic substance in ivory funnels, the alkaloid muscarine, is -- like many organic compounds -- chiral.  Naturally-occurring muscarine is all left-handed.  However, when it's synthesized artificially in the lab, you end up with a mixture of right- and left-handed molecules, in about equal numbers.  So when the contention is made that the victim hadn't mistakenly included a poisonous mushroom in with the edible ones, but had been deliberately poisoned by someone who'd added the chemical to his food, the investigators realize this is the key to solving the riddle of the man's death.

Chiral molecules have another odd property; if you shine a beam of polarized light through a crystal, right-handed ones rotate the polarization angle of the beam clockwise, and left-handed ones counterclockwise.  So when an extract from the victim's digestive tract is analyzed, and a polarized light beam shined through it splits in two -- part of the beam rotated clockwise, the other part counterclockwise -- there's no doubt he was poisoned by synthetic (mixed-chiral) muscarine, not by mistakenly eating a poisonous mushroom that would only have contained the left-handed form.

So specific chirality is ubiquitous in the natural world.  We have a particular handedness, all the way down to the molecular level.  What's a little puzzling, however, is why this tendency occurs.  Not chirality per se; that merely arises from the fact that if you bond four different atoms or groups around a central carbon atom, there are two ways you can do it, and they result in molecules that are mirror images of each other (as shown in the image above).  But why do living things all exhibit a preference for a certain handedness?  It must have evolved extremely early, because virtually all living things share the same preferences.  But what got this bias started -- especially given that left-handed and right-handed molecules are equally easy to make abiotically, and have nearly identical physical and chemical properties?

Well, a paper this week in the journal Advanced Materials may have just answered this long-standing question.  A group led by Karl-Heinz Ernst, at the Swiss Federal Laboratories for Materials Science and Technology, found that the selection for a particular handedness happened because of the interplay between the electromagnetic fields of metallic surfaces with the spin configuration of chiral molecules.

They created surfaces coated with patches of a thin layer of a magnetic metal, such as iron or cobalt, and analyzed the magnetic "islands" to determine the direction of orientation of the magnetic field of each.  They then took a solution of a chiral molecule called helicene, which had equal numbers of right and left-handed forms, and poured it over the surface.  The hypothesis was that the opposite patterns of spin of the electrons in the two different forms of helicene would allow them to bond only to a magnetic patch with a specific orientation. 

So after introducing the mixed helicene to the metal surfaces, they looked to see where the molecules adhered.

Sure enough -- depending on the direction of the magnetic field, one or the other form of helicene stuck to the metal surface.  The magnetic field was acting as a selecting agent on the spin, picking out the handedness that was compatible with the orientation of the patch.

This, of course, is only a preliminary study of a single chiral molecule in a very artificial setting.  However, it does for the first time provide a mechanism by which selective chirality could have originated.  "In certain surface-catalyzed chemical reactions," Ernst explained, "such as those that could have taken place in the chemical 'primordial soup' on the early Earth, a certain combination of electric and magnetic fields could have led to a steady accumulation of one form or another of the various biomolecules -- and thus ultimately to the handedness of life."

So a simple experiment (simple to explain, not to perform!) has taken the first step toward settling a question that chemistry Nobel laureate Vladimir Prelog called "one of the first questions of molecular theology" back in 1975.  It shows that science has the capacity for reaching back and explaining the earliest origins of biochemistry -- and how life as we know it came about.

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The language of Sark

The title of my master's thesis was The Linguistic and Cultural Effects of the Viking Invasions on England and Scotland.  I don't think many people read it other than me and my committee, but it did win the 1996 International Prize For Research With Absolutely No Practical Applications Whatsoever.  And it allowed me to learn valuable information such as the fact that there were two words in eleventh-century England for window -- one from Old English (eagþyrl, literally "eye-hole") and one from Old Norse (vindauga, literally "wind-eye") -- and for some reason the Old Norse one won and our word window comes from it rather than from Old English.

Which is a handy "fun fact" for me to bring out at cocktail parties, especially if I want everyone to back away slowly and then find other people to talk to for the rest of the evening.

