Drawing the line

A friend and loyal reader of Skeptophilia sent me a link to a YouTube video for my facepalming pleasure a couple of days ago, and being a generous sort, I wanted to share the experience will all of you.  The video is called "Nazca Lines Finally Solved!  The Answer is Amazing!", and is well worth watching in its entirety.  But if you understandably don't want to spend seven minutes of your life watching the video that you will never, ever get back, I'll provide you with a capsule summary and some editorial commentary from Yours Truly.

The Nazca Lines, you probably know, are a series of geoglyphs in southern Peru, which are large enough that their overall shape really can't be discerned except from the air.

[Image is in the Public Domain]

The relative impossibility of seeing the pattern except from above has led to wingnuts such as Erich von Däniken (of Chariots of the Gods fame) to propose that they were made to signal aliens visiting Earth from other planets.  Why aliens would be impressed by our drawing a giant monkey on the ground, I have no idea.  It also bears mention that Nazca is hardly the only place in the world that has geoglyphs, and none of them have much to do with flying saucers.  There's the Cerne Abbas Giant of Dorsetshire, England, for example, who is really really glad to see you:

[Image licensed under the Creative Commons PeteHarlow, Cerne-abbas-giant-2001-cropped, CC BY-SA 3.0]

Be that as it may, the guy in the video, one Damon T. Berry, thinks the Nazca lines are trying to tell us something.  What?  Well, he starts out with a bang by saying that "the universal language is constellations."  Whatever the fuck that means.  Given that the constellations are random assemblages of stars that would look completely different from another vantage point in space, it's hard to imagine anything "universal" about them except that they're, by default, part of the universe.

What Berry tells us then is that each of the glyphs has a code that points at a particular destination.  He starts with the glyph shaped like a bird, and then talks about birds representing flight (okay, I'm with you so far), and some of the glyphs being runways for flying machines (why the hell you'd make a runway shaped like a monkey, I have no idea), and then goes into a long part about how it's significant that the bird has four toes on one foot and five on the other.

"It is a bird," Berry says.  "It appears to be a bird.  But think like an alien.  Look closer at its feet."

I'm not sure why thinking like an alien involves looking at feet.  Maybe the aliens have some kind of weird foot fetish.  I dunno.

Anyhow, what does the fact of its having nine toes mean?  It means, Berry says, that "this is not a bird.  This is a constellation."  In fact, it's the constellation Aquila, a grouping of stars in the northern hemisphere which evidently looked like an eagle to some ancient Greeks who had just polished off their second bottle of ouzo.  The nine toes correspond to the nine brightest stars in the constellation, he says.

Then he moves on to another bird glyph, this one of a hummingbird.  Berry tells us in astonished tones that this bird has the same number of toes on each foot, as if that was an unusual condition or something.  He then says, and this is a direct quote: "The clue lies elsewhere... in the wings.  And the elongated wings are meant to draw your attention... to the wings."

I had to pause the video at this point to give myself a chance to stop guffawing.

We're then directed to count the feathers, and he comes up with eleven.  He includes the tail, but I'm not going to quibble about that because otherwise we'll be here all day.  He says that the number eleven can only mean one thing: the glyph points to the "constellation Columbia."

For the record, the constellation is actually Columba, not Columbia.  Cf. my comment about not quibbling.

The fact that Columba "has eleven stars" means there's an obvious correspondence.  Well, I have two things to say about that.

1. Do you really think that there's nothing else in the universe that is made up of eleven parts?
2. There are way more than eleven stars in Columba, it's just that the shape of the constellation (identified as a dove by the aforementioned ouzo-soaked Greeks) is generally outlined using the brightest eleven stars, just as Aquila was with the nine brightest as earlier described.

He then goes on to analyze the monkey glyph, and once again makes a big deal about the number of fingers and toes, which add to fifteen.  This points to the "constellation of the monkey," which he draws for us.  It's fortunate that he does, because as I do not need to point out to any astronomy buffs out there, there is no constellation of the monkey.  As far as I can tell, he just took some random dots and connected them with straight lines to look vaguely like a monkey.

Whether ouzo was involved, I don't know.

