Mudslinging

I've been writing here at Skeptophilia for twelve years, something that I find a little mind-boggling.

Even more astonishing is that despite the amount of time I've spent debunking crazy ideas, I still run into ones I'd never heard of before.  Such as the phenomenally loopy claim I bumped into yesterday, about the "Tartaria mud flood."

First, a little background.

The Tatars are a group of Turkic ethnic groups that now live mainly in Russia, Ukraine, Kazakhstan, and Turkey.  They were the predominant force in the "Golden Horde" that swept across Central Asia in the thirteenth century C.E., establishing a khanate there that would last for four centuries.  The Europeans -- as usual, not particularly concerned with accuracy in describing people they considered inferior -- picked up this name, and started calling pretty much anyone from Central Asia and eastern Siberia "Tatars" (more commonly misspelled as "Tartars").  And the entire region appears on old maps as "Tartary."

An English map from 1806 showing "Tartary" (note that they even include Japan under this name!) [Image is in the Public Domain]

Not to beat the point unto death, but the whole European concept of Tartary was wrong right from the get-go; it was lumping together dozens of groups of people who were not only not Tatars, but weren't even Turkic, and it was pretending that the whole lot of them were under some kind of unified central government.

So we're on shaky ground from the start, but it gets worse.

In 2016, a guy named Philipp Druzhinin started posting videos and articles claiming that not only was Tartary (which he called "Tartaria") real, it had been ascendant until the 1800s -- at which point, something catastrophic happened.  Some time in the early nineteenth century, there had been a worldwide "mud flood" that had buried Tartarian cities and effectively ended the theretofore thriving country of Tartaria.  At first, his videos got little notice, but then something happened in 2019 -- it's not entirely apparently what -- that made them suddenly gain traction.

A lot of traction.  And, as you'll see, started entangling them with something a lot darker.

But first, with regards to the claim itself, I have several questions.

First, what evidence is there that anything like this ever happened?

The most accurate answer is "almost none."  The main argument seems to be that in a lot of cities there are catacombs and underground passageways, which in Druzhinin's pretend world were the actual original street levels before all the mud came in and buried stuff.  (Amusingly, he includes the Seattle Underground City in this, despite the fact that (1) Seattle is on the other side of the world from "Tartaria," and (2) the Underground City was created from a thoroughly-documented reconstruction project designed to raise street levels after the Great Seattle Fire of 1889.)

Second, why doesn't this show up in any reputable history books?

Well, Druzhinin knows the answer to that.  The knowledge was suppressed.  Because of course it was.  The evil, scheming historians went and destroyed any record of the mud flood, cackling and rubbing their hands together the entire time.  Notwithstanding the impossibility of erasing every account of a supposedly worldwide event that only happened two centuries ago.  Historians are just that diabolical, apparently.  Why they did this is unclear.  Maybe just being eeeeee-vill is enough.

Third, where did all the mud come from?

Druzhinin is a little thin on this point.  (Truthfully, he's a little thin on every point.)  Considering that even a good-sized volcano can only cover a few square miles in lava during an eruption, it's hard to imagine any process that could produce enough mud to generate a mud flood worldwide.  But, hey... Noah's ark and everything, amirite?  So q.e.d., apparently.

The Tartarian mud flood claim is so patently ridiculous that you'd think an average middle schooler would recognize it as such, and yet -- since its first appearance seven years ago -- it has gained tremendous traction.  YouTube videos about it have been watched and downloaded millions of times.  Worse still, the whole thing has gotten tangled up in other, nastier conspiracy theories -- QAnon, the Illuminati, various antisemitic ideologies, all the One World Government nonsense, microchip implantation schemes, even climate change denialism -- because, as I've pointed out before, once you've abandoned hard evidence as the touchstone for understanding, you'll fall for damn near anything.

Or perhaps for everything, all at the same time.

What would be hilarious if it weren't so disturbing is that a big part of this crazy conglomeration of claims state that the Powers-That-Be want to silence all dissent and stop anyone from finding out about their nefarious dealings, and yet some tinfoil-hat-wearing twenty-something living in his mom's basement can make and upload hours of YouTube videos on the topic, and the response from the Powers-That-Be is: *crickets*

Almost drives you to the awkward conclusion that the whole lot of it is unadulterated horse waste, doesn't it?