In any case, I spent a good bit of my time in graduate school learning assorted random facts about western European linguistics, which was why I was a bit gobsmacked when I found out that there's a language in western Europe that I had never even heard of.  It's called Sarkese, and is only found on the tiny (1.5 by 3.5 kilometers) island of Sark, east of Guernsey in the Channel Islands.

The Channel Islands [Image licensed under the Creative Commons Aotearoa, Wyspy Normandzkie, CC BY-SA 3.0]

Sark is currently home to five hundred people, of whom only three learned Sarkese (known colloquially as patois) as their first language.  It's a Romance language -- the closest relative is French, but it's not mutually intelligible.  It came originally from medieval Norman French via the isle of Jersey; the ancestors of the people of Sark came over from Jersey in 1565 and it's been relatively isolated ever since.

The samples of Sarkese in the article I linked above illustrate how far the two have diverged in the close to a thousand years since it split from mainland French.  "Thank you very much," for example -- merci beaucoup in French -- is mérsî ben dê fê in Sarkese.  French has seventeen different vowel phonemes; Sarkese has over fifty.  Add to that the complication that the island is shaped like an hourglass, with a narrow isthmus (La Coupée) that is all but impassible during storms, and the two pieces (Big Sark and Little Sark) have different dialects.

Fortunately, a Czech linguist, Martin Neudörfl, is trying to document Sarkese, and has worked with the three remaining fluent speakers -- who are all over eighty years old -- and about fifteen semi-fluent individuals to produce a huge library of recordings, and reams of documents describing the morphology and syntax of Sarkese.  "We have hundreds of hours [of recordings] and our audio archive is outstanding," Neudörfl said.  "Even if I were to disappear, someone could revive the language just using the recordings.  We've only achieved this through years of exhaustive research.  It's all thanks to [the speakers] for sharing their knowledge."

It's always sad when a language goes extinct, and so many have done so without anyone ever recording them or writing them down.  In large part it's due to competition with more widely spoken languages; it's eye-opening to know that half of the world's individuals are native speakers of only fifteen different languages.  The other half speak one of the other seven-thousand-odd languages that currently exist in the world.  Sarkese is one of many languages that have fallen prey to the prevalence, convenience, and ubiquity of English.

On the one hand, I get why it happens.  If you want to be understood, you have to speak a language that the people around you can understand, and if you only spoke Sarkese you could communicate with eighteen other people on the island (and one Czech linguist).  But still, each language represents a trove of knowledge about the culture and history of a people, and it's a tragedy when that is lost.

So kudos to Martin Neudörfl, and the Sarkese speakers who are working with him to record this language before it's too late.  Makes me wish I'd tackled a project like this for my master's research.  I could be wrong, but I don't think Old Norse is coming back any time soon.

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Animalia paradoxa

Carl Linnaeus was born in Råshult, Sweden, on 23 May 1707.  His father Nils was the minister of the parish of Stenbrohult but was also an avid gardener, and the story goes that when Carl was young and got upset, Nils would bring him a flower and tell the little boy its name, and that always calmed him down.

The love of botany -- and of knowing the names of living things -- was to shape Carl Linnaeus's life.  Prior to his time, there was no systematic way of giving names to species; there were dozens of names in various languages for the same species, and sometimes several different names in the same language.  Additionally, the fact that this is before the recognition of the relatedness of all life meant that things were named simply by their superficial appearance, which may or may not indicate an underlying relationship.  We still have some leftovers from this haphazard practice, such as the various birds called buntings (from the Middle English buntynge, "small bird") that aren't necessarily related to each other.  (For example, the North American indigo bunting is in the cardinal family; the European pine bunting in the family Emberizidae.) 

Young Linnaeus was lucky enough not only to have supportive parents, but a variety of people who recognized his intellect and ability and nurtured him in his studies.  (Amongst them was the scientist and polymath Olof Celsius, whose nephew Anders gave us the Celsius temperature scale.)  He was primarily interested in botany, but quickly became frustrated with the fact that the same plant could have six different names in six different villages -- and worse still, it was impossible to communicate taxonomic information clearly to botanists in other countries, where the names would have come from their native language.

So he decided to do something about it.

