Adventures in solid geometry

I've always been a bit in awe at people who are true math-adepts.

Now, I'm hardly a math-phobe myself; having majored in physics, I took a great many math courses as an undergraduate.  And up to a point, I was pretty good at it.  I loved calculus -- partly because my teacher, Dr. Harvey Pousson, was a true inspiration, making complex ideas clear and infusing everything he did with curiosity, energy, and an impish sense of humor.  Likewise, I thoroughly enjoyed my class in differential equations, a topic that is often a serious stumbling block for aspiring math students.  Again, this was largely because of the teacher, a five-foot-one, eccentric, hypercharged dynamo named Dr. LaSalle, who was affectionately nicknamed "the Roadrunner" because she was frequently seen zooming around the halls, dodging and weaving around slow-moving students as if she were late for boarding a plane.

I recall Dr. LaSalle finishing up some sort of abstruse proof on the board, then writing "q.e.d."  She turned around, and said in a declamatory voice, "Quod erat demonstrandum.  Which is Latin for 'ha, we sure showed you.'"  It was only much later that I found out her translation was actually pretty accurate.

But other than those bright spots, my math career pretty much was in its final tailspin.  At some point, I simply ran into an intellectual wall.  My sense is that it happened when I stopped being able to picture what I was studying.  Calculating areas and slopes and whatnot was fine; so were the classic differential equations problems involving things like ladders slipping down walls and water leaking out of tanks.  But when we got to fields and matrices and tensors, I was no longer able to visualize what I was trying to do, and it became frustrating to the extent that now -- forty-five years later -- I still have nightmares about being in a math class, taking an exam, and having no idea what I'm doing.

Even so, I have a fascination for math.  There is something grand and cosmic about it, and it underpins pretty much everything.  (As Galileo put it, "Mathematics is the language with which God wrote the universe.")  It's no wonder that Pythagoras thought there was something holy about numbers; there are strange and abstruse patterns and correspondences you start to uncover when you study math that seem very nearly mystical.

The topic comes up because of a paper in the journal Experimental Mathematics that solved a long-standing question about something that also came out of the ancient Greek fascination with numbers -- the five "Platonic solids", geometrical figures whose sides are composed of identical regular polygons and which all have identical vertices.  The five are the tetrahedron (four triangular faces), the cube (six square faces), the octahedron (eight triangular faces), the dodecahedron (twelve pentagonal faces), and the icosahedron (twenty triangular faces).  And that's it.  There aren't any other possibilities given those parameters.

[Image is in the Public Domain]

The research had to do with a question that I had never considered, and I bet you hadn't, either.  Suppose you were standing on one corner of one of these shapes, and you started walking.  Is there any straight path you could take that would return you to your starting point without passing through another corner?  (Nota bene: by "straight," of course we don't mean "linear;" your path is still constrained to the surface, just as if you were walking on a sphere.  A "straight path" in this context means that when you cross an edge, if you were to unfold the two faces -- the one you just left and the one you just entered -- to make a flat surface, your path would be linear.)

Well, apparently it was proven a while back that for four of the Platonic solids -- the tetrahedron, cube, octahedron, and icosahedron -- the answer is "no."  If you launched off on your travels with the rules outlined above, you would either cross another corner or you'd wander around forever without ever returning to your starting point.  Put a different way: to return to your starting point you'd have to cross at least one other corner.

The recent research looks at the odd one out, the dodecahedron.  In the paper "Platonic Solids and High Genus Covers of Lattice Surfaces," mathematicians Jayadev Athreya (of the University of Washington), David Aulicino (of Brooklyn College), and W. Patrick Hooper (of the City University of New York) showed the astonishing result that alone of the Platonic solids, the answer for the dodecahedron is yes -- and in fact, there are 31 different classes of pathways that return you to your starting point without crossing another corner.

The way they did this started out by imagining taking the dodecahedron and opening it up and flattening it out.  You then have a flat surface made of twelve different pentagons, connected along their edges in some way (how depends on exactly how you did the cutting and unfolding).  You start at the vertex of one of the pentagons, and strike off in a random direction.  When you reach the edge of the flattened shape, you glue a second, identical flattened dodecahedron to that edge so you can continue to walk. This new grid will always be a rotation of the original grid by some multiple of 36 degrees.  Reach another edge, repeat the process. Athreya et al. showed that after ten iterations, the next flattened dodecahedron you glue on will have rotated 360 degrees -- in other words, it will be oriented exactly the same way the first one was.