He finishes up by basically saying that aliens are out there and will be coming to visit us from those constellations.  At this point, I started shouting at my computer, "You can't be 'from a constellation!'  The stars in a constellation have nothing to do with one another!"  This caused my dog, Rosie, to come into my office and give me the Canine Head Tilt of Puzzlement, meant to communicate the one concept she's capable of hosting in her brain ("What?").  I reassured her that I wasn't mad at her, that I was mad at the silly man on YouTube, and she accepted that and toddled off to interact with something on her intellectual level, like a dust bunny.

Anyhow.  At the end we're told we can learn more if we just watch his longer and more in-depth production, available on Amazon Prime, but I don't think I'm gonna.  I've heard enough.  Me, I'll go back to trying to figure things out through science instead of pulling random correspondences out of my ass.  Call me narrow-minded, but it seems in general like a better way to understand the universe, even if it doesn't involve counting an animal's toes and acting like it means something significant.

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Smile, and the world smiles with you

In the menagerie of weird creatures from urban legends we have such entities as the Men in Black, Slender Man, the Black-eyed Children, not to mention older creatures of the night such as the Evil Serial Killer With A Hook For A Hand that has been scaring the absolute shit out of kids around campfires for generations.

I just ran into a new member of the zoo yesterday, thanks to crypto-maven Nick Redfern over at Mysterious Universe.  Called "Grinning Man," he's a tall guy in an old-fashioned suit and fedora, with a creepy smile on his face.  His skin is supposedly "plastic-like," so believers think he's only masquerading as a human.  Redfern says he's an operative of the Men in Black; me, I'm thinking more of The Gentlemen from Buffy the Vampire Slayer:

But Grinning Man isn't followed around by guys with long, flailing arms who rip your ribcage open and steal your heart.  Apparently, Grinning Man just kind of stands there... grinning.  Thus the name.  Redfern tells the tale of a California family who saw a UFO while out driving, and the following day had a visitor.  He writes:

It was while one of the teenage children was sat [sic] on the porch and playing music that she caught sight of a man on the other side of the road.  He was dressed completely in black, aside from a white shirt.  He even wore black gloves, on what was a bright, summer day.  The girl was particularly disturbed by the fact that the man sported a weird grin and was staring right at her.  So unsettled was she that she went back into the home and told her father of what had just happened.  He quickly went to the door but – no surprise – the smiling MIB was gone.

John Keel, of "Mothman" fame, describes another encounter, this one near Point Pleasant, West Virginia (home of the original Mothman story):

[A] sewing machine salesman claims to have been stopped on a highway by a strange looking automobile.  A man appeared from a hatch on the side of the vehicle, and a tall, bald man wearing a blue metallic suit approached the man.  He could see the "man" had "slightly elongated" eyes and a demented grin that could be seen glinting in the cars headlights.  The grinning man identified himself as Indrid Cold, and the two had a bizarre telepathic conversation before the entity left, saying they would see each other again.

"Indrid Cold," eh?  A cousin of Mr. Freeze, perhaps?

Now that I think of it, the resemblance is pretty striking.

But unlike Mr. Freeze, "Indrid Cold" was a true alien, Keel said:

The salesman, Woodrow Derenberger, would go on to claim that Indrid Cold would visit him, and would reveal that he was an alien from a planet called Lanulos, situated in another galaxy.  Derenberger claimed to have visited Cold on his homeworld, and met many other beings like Indrid Cold in his travels.  He would write a book about his experiences, but would lose his job, his wife and some say his sanity in the years after, dying in 1990, some saying his obsession with his grinning friend cost him his life.

So that's kind of unfortunate.

Once again, we have the common thread that Grinning Man doesn't seem to do anything.  He doesn't freeze people, he doesn't abduct their children (like Slender Man), he doesn't threaten to kill them if they talk to the authorities (like the Men in Black), etc.  So as extraterrestrial villains go, he's pretty lame, although I have to say in all honesty that if I looked out of my window at night and saw a creepy, pasty-faced guy in a fedora grinning back at me, I'd probably have an aneurysm, so I guess that counts for something, evil-wise.