Linnaeus came up with the idea of binomial nomenclature -- the "two-name naming system," more commonly called "scientific names."  Each species would be assigned a unique and unambiguous name made of the genus and species names, each derived from Latin or Greek (which were the common languages of science at the time).  The genus would include various related species.  His determinations of who was related to whom were based upon appearance -- this is long before genetics became the sine qua non of systematics -- and some of Linnaeus's classifications have been revised in the 250-odd years since he wrote his magnum opus, the Systema Naturae.  But even so, the system he created is the one we still use today.

And this is why scientists the world over will know, if you say Mustela nigripes, that you are talking about the black-footed ferret.  (The scientific name translates to... "black-footed ferret."  Just because they're fancy-sounding Latin and Greek words doesn't mean they're all that revelatory.)

So Linnaeus took the first steps toward ordering the natural world.  But what is less well-known is that he included a few animals in his book that are more than a little suspect -- and labeled them as such, illustrating an admirable dedication to honoring hard evidence as the touchstone for scientific understanding.

In a section called "Animalia paradoxa," Linnaeus listed some "species" that had been reported by others, but for which there was no clear evidence.  From the tone of his writing, it's obvious he was doubtful they existed at all, and was only including them to point out that any reports of them were based upon hearsay.  These included the following genera, along with his description of them:

• Hydra: "body of a snake, with two feet, seven necks and the same number of heads, lacking wings, preserved in Hamburg, similar to the description of the Hydra of the Apocalypse of St.John chapters 12 and 13.  And it is provided by very many as a true species of animal, but falsely.  Nature for itself and always the similar, never naturally makes multiple heads on one body.  Fraud and artifice, as we ourselves saw [on it] teeth of a weasel, different from teeth of an Amphibian [or reptile], easily detected."
• Monoceros: "Monoceros of the older [generations], body of a horse, feet of a 'wild animal,' horn straight, long, spirally twisted.  It is a figment of painters.  The Monodon of Artedi [= narwhal] has the same manner of horn, but the other parts of its body are very different."
• Satyrus: "Has a tail, hairy, bearded, with a manlike body, gesticulating much, very fallacious, is a species of monkey, if ever one has been seen."
• Borometz: "The Borometz or Scythian Lamb is reckoned with plants, and is similar to a lamb; whose stalk coming out of the ground enters an umbilicus; and the same is said to be provided with blood from by chance devouring wild animals.  But it is put together artificially from roots of American ferns. But naturally it is an allegorical description of an embryo of a sheep, as has all attributed data."
• Manticora: "Has the face of a decrepit old man, body of a lion, tail starred with sharp points."

A manticore, from Johannes Jonston's Historiae Naturalis (1650) [Image is in the Public Domain]

I've always admired Linnaeus -- like him, I've been fascinated with the names of things since I was little, and started out with plants -- but knowing about his commitment to avoid getting drawn into the superstition and credulity of his time makes me even more fond of him.  He was unafraid to call out the Animalia paradoxa as probable hoaxes, and that determination to follow the rules of scientific skepticism still guides taxonomists to this day.

Of course, sometimes there are some bizarre "forms most beautiful and most wonderful" in the natural world, to borrow a phrase from Darwin.  When the first taxidermied pelts and skeletons of the duck-billed platypus were sent from Australia back to England, many English scientists thought they were a prank -- that someone had stitched together the remains of various animals in an attempt to play a joke.  And once convinced that they were real, the first scientific name given to the platypus was...

... Ornithorhynchus ("bird-billed") paradoxa.

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Shaky ground

A little less than six years apart -- on 1 November 1755 and 31 March 1761 -- two major earthquakes struck the country of Portugal, each time generating a tsunami that devastated the capital city of Lisbon.

They were both huge, although given that this was before the invention of the seismometer, we can only guess at how big; estimates are that the 1761 quake was around 8.5 on the Richter Scale, while the 1755 one may have been as high as 9.0.  Each time, the tremors were felt far from the epicenter.  The shaking from the 1755 quake was recorded as far away as Finland.

The effects in Portugal and nearby nations were devastating.  In 1755 the combined death toll in Portugal, Spain, and Morocco -- mostly from the tsunami -- is estimated at fifty thousand.  Over eighty percent of the buildings in Lisbon were damaged or completely destroyed -- and five and a half years later, many of the ones that had survived in 1755 collapsed.