And of course, the purveyors of this nonsense love it when people like me write stuff like this, because there's nothing for their sense of self-righteousness like also feeling persecuted.  Laughing at them just increases their certainty they're right, because otherwise... why would we be laughing?  It reminds me of the quote from Carl Sagan: "[T]he fact that some geniuses were laughed at does not imply that all who are laughed at are geniuses.  They laughed at Columbus, they laughed at Fulton, they laughed at the Wright brothers.  But they also laughed at Bozo the Clown."

Anyhow, keep an eye out for this.  One of the most recent additions to the long, ugly list of conspiracy theories.  Dating from when it really took off, the whole thing is only about four years old, and astonishingly -- considering the logical leaps you have to make to believe any of it -- is still gaining serious traction.

Which just pisses me off.  I work my ass off to get views here at Skeptophilia, and some wingnut claims that a magical mud flood wiped out a non-existent country two centuries ago, and it somehow gains wings.  It reminds me of the quote from Charles Haddon Spurgeon -- "A lie can go all the way around the world while truth is still lacing up its boots."

****************************************

Pitch perfect

I've been a music lover since I was little.  My mom used to tell the story of my being around four years old and begging her to let me put records on the record player.  At first, she was reluctant, but for once my persistence won the day, and she finally relented.  To my credit, despite my youth I was exceedingly careful and never damaged a record; the privilege was too important to me to risk revocation.  There were certain records I played over and over, such as Rimsky-Korsakov's Scheherazade (a piece I love to this day).

I've always been fascinated with the question of whether musicality is inborn or learned.  My parents, while they had a decent record collection, weren't musical themselves; they certainly didn't have anything like the passion for it I experienced.  While the capacity for appreciating music is still poorly understood, today I'd like to tell you about some research indicating that the way our brains interpret tone structure is inborn.

First, a little background.

While it may appear on first glance that the major key scale -- to take the simplest iteration of tone structure as an example -- must be arbitrary, there's an interesting relationship between the frequencies of the notes.  Middle C, for example, has a frequency of about 260 hertz (depending on how your piano is tuned), and the C above middle C (usually written C') has exactly twice that frequency, 520 hertz. Each note is half the frequency of the note one octave above.  The frequency of G above middle C (which musicians would say is "a fifth above") has a frequency of 3/2 that of the root note, or tonic (middle C itself), or 390 hertz.  The E above middle C (a third above) has a frequency of 5/4 that of middle C, or 325 hertz.  Together, these three make up the "major triad" -- a C major chord.  (The other notes in the major scale also have simple fractional values relative to the frequency of the tonic.)

[Note bene: Music theoretical types are probably bouncing up and down right now and yelling that this is only true if the scale is in just temperament, and that a lot of Western orchestral instruments are tuned instead in equal temperament, where the notes are tuned in intervals that are integer powers of the basic frequency increase of one half-tone.  My response is: (1) yes, I know, and (2) what I just told you is about all I understand of the difference, and (3) the technical details aren't really germane to the research I'm about to reference.  So you must forgive my oversimplifications.]

Because there are such natural relationships between the notes in a scale, it's entirely possible that our ability to perceive them is hard-wired.  It takes no training, for example, to recognize the relationship between a spring that is vibrating at a frequency of f (the lower wave on the diagram) and one that is vibrating at a frequency of 2f (the upper wave on the diagram).  There are exactly twice the number of peaks and troughs in the higher frequency wave as there are in the lower frequency wave.

Still, being able to see a relationship and hear an analogous one is not a given.  It seems pretty instinctive; if I asked you (assuming you're not tone deaf) to sing a note an octave up or down from one I played on the piano, you probably could do it, as long as it was in your singing range.

But is this ability learned because of our early exposure to music that uses that chord structure as its basis?  To test this, it would require comparing a Western person's ability to match pitch and jump octaves (or other intervals) with someone who had no exposure to music with that structure -- and that's not easy, because most of the world's music has octaves, thirds, and fifths somewhere, even if there are other differences, such as the use of quarter-tones in a lot of Middle Eastern music.

This brings us to a paper in the journal Current Biology called "Universal and Non-universal Features of Musical Pitch Perception Revealed by Singing," by Nori Jacoby (of the Max Planck Institute and Columbia University), Eduardo A. Undurraga, Joaquín Valdés, and Tomás Ossandón (of the Pontificia Universidad Católica de Chile), and Malinda J. McPherson and Josh H. McDermott (of MIT).  And what this team discovered is something startling; there's a tribe in the Amazon which has had no exposure to Western music, and while they are fairly good at mimicking the relationships between pairs of notes, they seemed completely unaware that they were singing completely different notes (as an example, if the researchers played a C and a G -- a fifth apart -- the test subjects might well sing back an A and an E -- also a fifth apart but entirely different notes unrelated to the first two).