Anyhow, that's latest member of the Pantheon of Creepiness.  As I've mentioned before, it's kind of amazing that given how long I've been writing Skeptophilia (twelve years as of last November), I still run into weird beliefs I'd never heard of before.  I still think for pure terror, you can't beat the Black-eyed Children, which is why I wrote a trilogy of novels based on the legend (Lines of Sight, Whistling in the Dark, and Fear No Colors).  Whether I did the Children justice is up to you to decide.

But maybe I'm thinking about this wrong.  Maybe Grinning Man is grinning because he is planning something he hasn't carried out yet.  If so, he'd better get at it, because Derenberger's encounter with "Indrid Cold" happened back in the 1960s.  If he wants people to keep being scared of him, he probably should wipe the silly smile off his face and get on with it.

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Start your day with kindness

When I was in my twenties, my parents got into watching the television series Cops.

Me, I never could see the draw.  The plot was the same every time:

• Bad guys do bad stuff.
• Cops get involved.
• Bad guys get arrested or shot.  Or both.
• Repeat x100.

I like my entertainment to have a few more in the way of unexpected twists.  But that's just me, apparently.

Anyhow, there came a point that Cops went into syndication, and on one station, it played every single night.  And my parents had it on.

Every single night.

At this point, I should explain that my parents, especially my mother, had a tremendous suspicion of the unknown.  If there's a word that means the opposite of "adventurous," that was my mom.  As an example, when I made my first trip overseas -- a one-month cross-country hike of England, from Blackpool to Whitby -- her last words to me on the night before I left were, "Don't trust anyone."

I know about correlation not implying causation and all, but I can't help but wonder how much her view of the world as a scary, unsafe place was reinforced by watching a television show that every single night showed the worst of humanity.  I'm guessing the causation probably went both ways -- she gravitated toward the series because she already had that attitude, and the series acted to reinforce the attitude, and round and round it went.

[Image licensed under the Creative Commons West Midlands Police from West Midlands, United Kingdom, 101 Non-Emergency Number - Cops and Robbers (8264612462), CC BY-SA 2.0]

Whatever the cause, her lack of comprehension of how I could possibly want to travel to Dangerous Foreign Countries Inhabited By Dangerous Foreign People (like the English, for fuck's sake) only got worse as she got older.  When we took our first trip to Ecuador back in 2001, not only going to (gasp!) South America, but (1) doing so three months after 9/11, and (2) bringing along both of our sons, at that point ages 11 and 13, she was aghast, but at least knew by then that it'd be futile to try to talk me out of it.

This all comes up because of a study in the Journal of Applied Psychology called, "Rude-Colored Glasses: The Contaminating Effects of Witnessed Morning Rudeness on Perceptions and Behaviors Throughout the Workday."  While on first glance, the study may not seem to have much to do with an overall perception of the world as dangerous, the two are connected.  The study shows pretty clearly that the behavior we are exposed to (or expose ourselves to) colors how we see everything -- and that the effect can last far beyond the time immediately after the incident in question.  The authors write:

Using an experimental experience sampling design, we investigate how witnessing morning rudeness influences workers’ subsequent perceptions and behaviors throughout the workday.  We posit that a single exposure to rudeness in the morning can contaminate employees’ perceptions of subsequent social interactions leading them to perceive greater workplace rudeness throughout their workday.  We expect that these contaminated perceptions will have important ramifications for employees’ work behaviors.  In a 10-day study of 81 professional and managerial employees, we find that witnessed morning rudeness leads to greater perceptions of workplace rudeness throughout the workday and that those perceptions, in turn, predict lower task performance and goal progress and greater interaction avoidance and psychological withdrawal.

I can vouch for this from my own personal experience.  When the first thing I'm faced with in the morning is a news story about how horrible people are, or -- worse -- someone online being awful to me or to a friend, I'm set up to be grouchy and irritable for the rest of the day.

However. I've found that the reverse is also true.  When I'm in a sour mood and something unexpectedly good happens, my frame of mind can flip just as quickly.  All of which is yet another indication that we should strive to be as polite and kind as we can; you never know whose life you may be touching.