Ruins of the Convento do Carmo, which was destroyed in the Great Lisbon Earthquake of 1755 [Image licensed under the Creative Commons Chris Adams, Convento do Carmo ruins in Lisbon, CC BY-SA 3.0]

What's curious is that Portugal isn't ordinarily thought to be high on the list of seismically-active nations.  It's not on the Ring of Fire, where the majority of the world's earthquakes and volcanoes occur.  The fact is, though, there is a poorly-studied (and poorly-understood) fault zone offshore -- the Azores-Gibraltar Transform Fault -- that is thought to have been responsible for both of the huge eighteenth century quakes, as well as a smaller (but still considerable) earthquake in 1816.

The AGTF, and how it's evolving, was the subject of a paper in the journal Geology last week.  The big picture here has to do with the Wilson Cycle -- named after plate tectonics pioneer John Tuzo Wilson -- which has to do with how the Earth's crust is formed, moved, and eventually destroyed.

At its simplest level, the Wilson Cycle has two main pieces -- divergent zones (or rifts) where oceanic crust is created, pushing plates apart, and convergent zones (or trenches) where oceanic crust is subducted back into the mantle and destroyed.  Right now, one of the main divergent zones is the Mid-Atlantic Rift, which is why the Atlantic Ocean is gradually widening; the Pacific, on the other hand, is largely surrounded by convergent zones, so it's getting smaller.

Of course, the real situation is considerably more complex.  In some places the plates are moving parallel to the faults; these are transform (or strike-slip) faults, like the AGTF and the more famous San Andreas Fault.  And what the new paper found was that the movement along the AGTF doesn't just involve side-by-side movement, but there's a component of compression.

So the Azores-Gibraltar Transform Fault, in essence, is trying to turn into a new subduction zone.

"[These are] some of the oldest pieces of crust on Earth, super strong and rigid -- if it were any younger, the subducting plate would just break off and subduction would come to a halt," said João Duarte, of the University of Lisbon, who lead the research, in an interview with Science Daily.  "Still, it is just barely strong enough to make it, and thus moves very slowly."

The upshot is that subduction appears to be invading the eastern Atlantic, a process that (in tens or hundreds of millions of years) will result in the Atlantic Ocean closing up once more.  The authors write:

[T]he Atlantic already has two subduction zones, the Lesser Antilles and the Scotia arcs.  These subduction zones have been forced from the nearby Pacific subduction zones.  The Gibraltar arc is another place where a subduction zone is invading the Atlantic.  This corresponds to a direct migration of a subduction zone that developed in the closing Mediterranean Basin.  Nevertheless, few authors consider the Gibraltar subduction to be still active because it has significantly slowed down in the past millions of years.  Here, we use new gravity-driven geodynamic models that reproduce the evolution of the Western Mediterranean, show how the Gibraltar arc formed, and test if it is still active.  The results suggest that the arc will propagate farther into the Atlantic after a period of quiescence.  The models also show how a subduction zone starting in a closing ocean (Ligurian Ocean) can migrate into a new opening ocean (Atlantic) through a narrow oceanic corridor.

So the massive Portugal quakes of the eighteenth and nineteenth centuries seem to be part of a larger process, where compression along a (mostly) transform fault is going to result in the formation of a trench.  It's amazing to me how much we've learned in only sixty-odd years -- Wilson and his colleagues only published their seminal papers that established the science of plate tectonics between 1963 and 1968 -- and how much we are still continuing to learn.

And along the way elucidating the processes that generated some of the biggest earthquakes ever recorded.

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Dream a little dream of me

In one of my favorite novels, The Lathe of Heaven by Ursula LeGuin, the main character -- an unassuming man named George Orr -- figures out that when he dreams, his dream changes reality.  The problem is, since when the change occurs, it alters everyone else's memories of what had happened, the only one who realizes that anything has changed is him.

At first, of course, he doesn't believe it.  He must be remembering wrong.  Then, when he becomes convinced it's actually happening, he starts taking drugs to try to stop him from dreaming, but they don't work.  As a last resort, he tries to get help from a psychologist...