The authors write:

Musical pitch perception is argued to result from nonmusical biological constraints and thus to have similar characteristics across cultures, but its universality remains unclear.  We probed pitch representations in residents of the Bolivian Amazon—the Tsimane', who live in relative isolation from Western culture—as well as US musicians and non-musicians.  Participants sang back tone sequences presented in different frequency ranges.  Sung responses of Amazonian and US participants approximately replicated heard intervals on a logarithmic scale, even for tones outside the singing range.  Moreover, Amazonian and US reproductions both deteriorated for high-frequency tones even though they were fully audible.  But whereas US participants tended to reproduce notes an integer number of octaves above or below the heard tones, Amazonians did not, ignoring the note “chroma” (C, D, etc.)...  The results suggest the cross-cultural presence of logarithmic scales for pitch, and biological constraints on the limits of pitch, but indicate that octave equivalence may be culturally contingent, plausibly dependent on pitch representations that develop from experience with particular musical systems.

Which is a very curious result.

It makes me wonder if our understanding of a particular kind of chord structure isn't hardwired, but is learned very early from exposure -- explaining why so much of pop music has a familiar four-chord structure (hilariously lampooned by the Axis of Awesome in this video, which you must watch).  I've heard a bit of the aforementioned Middle Eastern quarter-tone music, and while I can appreciate the artistry, there's something about it that "doesn't make sense to my ears."

Of course, to be fair, I feel the same way about jazz.

In any case, I thought this was a fascinating study, and like all good science, opens up a variety of other angles of inquiry.  Myself, I'm fascinated with rhythm more than pitch or chord structure, ever since becoming enthralled by Balkan music about thirty years ago.  Their odd rhythmic patterns and time signatures -- 5/8, 7/8, 11/16, 13/16, and, no lie, 25/16 -- take a good bit of getting used to, especially for people used to good old Western threes and fours.

So to conclude, here's one example -- a lovely performance of a dance tune called "Gankino," a kopanica in 11/16.  See what sense you can make of it.  Enjoy!

****************************************

The registry of dissent

I wonder if you've heard about the latest attempt to turn the state of Florida into an autonomous authoritarian oligarchy.

No, I'm not talking about Governor Ron DeSantis's virtual takeover of Disney, although for a party that is supposedly staunchly pro-corporation, it seems like a hypocritical thing to do.  "We're staunchly pro-corporation as long as the corporation toes the far-right line" is nearer the mark.

The particular move I'm thinking of today struck closer to the bone for me, because it's targeted specifically at bloggers.  A bill called "Information Dissemination" proposed by Senator Jason Brodeur would, if passed, require bloggers who post anything critical of Governor DeSantis or other elected officials to sign onto a state registry -- or face fines of up to $2,500.  It's unclear from the wording of the bill if this would apply to bloggers out of state who criticize Florida officials.  This certainly doesn't seem to be overtly excluded, but if so, it raises serious issues of jurisdiction.

The bill tries to dodge First Amendment concerns by limiting itself to bloggers who are financially compensated for their writing -- ostensibly to restrict people from taking money from lobbyists and engaging in criticism-for-pay -- but just about all bloggers get compensated in some way, even if it's just through ad monetization.  So the fact is, this bill is meant to do only one thing: stifle dissent.  

The spirit, and even the wording, of the bill have drawn speculation that it was inspired by a similar law passed by the authoritarian régime of President Viktor Orbán of Hungary in 2010.  This may sound far-fetched, but Orbán is a revered figure amongst the far right, and the elected leaders of Florida have praised him before.  Right-wing commentator Rod Dreher, who is currently living in Budapest, described in an interview a conversation with a reporter who had "talked to the press secretary of Governor Ron DeSantis of Florida and she said, 'Oh yeah, we were watching the Hungarians, so yay Hungary.'"  Steve Bannon calls Orbán "one of the great moral leaders of our time."  It's not certain if Brodeur's bill is a case of imitation or just parallel processes from like minds -- but either way, it's horrifying.