Okay, that's kind of when my brain pooped out.  From there, they took the ten linked, flattened dodecahedrons and folded that back up to make a shape that is like a polygonal donut with eighty-one holes.  And that surface is related mathematically to a well-studied figure called a double pentagon, which allowed the researchers to prove that not only was a straight line returning to your origin without crossing another corner possible, there were 31 ways to do it.

"This was one of the most fun projects I've worked on in my entire career," lead author Jayadev Athreya said, in an interview with Quora.  "It's important to keep playing with things."

But it's also pretty critical to have a brain powerful enough to conceptualize the problem, and I'm afraid I'm not even within hailing distance.  I'm impressed, intrigued, and also convinced that I'd never survive in such rarified air.

So on the whole, it's good that I ended my pursuit of mathematics when I did.  Biology was probably the better choice.  I think I'm more suited to pursuits like ear-tagging fruit bats than calculating straight paths on Platonic solids, but I'm glad there are people out there who are able to do that stuff, because it really is awfully cool.

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

Non-trivial donuts

In the New Research That Sounds Crazy But Isn't department, we have: an inquiry into whether the universe is actually shaped like a donut.

[Image credit: J. Law, ESO]

The overall shape of spacetime is something that is nowhere near as obvious as it might seem to a layperson.  From the look of it, we seem to live in a completely Euclidean universe; perpendicular lines meet at a perfect ninety degree angle, parallel ones never intersect, and all of the other happy stuff you learned in high school geometry class.  But as mathematicians Leonhard Euler and Nikolai Lobachevsky showed, this isn't the only possibility.  The fabric of space could have an overall spherical shape, where there are no parallel lines (a 2-D example of a spherical geometry is the surface of the Earth).  On the other hand, in a hyperbolic space, given a line and a point not on that line, there is an infinite number of parallel lines passing through that point.  (It's harder to picture, for me at least, but a 2-D analog to a hyperbolic space is the surface of a saddle -- or a Pringle's potato chip.)

To our best measurements thus far, however, it looks like the simple solution -- that spacetime is flat and Euclidean -- is correct.  (That's on the largest scales; on small scales, anything with mass warps the geometry of spacetime.  However, it appears that those local divots and dimples are in a spacetime which is, overall, flat.)  

But according to a paper in the journal Physical Review Letters, there might be other possibilities we haven't considered -- ones even more mindblowing than a spherical or hyperbolic universe.

Theoretical physicist Glenn Starkman, of Case Western Reserve University, has proposed that the universe's geometry might have a nontrivial topology.  Euclidean spaces -- and also spheres and saddles -- have what topologists call a trivial topology; the simplest way to think about this is to consider what happens if you draw a closed loop anywhere on one of those surfaces, and then make it shrink.  On a surface with a trivial topology, no matter where you draw it, you can continue to shrink the loop all the way down to a single point.  On one with nontrivial topology, there are at least some loops that you can't do that to without deforming the shape of the surface.

Consider, for example, a donut.  A loop that goes around the donut longitudinally (i.e. through the hole and back around again) can't be shrunk indefinitely; neither can one that runs all the way around the hole.  Shapes with a single hole all the way through are called genus one tori.  A donut is a genus one torus, as is a mug with a single handle.  (Giving rise to the old joke that topologists are so smart that at breakfast, they can't tell their coffee cups from their donuts.)

This may seem like nothing more than intellectual noodling about, but if the universe has a weird non-trivial topology, it could explain ongoing mysteries like the asymmetries (and unexpected symmetries) in the cosmic microwave background radiation.  One possibility is that the geometry of the universe is some kind of multiply-connected hypertorus -- a bit like a three-dimensional version of the old game PacMan, where if you exit the screen on one side, you reappear on the opposite side.  This would mean when you look out into space in one direction, your sight line comes back at you from the other direction.  This could potentially explain another long-standing and vexing problem in physics, the horizon problem -- which is the question of why space is so homogeneous, despite the fact that there are regions of space that, if space has a trivial topology, have been causally disconnected since the time of the Big Bang.  If when you peer one direction into the night sky, your visual line travels in a gigantic loop, the horizon problem kind of goes away; you're seeing the same stuff out at the edges of the universe no matter which way you look.