And I think the same thing applies more globally to the people, media, and general context we're exposed to every day.  If you allow yourself to be constantly bombarded by rudeness, negativity, and bad news, it's kind of inevitable that you'll eventually get swallowed up by it.

I'm not trying to turn us into some modern-day version of Dr. Pangloss from Voltaire's Candide -- smiling blandly and chanting, "Everything happens for the best in the best of all possible worlds."  We shouldn't blind ourselves to the ills of society.  (Witness yesterday's post about the urgency with which we should be addressing climate change.)  But this is no excuse for meanness and cynicism.  Looking at the world honestly, and keeping in mind that the vast majority of people are kind, compassionate, and friendly, are equally important.  You certainly aren't going to do yourself or the world any favors by allowing yourself to be driven to the conclusion that humanity is irredeemably evil.

As author Ken Keyes put it, "A loving person lives in a loving world.  A hostile person lives in a hostile world.  Everyone you meet is your mirror."

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Shattering the records

As I write this, large chunks of the states of California, Oregon, Washington, Utah, Nevada, Idaho, Arizona, New Mexico, Texas, Louisiana, Mississippi, Alabama, and Florida are under NOAA Heat Advisories.  June saw over a thousand temperature records set, and conditions this week are predicted to break at least some of those records in the next few days.

For the third time in the last six weeks, out-of-control wildfires in Canada are dumping smoke across the Midwest and Northeast.  Montana, Nebraska, Minnesota, Iowa, Illinois, Indiana, Michigan, New York, Pennsylvania, New Jersey, and Ohio are all under Air Quality Advisories, with many areas posting AQIs of over 200 -- "Very Unhealthy For All Individuals."

The eastern parts of New York, Pennsylvania, and Maryland, and all of Delaware, New Jersey, Connecticut, Massachusetts, Rhode Island, Vermont, New Hampshire, and Maine are under Flood Watches.  Last week, storms dumped an unprecedented amount of rain in the area, resulting in floods in much of Vermont, New Hampshire, and eastern New York, the likes of which have not been seen in recent history.  More torrential downpours are expected into this week.

The European Space Agency released an alarming forecast for a huge swath of Europe, including much of Italy, Spain, France, Germany, and Poland, where a combination of high heat and humidity is predicted to result in life-threatening conditions.  Sixteen cities in Italy, including Rome and Florence, posted "Extreme Heat Warnings" -- the highest level of heat advisory the ESA issues -- with the temperatures in Sicily and Sardinia predicted to reach 48 C (118 F).  If this forecast pans out, it will be an all-time temperature record for the entire continent of Europe.

A heat wave in India and Pakistan in June crossed what one study called "the limits of survivability," reaching 47 C (116 F) with extreme humidity.  The heat was only broken when it started to rain -- but then it didn't stop.  The resulting flooding has caused damage estimated in the millions.  This followed a "once in two hundred years" heat wave in Thailand, Myanmar, Laos, Vietnam, and Malaysia in April.

Sea surface temperatures are the hottest ever recorded.  We're talking pretty much worldwide, here.  Antarctic sea ice is at its lowest level for June -- middle of the Antarctic winter -- since measurements began.  The Atlantic Ocean is so hot it's got the scientists struggling to find words to describe how bad things are.  "The temperatures in the North Atlantic are unprecedented and of great concern," said Michael Sparrow, head of the World Meteorological Organization's World Climate Research Department.  "They are much higher than anything the models predicted."  The European Centre for Medium-Range Weather Forecasting called them "off the charts."  This raises the specter of a bad Atlantic hurricane year, although how the high temperatures will interact with other factors -- such as wind shear and the fact that we're going into an El Niño, usually an Atlantic storm suppressor -- are unknown.

How much evidence do people need?

It's not so hard to say, you know?  Give it a try, climate-change deniers.  "Well, I guess we were wrong, then."  "Maybe we should have listened to the scientists, who have been warning us about this for forty fucking years."  

But no.  Just yesterday I saw someone post a photograph of a buckled road surface in Louisiana...

... and blamed it on the fact that the contractors hired to build roads don't give a damn and are doing slipshod work.