[Image licensed under the Creative Commons Madelgarius, Freedom of speech (3), CC BY-SA 4.0]

Even some GOP members seem to realize Brodeur's bill is a case of serious governmental overreach.  In a statement that would be funny if it weren't so appalling, none other than Newt Gingrich tweeted, "The idea that bloggers criticizing a politician should register with the government is insane.  It is an embarrassment that it is a Republican state legislator in Florida who introduced a bill to that effect.  He should withdraw it immediately."  Which brought to mind the trenchant quote from Stephen King: "Conservatives who for years sowed the dragon's teeth of partisan politics are horrified to discover they have grown an actual dragon."  Gingrich, perhaps more than any other single individual, is the architect of the far right; the fact that the careening juggernaut he created has lurched into authoritarian neo-fascism should come as no surprise to him, or to anyone else.  The subtext has always been "We're the party of small hands-off government until we want big intrusive government;" bills like Brodeur's, and (even more strikingly) the current tsunami of anti-trans legislation being passed in red states across the country, just pull the mask off the ugly agenda that was there from the very beginning.

The optimists say that even if Brodeur's bill passes, it'll be struck down on First Amendment grounds almost immediately.  Me, I wonder.  DeSantis and his ilk are in ascendency, and I'm perhaps to be excused if I suspect it's not so certain as all that.  Here I sit, in upstate New York, far away from the epicenter; but I hope my writer colleagues in Florida will not be cowed into silence.  Believe me, if I did live in Florida, I'd be criticizing Brodeur, DeSantis, and the proposed legislation for all I'm worth.  I'm not usually a "come at me, bro" type, but we can't keep quiet about it and hope that the First Amendment will shield us.  If this bill passes -- and I think it probably will -- it will act as a template for other state legislatures intent on crushing dissenting voices.

If you think this kind of thing can't spread like a contagion, I have only refer you to the history of Germany in the 1930s for a counterexample.

Whatever the legality of extending this law to apply to out-of-state bloggers criticizing Florida legislators, allow me to go on record as stating that this is me, criticizing the absolute shit out of the whole lot of them.  And as far as my ever signing onto a registry for doing so, I am also going on record as stating that Brodeur can take his blogger registry and stick it up his ass.

Sideways.

****************************************

The lost catalog

After yesterday's rather elegiac post about the breadth of creative work we've lost over the ages, a friend and long-time loyal reader of Skeptophilia sent me a link suggesting that in some fortunate cases, maybe "lost" doesn't mean forever.

The ancients knew all too well that the vagaries of time made books and scrolls precious, easily-damaged treasures.  Fire, damage from insects and mice, and even just the wear-and-tear from repeated use all took their toll on written work.  Add to that the fact that before the invention of movable type, hand-copying manuscripts was a laborious and time-consuming occupation, and it's no wonder that books were rare and expensive, often only to be found in libraries, monasteries, and the homes of the very wealthy.

This awareness of how much could perish forever if a single library was destroyed prompted some scholars to try and catalog manuscripts, to create a record of the rich diversity of books out there in the world.  One of these was named Hernando (also known as Ferdinand) Colón, the illegitimate son of none other than Christopher Columbus.

Colón was a fascinating character.  Uninterested in his father's passion of establishing trade routes, exploration, and colonization, he preferred instead to travel around Europe and buy books.  He founded a personal library in Seville where he welcomed visits from other scholars, and at its height it contained over fifteen thousand books.  His library contained all sorts of books -- unlike many of his time, he didn't consider books by non-Christians to be worthless "works of infidels" -- and his library became one of the best-known in western Europe.

It was also unwieldy.  Imagine trying to find a particular piece of information in a library that big, with no indexing system.  Back then, there was no such thing as a card catalog, much less a search engine.  So Colón set about writing the sixteen immense volumes of Libros de los Epitomes, a bibliography and short summary of every single one of the books in the library.

It's a good thing he did, because (like the Library of Cologne I wrote about yesterday) Colón's library wasn't to last.  Besides the aforementioned hazards all books are subject to, Colón came to the attention of the narrow-minded zealous religious bigotry of the Inquisition, and a number of his books -- the ones judged to be heretical -- were seized and burned.  But by that time they had been catalogued, so we have at least a glimpse of what lay inside them.

Fourteen of the sixteen Libros were known to have survived, and reside at the Biblioteca Colombina de Sevilla, along with what is left of Colón's book collection.  But now, quite by accident, the fifteenth volume was found to be still in existence as well -- somehow it had made its way to the Arnamagnæan Institute at the University of Copenhagen, which houses the huge book collection of eighteenth-century Icelandic scholar Árni Magnússon.  The three-thousand-odd books in the Institute have only recently been studied in any sort of detail, and it was quite a shock when Guy Lazure, of the University of Windsor (Canada), was working there and found a thirty-centimeter-thick, two thousand page book that turned out to be one of the lost volumes of the Libros de los Epitomes.