... but the psychologist realizes how powerful this ability could be, and starts guiding George into dreams that will shape the world into what he wants it to be.

It's a powerful cautionary tale about what happens when an unscrupulous person gains control over someone with a valuable talent.  Power corrupts, as the oft-quoted line from John Dalberg-Acton goes, and absolute power corrupts absolutely.

I couldn't help thinking about The Lathe of Heaven when I read about some new exploration of lucid dreaming taking place at REMSpace, a California startup, that will be featured in a paper in The International Journal of Dream Research soon (a preprint is available at the link provided).  A lucid dream is one in which you are aware that you're dreaming while you're dreaming, and often have some degree of control over what happens.  Around twenty percent of people report regular lucid dreaming, but there is some research that suggests many of us can learn to lucid dream.

Dickens's Dream by Robert W. Buss (1875) [Image is in the Public Domain]

At this point, I'll interject that despite a long history of very vivid dreams, I've never had a lucid dream.  I did have an almost-lucid dream, once; it was a weird and involved story about being a groomsman in a wedding in a big cathedral, and when the priest said the whole "does anyone have any objections?" thing, a gaudily-dressed old lady in the front row stood up and started shouting about what an asshole the groom was and how the bride could do way better.  And I'm standing there, feeling horrified and uncomfortable, and I thought, "This is bizarre!  How could this be happening?  Is this a dream?"  So I kind of looked around, then patted myself to reassure myself that I was solid, and thought, "Nope.  I guess this is real."

So the one time I actually considered the question of whether I was dreaming, I got the wrong answer.

But I digress.

Anyhow, the researchers at REMSpace took a group of test subjects who all reported being able to lucid dream, and hooked them up to electromyography and electroencephalography sensors -- which, respectively, measure the electrical discharge from voluntary muscle contractions and neural firing in the brain -- and gave them the pre-sleep suggestion that they would dream about driving a car.  Using the output from the sensors, they created a virtual avatar of the person on a computer screen, and found that they were able to use tiny motions of their hands to steer it, and even avoid obstacles.

"Two-way interaction with a computer from dreams opens up a whole area of new technologies," said Michael Raduga, who led the experiment.  "Now, these developments are crude, but soon they will change the idea of human capabilities."

Maybe so, but it also puts the dreamer in the hands of the experimenter.  Now, I'm not saying Michael Raduga and his team are up to anything nefarious; and obviously I don't believe anyone's got the George-Orr-like ability to change reality to conform to what they dream.  But does anyone else have the feeling that "two-way interaction" into your dreams is potentially problematic?  I've heard a lot of people say things like, "hypnosis isn't dangerous, you can't be given a post-hypnotic suggestion that induces you to do something you wouldn't ordinarily do," but if there's one thing my knowledge of neuroscience has taught me, it's that the human brain is highly suggestible.

So as interested as I am in lucid dreaming, I'm not ready to sign up to have my dreams interacted with by a computer controlled by someone else.  And I hope like hell that when Raduga and his group at REMSpace start "changing the idea of human capabilities," they are extremely careful.

Anyway, that's our interesting-but-a-little-scary research for today.  Me, I'm gonna stick with my ordinary old dreams, which are peculiar enough.  And given my failure at detecting a potentially lucid dream when I had the chance, I doubt I'd be all that good at it in any case.  I'd probably drive my virtual dream car right into a telephone pole.

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The viral accelerator

It's virus season, which thus far I've been able to avoid participating in, but seems like half the people I see are hacking and snorting and coughing so even with caution and mask-wearing I figure it's only a matter of time.  Viruses are odd beasts; they're obligate intracellular parasites, doing their evil work by hijacking your cellular machinery and using it to make more viruses.  Furthermore, they lack virtually all of the structures that cells have, including cell membranes, cytoplasm, and organelles.  They really are more like self-replicating chemicals than they are like living things.

Simian Polyoma Virus 40 [Image licensed under the Creative Commons Phoebus87 at English Wikipedia, Symian virus, CC BY-SA 3.0]

What is even stranger about viruses is that while some of the more familiar ones, such as colds, flu, measles, invade the host, make him/her sick, and eventually (with luck) are cleared from the body -- some of them leave behind remnants that can make their presence known later.  This behavior is what makes the herpes family of viruses so insidious.  If you've been infected once, you are infected for life, and the latent viruses hidden in your cells can cause another eruption of symptoms, sometimes decades later.