Yes, I know, all of the information I posted above is weather, and "weather is not climate," a phrase the climate change deniers like to trot out when it's convenient and then proceed to forget about when they gleefully point out there's been a cold snap in Minnesota in January.  I'm not exaggerating; James Inhofe, retired (thank heaven) senator from Oklahoma, set a new record himself -- for the stupidest thing ever said in the halls of the United States Senate -- when he brought a snowball inside in December and claimed it was proof that anthropogenic climate change is a hoax.  

Any individual record that's been broken this year is "weather."  Taken all together, what we have is "climate."

Not to mention a crisis that is threatening the long-term habitability of the planet.

Look, it's time we stop playing nice, here.  There's a point at which giving a forum to people who are either ignorant, or else have a vested interest in hoodwinking the gullible, isn't "giving the other side a chance to speak their views," it's a well-nigh suicidal waste of time we don't have.  I've quoted Isaac Asimov many times, but we cannot continue to allow the control of the planet to be hijacked by people who believe that "my ignorance is just as good as your knowledge."  Contrary to what they say, the climate change deniers didn't "do their research;" they were bamboozled by Fox News, Newsmax, and other media in the pockets of the fossil fuels industry.  At best, they spent fifteen minutes cherry-picking websites that agreed with what they already believed and completely ignored the actual research done by actual scientists.

The result?  A populace who sees a buckled road surface in the middle of a catastrophic, life-threatening heat wave, and blames it on inept road workers.

Is it already too late?  I honestly don't know.  Doesn't abrogate our responsibility to do what we can.  I don't know of anyone who, if their house was on fire, would tell the firemen, "Don't bother trying to save it."  At this point, though, I'm sure of one thing; the only solution is to get to the ballot box and vote out the fossil-fuel-funded political hacks who have spent decades pulling the wool over our eyes and fooling us into believing nothing is wrong. 

If we don't, I can nearly guarantee that this blisteringly hot summer will be the coolest one we'll have for a very long time.

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Circles, bursts, and transients

I hate it when popular media reports on science stories with headlines like, "New Discovery Has Researchers Stumped!" and "This Will Rewrite Every Textbook On The Subject!" and "Recent Find Sends Scientists Back To The Drawing Board!"

The truth is that it's very seldom that real, honest-to-goodness paradigm shifts happen in science.  We've been at this long enough that most of the basic theory, in just about every branch of science, is on rock-solid footing.  It's highly doubtful that much of anything will "rewrite all the textbooks," and as far as the last one, I tend to agree with eminent astrophysicist Neil deGrasse Tyson.  "As scientists, we're always at the drawing board.  If you're not at the drawing board, you're not doing science."

As we've seen over the last few days' posts, however, that doesn't mean the experts have everything figured out.  Even if the overall edifice of science is on a firm enough foundation that it's doubtful it'll ever be significantly overturned, there's still plenty of area to explore around what NdGT calls "the perimeter of our ignorance."

So appropriately enough, given our recent theme of "Stuff We Haven't Figured Out Yet," today we're going to look at three recently discovered astronomical phenomena that thus far, have eluded astrophysicists' best attempts at an explanation.

First, we have the aptly-named odd radio circles that were discovered through work at the Australian Square Kilometre Array Pathfinder Telescope (ASKAP).  These structures, which "do not seem to correspond to any... known object or artefact," resemble gossamer soap bubbles in space, glowing faintly in the radio region of the electromagnetic spectrum (thus the name).  As far as astronomers have found, none of them seem to have anything at the center, which seems to rule out something like a planetary nebula, which is the (usually) spherical shell of ionized gas blown off the surface of a red giant star as it nears the end of its life.

Odd radio circle ORC J-2103-6200, in an image from the MeerKAT Radio Telescope in South Africa [Image licensed under the Creative Commons Jayanne English MeerKAT, ORC J2103-6200 2022, CC BY-SA 4.0]

Thus far, five odd radio circles have been identified, but astrophysicists have no good explanation of how they form.