The recently rediscovered fifteenth volume of Libros de los Epitomes [Image courtesy of the Arnamagnæan Institute and the University of Copenhagen]

"It’s a discovery of immense importance, not only because it contains so much information about how people read five hundred years ago, but also, because it contains summaries of books that no longer exist, lost in every other form than these summaries," said Edward Wilson-Lee of Cambridge University, who wrote a biography of Colón called The Catalogue of Shipwrecked Books.  Wilson-Lee emphasizes that Colón was qualitatively different from other book collectors of the time, because he didn't limit his acquisitions to scholarly tomes and the classics.  "This was someone who was, in a way, changing the model of what knowledge is.  Instead of saying 'knowledge is august, authoritative things by some venerable old Roman and Greek people', he’s doing it inductively: taking everything that everyone knows and distilling it upwards from there.  It’s much more resonant with today, with big data and Wikipedia and crowdsourced information.  This is a model of knowledge that says, 'We’re going to take the breadth of print – ballads and pornography and newsletters – and not exclude that from the world of information.'"

It will be fascinating to see what lost gems of antiquity will show up -- in summary form, at least -- in the fifteenth volume of the Libros.  Not as good, perhaps, as having the actual copies as they were before they fell prey to time and the Inquisition, but far better than nothing.  At least it will give us an idea of the scope of what was lost -- and raise the hope that maybe, in other obscure collections somewhere out there, some lost masterpieces of the past are still waiting to be found.

****************************************

A library of ghosts

I'm currently working on a trilogy about the fall of civilization that is not, I hasten to state, inspired by current events.

It's actually a story I've been cogitating on since I was in college.  How would ordinary people cope with the collapse of the comfortable support network we're all so very used to?  The three books of the trilogy are set about five hundred years apart, and center around (respectively) the time when everything fell apart, a period of "Dark Ages" during which a significant chunk of what's left of humanity has lost technology and even literacy, and the time during which things come full circle and people begin to rediscover science and mathematics and all that comes with it.  In the second book, The Scattering Winds, there's a sequence when the main character comes across the mostly-intact remnants of a library from before the fall -- and is overwhelmed by the magnitude of what was lost:

"Do these books come from the Before Time?" Kallian asked in a near whisper.

Kasprit Seely nodded, looking around them at the shadowed shelves, laden with dust-covered books.  "Before the flood, you mean?  I’ve no doubt that many of them do.  During the Black Years, with the floods and the plagues, people were trying their hardest just to survive.  A lot of them didn’t, of course.  From what I’ve read, in the times before, there were a thousandfold more people than there are now, and they had ample food and living space and comfort and could spend their time reading and writing books.  But when a hundred years passes with deprivation and famine and death on your doorstep every day, a lot is forgotten.  You’ll see in some books there are numbers that I believe were some sort of system of keeping track of the passage of years.  But I’ve not been able to decipher how it’s to be read, nor how it relates to the present day.  Nowadays we simply track time by the year of the reign of the current king.  So this is the twenty-first year of the reign of High King Sweyn VII, long may he live."  Kasprit pulled a book off a shelf in the room they’d entered—the cover said The Diversity of Life by E. O. Wilson, and was adorned with a design of a brightly-colored beetle with long antennae.  He blew the dust off the top and opened the cover, flipped a couple of pages in, and rested the tip of his long index finger on a line that said, "Copyright 1992."

I thought about this scene when I came upon an article about an archaeological discovery made in 2017 in the center of the German city of Cologne.  Cologne is immensely old; it was the main settlement of the Ubii, a Germanic tribe that (unlike many of their neighbors) forged a strong and long-lasting alliance with the Romans.  Eventually, the place got so thoroughly Romanized that it was renamed Colonia Claudia Ara Agrippinensium -- "Colony of Claudius and the Altar of the Agrippinians."  This proved to be a clumsy appellation, and it was shortened to Colonia, which is where the modern name of Cologne comes from.

Well, it turns out in the center of modern Cologne, a city with a million inhabitants, are the remnants of what used to be the Library of Colonia.  At first, it was thought that the foundation was part of a stone-walled fortification, but when the archaeologists began to discover deep niches in the walls, they realized that its purpose was something altogether different.

"It took us some time to match up the parallels – we could see the niches were too small to bear statues inside," said Dirk Schmitz, of the Roman-Germanic Museum of Cologne, who participated in the research.  "But what they are are kind of cupboards for the scrolls.  They are very particular to libraries – you can see the same ones in the library at Ephesus."