Even weirder is when those latent viral remnants cause havoc in a completely different way than the original infection did.  There's a piece of a virus left in the DNA of many of us called HERV-W (human endogenous retrovirus W) which, if activated, can trigger multiple sclerosis or schizophrenia.  Another one, Coxsackie virus, has an apparent connection to type-1 diabetes and Sjögren's syndrome.  The usual sense is that all viral infections, whether or not they're latent, are damaging to the host.  So it was quite a shock to me to read a piece of recent research that there's a viral remnant that not only is beneficial, but is critical for creating myelin -- the coating of our nerve cells that is essential for speeding up nerve transmission!

The paper -- which appeared last week in the journal Cell -- is by a team led by Tanay Ghosh of the Cambridge Institute of Science, and looked at a gene called RetroMyelin.  This gene is one of an estimated forty (!) percent of our genome that is made up of retrotransposons, DNA that was inserted by viruses during evolutionary history.  Or, looking at it another way, genes that made their way to us using a virus as a carrier.  Once inside our genome, transposons begin to do what they do best -- making copies of themselves and moving around.  Most retrovirus-introduced elements are deleterious; HIV and feline leukemia, after all, are caused by retroviruses.  But sometimes, the product of a retroviral gene turns out to be pretty critical, and that's what happened with RetroMyelin.

Myelin is a phosopholipid/protein mixture that surrounds a great many of the nerves in vertebrates.  It not only acts as an insulator, preventing the ion distribution changes that allow for nerve conduction to "short-circuit" into adjacent neurons, it is also the key to saltatory conduction -- the jumping of neural signals down the axon, which can increase transmission speed by a factor of fifty.  So this viral gene acted a bit like a neural accelerator, and gave the animals that had it a serious selective advantage.

"Retroviruses were required for vertebrate evolution to take off," said senior author and neuroscientist Robin Franklin, in an interview in Science Daily.  "There's been an evolutionary drive to make impulse conduction of our axons quicker because having quicker impulse conduction means you can catch things or flee from things more rapidly.  If we didn't have retroviruses sticking their sequences into the vertebrate genome, then myelination wouldn't have happened, and without myelination, the whole diversity of vertebrates as we know it would never have happened."

The only vertebrates that don't have myelin are the jawless fish, such as lampreys and hagfish -- so it's thought that the retroviral infection that gave us the myelin gene occurred around the same time that jaws evolved on our branch of the vertebrate family tree, on the order of four hundred million years ago.

So even some fundamental (and critical) traits shared by virtually all vertebrates, like the myelin sheaths that surround our neurons, are the result of viral infections.  Just proving that not all of 'em are bad.  Something to think about the next time you feel a sore throat coming on.

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All set

How long is the coastline of Britain?

Answer: as long as you want it to be.

This is not some kind of abstruse joke, and if it sounds like it, blame the mathematicians.  This is what's known as the coastline paradox, which is not so much a paradox as it is the property of anything that is a fractal.  Fractals are patterns that never "smooth out" when you zoom in on them; no matter how small a piece you magnify, it still has the same amount of bends and turns as the larger bit did.

And coastlines are like that.  Consider measuring the coastline of Britain by placing dots on the coast one hundred kilometers apart -- in other words, using a straight ruler one hundred kilometers long.  If you do this, you find that the coastline is around 2,800 kilometers long.

[Image licensed under the Creative Commons Britain-fractal-coastline-100km , CC BY-SA 3.0]

But if your ruler is only fifty kilometers long, you get about 3,400 kilometers -- not an insignificant difference.

[Image licensed under the Creative Commons Britain-fractal-coastline-50km, CC BY-SA 3.0]

The smaller your ruler, the longer your measurement of the coastline.  At some point, you're measuring the twists and turns around every tiny irregularity along the coast, but do you even stop there?  Should you curve around every individual pebble and grain of sand?