Second, we have high energy neutrino bursts.  You probably know that neutrinos are tiny, electrically-neutral particles with such a vanishingly small rest mass that they almost never interact with matter at all.  As you read that last sentence, literally billions of neutrinos went right through you, and very likely not a single one affected any of your atoms in the slightest.  

So as you might imagine, studying such an aloof particle isn't easy.  But that's exactly what the IceCube Neutrino Observatory at Amundsen-Scott South Pole Station in Antarctica does -- uses highly sensitive detectors, dropped into deep holes bored into the Antarctic ice sheet, to catch the elusive motes of energy when they do interact with the matter they're flying through.

And at IceCube, they found twenty-eight separate events that defy explanation -- neutrinos that carried an astonishing energy of 50 trillion electron volts.  "The events cannot be explained by other neutrino fluxes, such as those from atmospheric neutrinos, nor by other high-energy events, such as muons produced by the interaction of cosmic rays in the atmosphere," the researchers said.  "The neutrinos are known to be extra-galactic in origin, and reach such extreme energies that, according to current physics, they must be generated in the equivalent of a huge-scale natural particle accelerator of some kind -- possibly black-hole driven."

But what process could give neutrinos such ridiculously high energies is thus far unknown.

If that's not extreme enough for you, consider a newly-discovered class of astronomical objects called fast blue optical transients.  FBOTs, as they're called, create sudden bursts of energy peaking in the blue region of the visible light spectrum, but (true to their name) fade almost as soon as they peak.  This makes spotting them tricky; you have to have your telescope pointing exactly the right direction at exactly the right time to see them.  The result is that only three have been observed thus far, but what we've seen is nothing short of astonishing.

FBOTs are high on the list of the most energetic phenomena ever studied.  In a fraction of a second, they eject material with a mass of around one-tenth that of the Sun -- at a velocity of 55%  of the speed of light.  The study, which appeared in Astrophysical Journal Letters, reflects how hard it is to talk about these things without lapsing into superlatives.

"This was unexpected," said Northwestern University's Deanne Coppejans, first author of the study, which is such an understatement it's kind of funny.  "We know of energetic explosions that can eject material at almost the speed of light, specifically gamma ray bursts, but they only launch a small amount of mass -- about one millionth the mass of the sun.  CSS161010 [one of the FBOTs Coppejans and her team studied] launched between one and ten percent the mass of the Sun at more than half the speed of light -- evidence that this is a new class of transient."

"We thought we knew what produced the fastest outflows in nature," said Raffaella Margutti, also of Northwestern, and a senior author of the study.  "We thought there were only two ways to produce them -- by collapsing a massive star with a gamma ray burst or two neutron stars merging.  We thought that was it.  With this study, we are introducing a third way to launch these outflows.  There is a new beast out there, and it's able to produce the same energetic phenomenon."

However, the mechanism by which these objects propel that kind of mass at such phenomenal speeds is completely unknown.

So there we are.  Three astrophysical puzzles that are on the other side of the "perimeter of our ignorance."  Thus illustrating what I've said many times before, which is that if you're interested in science, you'll never be bored.  Also shows that Shakespeare had it spot-on four-hundred-odd years ago, doesn't it?  "There are more things in heaven and earth, Horatio/ Than are dreamt of in your philosophy."

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The halting problem

A couple of months ago, I wrote a post about the brilliant and tragic British mathematician, cryptographer, and computer scientist Alan Turing, in which I mentioned in passing the halting problem.  The idea of the halting problem is simple enough; it's the question of whether a computer program designed to determine the truth or falsity of a mathematical theorem will always be able to reach a definitive answer in a finite number of steps.  The answer, surprisingly, is a resounding no.  You can't guarantee that a truth-testing program will ever reach an answer, even about matters as seemingly cut-and-dried as math.  But it took someone of Turing's caliber to prove it -- in a paper mathematician Avi Wigderson called "easily the most influential math paper in history."

What's the most curious about this result is that you don't even need to understand fancy mathematics to find problems that have defied attempts at proof.  There are dozens of relatively simple conjectures for which the truth or falsity is not known, and what's more, Turing's result showed that for at least some of them, there may be no way to know.