The foundations of the Roman Library of Cologne [Image courtesy of the Romano-Germanic Museum of Cologne]

The library of Cologne in its heyday -- the middle of the second century C.E. -- is thought to have housed around twenty thousand scrolls, of which not a single one survives.  All that remains are the spaces they occupied, now inhabited only by the ghosts of long-gone books whose titles we'll never know.

When I read this article, I was struck with same feeling of longing and grief I get whenever I think about the Great Library of Alexandria and the other repositories of human knowledge.  It's what I tried to communicate in Kallian Dorn's character in The Scattering Winds; perhaps lost knowledge can be regained, but the creativity, hearts, and voices of the people who wrote these scrolls are gone forever.  Impermanence is part of reality, and -- in the words of the band Kansas -- "Nothing lasts forever but the Earth and sky."  But seeing the remains of this once-great library makes me mourn for what was housed there, even so.

I suspect I'm not the only one who feels this way.  And if time travel is ever invented, I think the Great Libraries of Antiquity tour is going to be sold out indefinitely.

****************************************

Weird math

When I was in Calculus II, my professor, Dr. Harvey Pousson, blew all our minds.

You wouldn't think there'd be anything in a calculus class that would have that effect on a bunch of restless college sophomores at eight in the morning.  But this did, especially in the deft hands of Dr. Pousson, who remains amongst the top three best teachers I've ever had.  He explained this with his usual insight, skill, and subtle wit, watching us with an impish grin as he saw the implications sink in.

The problem had to do with volumes and surface areas.  Without getting too technical, Dr. Pousson asked us the following question. If you take the graph of y = 1/x:

And rotate it around the y-axis (the vertical bold line), you get a pair of funnel-shapes.  Not too hard to visualize.  The question is: what are the volume and surface area of the funnels?

Well, calculating volumes and surface areas is pretty much the point of integral calculus, so it's not such a hard problem.  One issue, though, is that the tapered end of the funnel goes on forever; the red curves never strike either the x or y-axis (something mathematicians call "asymptotic").  But calc students never let a little thing like infinity stand in the way, and in any case, the formulas involved can handle that with no problem, so we started crunching through the math to find the answer.

And one by one, each of us stopped, frowning and staring at our papers, thinking, "Wait..."

Because the shapes end up having an infinite surface area (not so surprising given that the tapered end gets narrower and narrower, but goes on forever) -- but they have a finite volume.

I blurted out, "So you could fill it with paint but you couldn't paint its surface?"

Dr. Pousson grinned and said, "That's right."

We forthwith nicknamed the thing "Pousson's Paint Can."  I only found out much later that the bizarre paradox of this shape was noted hundreds of years ago, and it was christened "Gabriel's Horn" by seventeenth-century Italian physicist and mathematician Evangelista Torricelli, who figured it was a good shape for the horn blown by the Archangel Gabriel on Judgment Day.

There are a lot of math-phobes out there, which is a shame, because you find out some weird and wonderful stuff studying mathematics.  I largely blame the educational system for this -- I was lucky enough to have a string of fantastic, gifted elementary and middle school math teachers who encouraged us to play with numbers and figure out how it all worked, and I came out loving math and appreciating the cool and unexpected bits of the subject.  It's a pity, though, that a lot of people have the opposite experience.  Which, unfortunately, is what happened with me in my elementary and middle school social studies and English classes -- with predictable results.

So math has its cool bits, even if you weren't lucky enough to learn about 'em in school.  Here are some short versions of other odd mathematical twists that your math teachers may not have told you about.  Even you math-phobes -- try these on for size.

1. Fractals

A fractal is a shape that is "self-similar;" if you take a small piece of it, and magnify it, it looks just like the original shape did.  One of the first fractals I ran into was the Koch Snowflake, invented by Swedish mathematician Helge von Koch, which came from playing around with triangles.  You take an equilateral triangle, divide each of its sides into three equal pieces, then take the middle one and convert it into a (smaller) equilateral triangle. Repeat. Here's a diagram with the first four levels:

And with Koch's Snowflake -- similar to Pousson's Paint Can, but for different reasons -- we end up with a shape that has an infinite perimeter but a finite area.

Fractals also result in some really unexpected patterns coming out of perfectly ordinary processes.  If you have eight minutes and want your mind completely blown, check out how what seems like a completely random dice-throwing protocol generates a bizarre fractal shape called the Sierpinski Triangle.  (And no, I don't know why this works, so don't ask.  Or, more usefully, ask an actual mathematician, who won't just give you what I would, which is a silly grin and a shrug of the shoulders.)