At some point, the practical aspects get a little ridiculous.  The movement of the ocean makes the exact position of the coastline vague anyhow.  But with a true fractal, we get into one of the weirdest notions there is: infinity.  True fractals, such as the ones investigated by Benoit B. Mandelbrot, have an infinite length, because no matter how deeply you plunge into them, they have still finer structure.

Oh, by the way: do you know what the B. in "Benoit B. Mandelbrot" stands for?  It stands for "Benoit B. Mandelbrot."

Thanks, you're a great audience.  I'll be here all week.

The idea of infinity has been a thorn in the side of mathematicians for as long as anyone's considered the question, to the point that a lot of them threw their hands in the air and said, "the infinite is the realm of God," and left it at that.  Just trying to wrap your head around what it means is daunting:

Teacher: Is there a largest number?
Student: Yes. It's 10,732,210.
Teacher: What about 10, 732,211?
Student: Well, I was close.

It wasn't until German mathematician Georg Cantor took a crack at refining what infinity means -- and along the way, created set theory -- that we began to see how peculiar it really is.  (Despite Cantor's genius, and the careful way he went about his proofs, a lot of mathematicians of his time dismissed his work as ridiculous.  Leopold Kronecker called Cantor not only "a scientific charlatan" and a "renegade," but "a corrupter of youth"!)

Cantor started by defining what we mean by cardinality -- the number of members of a set.  This is easy enough to figure out when it's a finite set, but what about an infinite one?  Cantor said two sets have the same cardinality if you can find a way to put their members into a one-to-one correspondence in a well-ordered fashion without leaving any out, and that this works for infinite sets as well as finite ones.  For example, Cantor showed that the number of natural numbers and the number of even numbers is the same (even though it seems like there should be twice as many natural numbers!) because you can put them into a one-to-one correspondence:

1 <-> 2
2 <-> 4
3 <-> 6
4 <-> 8
etc.

Weird as it sounds, the number of fractions (rational numbers) has exactly the same cardinality as well -- there are the same number of possible fractions as there are natural numbers.  Cantor proved this as well, using an argument called Cantor's snake:

Because you can match each of them to the natural numbers, starting in the upper left and proceeding along the blue lines, and none will be left out along the way, the two sets have exactly the same cardinality.

It was when Cantor got to the real numbers that the problems started.  The real numbers are the set of all possible decimals (including ones like π and e that never repeat and never terminate).  Let's say you thought you had a list (infinitely long, of course) of all the possible decimals, and since you believe it's a complete list, you claimed that you could match it one-to-one with the natural numbers.  Here's the beginning of your list:

7.0000000000...
0.1010101010....
3.1415926535...
1.4142135623...
2.7182818284...

Cantor used what is called the "diagonal argument" to show that the list will always be missing members -- and therefore the set of real numbers is not countable.  His proof is clever and subtle.  Take the first digit of the first number in the list, and add one.  Do the same for the second digit of the second number, the third digit of the third number, and so on.  (The first five digits of the new number from the list above would be 8.2553...)  The number you've created can't be anywhere on the list, because it differs from every single number on the list by at least one digit.

So there are at least two kinds of infinity; countable infinities like the number of natural numbers and number of rational numbers, and uncountable infinities like the number of real numbers.  Cantor used the symbol aleph null -- ℵ₀ -- to represent a countable infinity, and the symbol c (for continuum) to represent an uncountable infinity.

Then there's the question of whether there are any types of infinity larger than ℵ₀ but smaller than c.  The claim that the answer is "no" is called the continuum hypothesis, and proving (or disproving) it is one of the biggest unsolved problems in mathematics.  In fact, it's thought by many to be an example of an unprovable but true statement, one of those hobgoblins predicted by Kurt Gödel's Incompleteness Theorem back in 1931, which rigorously showed that a consistent mathematical system could never be complete -- there will always be true mathematical statements that cannot be proven from within the system.

So that's probably enough mind-blowing mathematics for one day.  I find it all fascinating, even though I don't have anywhere near the IQ necessary to understand it at any depth.  My brain kind of crapped out somewhere around Calculus 3, thus dooming my prospects of a career as a physicist.  But it's fun to dabble my toes in it.

Preferably somewhere along the coastline of Cornwall.  However long it actually turns out to be.

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