One of these is the Collatz conjecture, named after German mathematician Lothar Collatz, who proposed it in 1937.  It's so simple to state that a bright sixth-grader could understand it.  It goes like this:

Start with any positive integer you want.  If it's even, divide it by two.  If it's odd, multiply it by three and add one.  Repeat.  Here's a Collatz sequence, starting with the number seven:

7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1.

Collatz's conjecture is that if you do this for every positive integer, eventually you'll always reach one.

The problem is, the procedure involves a rule that reduces the number you've got (n/2) and one that grows it (3n + 1).  The sequence rises and falls in an apparently unpredictable way.  For some numbers, the sequence soars into the stratosphere; starting with n = 27, you end up at 9,232 before it finally hits a number that allows it to descend to one.  But the weirdness doesn't end there.  Mathematicians studying this maddening problem have made a graph of all the numbers between one and ten million (on the x axis) against the number of steps it takes to reach one (on the y axis), and the following bizarre pattern emerged:

[Image licensed under the Creative Commons Kunashmilovich, Collatz-10Million, CC BY-SA 4.0]

So it sure as hell looks like there's a pattern to it, that it isn't simply random.  But it hasn't gotten them any closer to figuring out if all numbers eventually descend to one -- or if, perhaps, there's some number out there that just keeps rising forever.  All the numbers tested eventually descend, but attempts to figure out if there are any exceptions have failed.

Despite the fact that in order to understand it, all you have to be able to do is add, multiply, and divide, American mathematician Jeffrey Lagarias lamented that the Collatz conjecture "is an extraordinarily difficult problem, completely out of reach of present-day mathematics."

Another theorem that has defied solution is the Goldbach conjecture, named after German mathematician Christian Goldbach, who proposed it to none other than mathematical great Leonhard Euler.  The Goldbach conjecture is even easier to state:

All positive integers greater than two can be expressed as the sum of two prime numbers.

It's easy enough to see that the first few work:

3 = 1 + 2
4 = 1 + 3
5 = 2 + 3
6 = 3 + 3 (or 1 + 5)
7 = 2 + 5
8 = 3 + 5

and so on.

But as with Collatz, showing that it works for the first few numbers doesn't prove that it works for every number, and despite nearly three centuries of efforts (Goldbach came up with it in 1742), no one's been able to prove or disprove it.  They've actually brute-force tested all numbers between 3 and 4,000,000,000,000,000,000 -- I'm not making that up -- and they've all worked.

But a general proof has eluded the best mathematical minds for close to three hundred years.

The bigger problem, of course, is that Turing's result shows that not only do we not know the answer to problems like these, there may be no way to know.  Somehow, this flies in the face of how we usually think about math, doesn't it?  The way most of us are taught to think about the subject, it seems like the ultimate realm in which there are always definitive answers.

But here, even two simple-to-state conjectures have proven impossible to solve.  At least so far.  We've seen hitherto intractable problems finally reach closure -- the four-color map theorem comes to mind -- so it may be that someone will eventually solve Collatz and Goldbach.

Or maybe -- as Turing suggested -- the search for a proof will never halt.

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Mystery relics

I was cleaning up my garage a while back, and I found this.

There are two holes on the squared-off lower edge, so it was evidently meant to be attached to something else by screws or bolts.  There was no context; it wasn't with anything else that might have given me a clue to what its purpose was.

It took way longer than it should have for me to figure out that it's a toe clip from a bicycle pedal.

This got me to thinking about how hard archaeologists have it.  They dig stuff up, often damaged or fragmentary, and have to figure out what it is, why it was created, what its uses may have been.  And if a relatively simple artifact from a device I use frequently left me scratching my head, how much harder is it when it's a creation of a long-dead culture about which we know very little?

I thought it might be entertaining to look at a few artifacts that have even the experts stumped -- where, like my pedal toe clip, we actually have the thing in hand and still can't figure out what it's used for.

1.  South Asian ringstones

In India and Pakistan, a number of beautifully-carved stone artifacts have been found.  They're circular, flat, with a hole in the center, and have fine decorative relief on one side and a polished surface on the other.