2. The Four-Color-Map Theorem

In 1852, a man named Francis Guthrie was coloring in a map of the counties of England, and noticed that he could do the entire map, leaving no two adjacent counties the same color, using only four different colors. Guthrie wondered if that was true of all maps.

Turns out it is -- something that wasn't proven for sure until 1976.

Oh, but if you're talking about a map printed onto a Möbius Strip, it takes six colors.  A map printed on a torus (donut) would take seven.

Once again, I don't have the first clue why.  Probably explaining how it took almost a hundred years to prove. But it's still pretty freakin' cool.

3. Brouwer's Fixed-Point Theorem

In the 1950s, Dutch mathematician Luitzen Brouwer came up with an idea that -- as bizarre as it is -- has been proven true.  Take two identical maps of Scotland.  Deform one any way you want to -- shrink it, expand it, rotate it, crumple it, whatever -- and then drop it on top of the other one.

Brouwer said that there will be one point on the deformed copy of the map that is exactly on top of the corresponding point on the other map.

[Nota bene: it works with any map, not just maps of Scotland.  I just happen to like Scotland.]

It even works on three dimensions.  If I stir my cup of coffee, at any given time there will be at least one coffee molecule that is in exactly the same position it was in before I stirred the cup.

Speaking of which, all this is turning my brain to mush.  I think I need to get more coffee before I go on to...

4. The types of infinity

You might think that infinite is infinite.  If something goes on forever, it just... does.

Turns out that's not true.  There are countable infinities, and uncountable infinities, and the latter is much bigger than the former.

Infinitely bigger, in fact.

Let's define "countable" first.  It's simple enough; if I can uniquely assign a natural number (1, 2, 3, 4...) to the members of a set, it's a countable set.  It may go on forever, but if I took long enough I could assign each member a unique number, and leave none out.

So, the set of natural numbers is itself a countable set.  Hopefully obviously.

So is the set of odd numbers.  But here's where the weirdness starts.  It turns out that the number of natural numbers is exactly the same as the number of odd numbers.  You may be thinking, "Wait... that can't be right, there has to be twice as many natural numbers as odd numbers!"  But no, because you can put them in a one-to-one correspondence and leave none out:

1-1
2-3
3-5
4-7
5-9
6-11
7-13
etc.

So there are exactly the same number in both sets.

Now, what about real numbers?  The real numbers are all the numbers on the number line -- i.e. all the natural numbers plus all of the possible decimals in between.  Are there the same number of real and natural numbers?

Nope.  Both are infinite, but they're different kinds of infinite.

Suppose you tried to come up with a countable list of real numbers between zero and one, the same as we came up with a countable list of odd numbers above.  (Let's not worry about the whole number line, even.  Just the ones between zero and one.)  As I mentioned above, if you can do a one-to-one correspondence between the natural numbers and the members of that list, without leaving any out, then you've got a countable infinity. So here are a few members of that list:

0.1010101010101010...
0.3333333333333333...
0.1213141516171819...
0.9283743929178394...
0.1010010001000010...
0.13579111315171921...

And so forth.  You get the idea.

German mathematician Georg Cantor showed that no matter what you do, your list will always leave some out.  In what's called the diagonal proof, he said to take your list, and create a new number -- by adding one to the first digit of the first number, to the second digit of the second number, to the third digit of the third number, and so on.  So using the short list above, the first six decimal places will be:

0.242413...

This number can't be anywhere on the list.  Why?  Because its first digit is different from the first number on the list, the second digit is different from the second number on the list, the third digit is different from the third number of the list, and so forth.  And even if you just artificially add that new number to the end of the list, it doesn't help you, because you can just do the whole process again and generate a new number that isn't anywhere on the list.

So there are more numbers between zero and one on the number line than there are natural numbers.  Infinitely more.

5. Russell's Paradox

I'm going to end with one I'm still trying to wrap my brain around.  This one is courtesy of British mathematician Bertrand Russell, and is called Russell's Paradox in his honor.

First, let's define two kind of sets:

• A set is normal if it doesn't contain itself.  For example, the "set of all trees on Earth" is normal, because the set itself is not a tree, so it doesn't contain itself.
• A set is abnormal if it contains itself.  The "set of everything that is not a tree" is abnormal, because the set itself is not a tree.

Russell came up with a simple idea: he looked at "the set of all possible normal sets."  Let's call that set R.  Now here's the question:

Is R normal or abnormal?

Thanks, I'll show myself out.

****************************************

A refuge from the cold

I've always wondered how our distant ancestors survived during the various ice ages.

After all, we're mostly-hairless primates evolved on the warm, comfy African savanna, and it's hard to imagine how we coped with conditions like you often see depicted in books on early humans:

Le Moustier Neanderthals by Charles Knight (1920) [Image is in the Public Domain]

Despite the bear pelts around their nether regions, I've always wondered how they didn't all freeze to death.  When the weather's nice, bare skin is fine; I only wear a shirt during the summer under duress, and can't remember the last time I wore swim trunks when I went swimming in my pond.  But when the weather's cold -- which, here in upstate New York, is more often than not -- I'm usually wearing layers, and that's even indoors with our nice modern heating system.  Okay, admittedly I'm a wuss about the cold, but the fact remains that we're evolved to dwell in temperate regions.  Which, for a significant part of the Pleistocene Epoch, most of the world was not.

In particular, during the Last Glacial Maximum, between twenty-six and twenty thousand years ago, much of the Northern Hemisphere was experiencing a climate that the word "unpleasant" doesn't even begin to describe.  The average temperature was 6 C (11 F) colder than it is today, which was enough to cause ice sheets to spread across much of North America and northern Europe (where I currently sit, in fact, was underneath about thirty meters of ice).  Much of the non-glaciated land experienced not only dreadful cold, but long periods of drought.  The combined result is that the sea level was an estimated 130 meters lower than it is today, and broad dry valleys lay across what are now the bottoms of the Bering Sea, the North Sea and English Channel, and the Gulf of Carpentaria.

These conditions opened up passageways for some people, and closed off living space for others.  This was the time that the various pulses of immigrants crossed from Siberia through Beringia and into North America, where they became the ancestors of today's Indigenous Peoples of North and South America.  (If you want to read a brilliant account of how this happened, and some of the science behind how we know, you must read Jennifer Raff's wonderful book Origin: A Genetic History of the Americas.)  The same sort of thing happened from southeast Asia into what is now Australia.

In Europe, though, things got dicey to the point that it's a wonder anyone survived at all.  In fact, what brings this up is a study that appeared in Nature last week by a humongous team led by paleogeneticist Cosimo Posth of the Max Planck Institute of Evolutionary Anthropology.  The team did a complete genomic analysis of 356 individuals whose remains range from thirty-five thousand to five thousand years of age -- so right across that awful Last Glacial Maximum period -- to try to figure out how groups moved when the ice started coming in, and afterwards, once it retreated.

What they found was that only one part of Europe showed a consistent human genetic signature throughout the time period: the Iberian Peninsula.  What this indicates is that modern Spain and Portugal were a "climate refugium" during the worst of the glaciation, where people came to stay when the climate turned very cold, and pretty much stayed put.  Other areas that you might think were possible candidates for comparatively warm hideouts, such as what are now Italy and Greece, show a significant genomic shift across the Last Glacial Maximum, indicating that the people there before the cold set in either migrated or else died out, and were replaced by immigrants who moved in after things warmed and the area once again became more hospitable for humans.

"At that time, the climate warmed up quickly and considerably and forests spread across the European continent," said Johannes Krause, senior author of the study, in an interview with Science Daily.  "This may have prompted people from the south to expand their habitat.  The previous inhabitants may have migrated to the north as their habitat, the 'mammoth' steppe, dwindled,.  It is possible that the migration of early farmers into Europe triggered the retreat of hunter-gatherer populations to the northern edge of Europe.  At the same time, these two groups started mixing with each other, and continued to do so for around three thousand years."

Me, I'm curious what happened to these people afterward.  As a linguist, not to mention a white guy of western European descent, I've wondered if we're talking about my forebears, here -- and what languages they spoke.  My suspicion is that we're looking at the ancestors of today's Basques, who still live in northern Spain; they speak a non-Indo-European language that is usually considered a relic of the earliest languages spoken in Europe.  The Indo-European-speaking peoples (therefore the ancestors of the majority of today's Europeans) didn't reach Europe until about four thousand years ago, so long after the heyday of the people who were the subjects of the Posth et al. paper.

So you have to wonder who the descendants of these very early Europeans are.  "Not me" is my general assessment, considering my general cold-hardiness.  Drop me into an ice age where I had to live in a cave, hike on glaciers, hunt mammoth, and fend off cave bears, and I'd last maybe three days, tops.  I'm highly impressed by the ability of these ancient humans to survive, but given a choice I'll stick with my warm house, indoor plumbing, electric stove, and coffee maker.

****************************************