Indian ringstone, approximately 2,200 years old, in the New York Metropolitan Museum of Art [Image is in the Public Domain]

Over seventy ringstones have been found, but their purpose is entirely unknown.  They're too heavy to be jewelry.  It's possible they were some sort of object of veneration, but that's entirely speculation.  Another possibility is that they were used as a pattern mold for impressing another substance (perhaps clay or gold foil) to make jewelry or decorative objects, but there's no particularly good evidence for that, either; and if they're molds, why are they always circular, with a hole through the center?

2. The Disquis spheres

In the Disquis delta region of Costa Rica, there are over three hundred nearly perfect stone spheres, most of which are made of a hard rock called diorite.  They range from a few centimeters to over two meters in diameter; the largest weigh more than fifteen tons.

[Image licensed under the Creative Commons Axxis10, Parque de las Esferas de Costa Rica, CC BY-SA 3.0]

Whoever made them put an incredible amount of work into them.  Stone artifacts are hard to date accurately, but nearby archaeological sites are about a thousand years old, so it's presumed that whoever made them came from around that era.  What purpose did they serve?

No one knows.

3. Erdstalls

Sometimes an artifact being both widespread and relatively recent doesn't help much.  This is the situation with erdstalls -- low, narrow tunnels found throughout central Europe, and which are believed to date from the Middle Ages.  

An erdstall in Austria [Image licensed under the Creative Commons Pfeifferfranz, Erdstall Ratgöbluckn Perg Eingang, CC BY-SA 3.0 AT]

Some have theorized that they were hiding places or escape tunnels, but this doesn't seem very plausible.  Although they can be up to fifty meters in length, they average under a meter and a half tall and only sixty centimeters wide.  Any escape tunnel is good enough if you're desperate, I suppose, but it seems like if they were deliberately constructed for that purpose, the makers would have dug them to be a little more spacious.  They're mentioned a couple of times in medieval manuscripts, but their purpose is never specified -- so it's uncertain if even the people who wrote about them knew what they were used for.

4.  "Frying pans"

In graves from the Early Cycladic Period of ancient Greek history (ca. 3100-1000 B.C.E.), archaeologists have found over two hundred shallow ceramic bowls, decorated on the outside, with short handles.

[Image is in the Public Domain]

They were nicknamed "frying pans" because of the shape, although they show none of the wear you'd expect from a cooking implement (and are really too shallow to be useful for that anyhow).  Other than the general fallback of unspecified "ceremonial uses," one suggestion is that they might have been filled with a thin layer of water or oil and used as mirrors, although that seems to be a little awkward to be practical.  Others have suggested that they were used to evaporate sea water to produce salt -- but they've only been found in burial sites, and none of them have shown any traces of salt.

5. Perforated batons

These are carved pieces of deer antler, widely distributed across Europe, and dating from 12,000 to 23,000 years of age -- so whatever they were for, people made them for over ten thousand years.  

[Image licensed under the Creative Commons Johnbod, Perforated baton with low relief horse, CC BY-SA 3.0]

They're intricately carved, and all of them have a nearly perfect circular hole cut through the middle.  Despite one researcher's claim that the wear around the inside of the hole shows they were tools (possibly for fashioning or straightening arrows), there are lots of other explanations that have been suggested -- that they're cloak or scarf fasteners, calendars, jewelry, or phallic symbols (not seeing that last one, honestly).  A paper in the journal Archaeological and Anthropological Sciences in 2019 said, "Despite the large number of batons found (> 400), their use still remains enigmatic.  No fewer than forty functional hypotheses have been proposed, following debates that have persisted for over 150 years; the perforated baton has consequently become emblematic of our misunderstanding of some ancient objects’ functions."

Which seems a fitting place to end.  I wonder what future archaeologists will make of the stuff we leave behind -- which bits they'll figure out immediately, and which ones will baffle them?  And as far as the relics that today's archaeologists are frowning over, I've barely scratched the surface.  There are dozens of other kinds of artifacts that have even the experts saying "damned if we know."  Which is not a problem, honestly; being open about the perimeter of your own ignorance is absolutely essential in research of any kind.

But it does set up a lovely bunch of puzzles for us interested laypeople to think about, doesn't it?

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