Of course, even that is not as complicated as it can get.  Starkman and his colleagues have proposed a total of seventeen different possible geometries that aren't ruled out by the observational evidence.  In some, the universe twists as it loops around, so that (using our PacMan analogy) when you exit the screen and reappear on the other side, you're now upside-down.  They are currently proposing looking for similar patterns in regions of space on opposite sides of the universe, but also have to consider that the pattern on one side may be inverted with respect to the other.

As you might imagine, doing this kind of comparison work is way beyond the scope of human analysts; it's going to require some heavy-duty computational firepower.  They're planning on turning over new survey data from the JWST and ESO to rapid machine-learning software for analysis, and we might actually have some preliminary answers by the end of the year.

If they get positive results, it'll be an incredible coup -- not only proposing a whole bunch of new physics, but simultaneously making inroads into solving the long-standing flatness and horizon problems.  I'm not holding my breath -- it's all too often these odd ideas fail the test of empirical evidence -- but wouldn't it be wonderful if it holds up?

I know I'd celebrate by eating a donut.

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

Adventures in solid geometry

I've always been a bit in awe at people who are true math-adepts.

Now, I'm hardly a math-phobe myself; having majored in physics, I took a great many math courses as an undergraduate.  And up to a point, I was pretty good at it.  I loved calculus -- partly because my teacher, Dr. Harvey Pousson, was a true inspiration, making complex ideas clear and infusing everything he did with curiosity, energy, and an impish sense of humor.  Likewise, I thoroughly enjoyed my class in differential equations, a topic that is often a serious stumbling block for aspiring math students.  Again, this was largely because of the teacher, a five-foot-one, eccentric, hypercharged dynamo named Dr. LaSalle, who was affectionately nicknamed "Roadrunner" because she was frequently seen zooming around the halls, dodging and weaving around slow-moving students as if she were late for boarding a plane.

But at some point, I simply ran into an intellectual wall.  My sense is that it happened when I stopped being able to picture what I was studying.  Calculating areas and slopes and whatnot was fine; so were the classic differential equations problems involving things like ladders slipping down walls and water leaking out of tanks.  But when we got to fields and matrices and tensors, I was no longer able to visualize what I was trying to do, and it became frustrating to the extent that now -- forty years later -- I still have nightmares about being in a math class, taking an exam, and having no idea what I'm doing.

Even so, I have a fascination for math.  There is something grand and cosmic about it, and it underpins pretty much everything.  (As Galileo put it, "Mathematics is the language with which God wrote the universe.")  It's no wonder that Pythagoras thought there was something holy about numbers; there are strange and abstruse patterns and correspondences you start to uncover when you study math that seem very nearly mystical.

The topic comes up because of a recent paper in Experimental Mathematics that solved a long-standing question about something that also came out of the ancient Greek fascination with numbers -- the five "Platonic solids", geometrical figures whose sides are composed of identical regular polygons and which all have identical vertices.  The five are the tetrahedron (four triangular faces), the cube (six square faces), the octahedron (eight triangular faces), the dodecahedron (twelve pentagonal faces), and the icosahedron (twenty triangular faces).  And that's it.  There aren't any other possibilities given those parameters.

[Image is in the Public Domain]

The research had to do with a question that I had never considered, and I bet you hadn't, either.  Suppose you were standing on one corner of one of these shapes, and you started walking.  Is there any straight path you could take that would return you to your starting point without passing through another corner?  (Nota bene: by "straight," of course we don't mean "linear;" your path is still constrained to the surface, just as if you were walking on a sphere.  A "straight path" in this context means that when you cross an edge, if you were to unfold the two faces -- the one you just left and the one you just entered -- to make a flat surface, your path would be linear.)

Well, apparently it was proven a while back that for four of the Platonic solids -- the tetrahedron, cube, octahedron, and icosahedron -- the answer is "no."  If you launched off on your travels with the rules outlined above, you would either cross another corner or you'd wander around forever without ever returning to your starting point.  Put a different way: to return to your starting point you'd have to cross at least one other corner.

The recent research looks at the odd one out, the dodecahedron.  In the paper "Platonic Solids and High Genus Covers of Lattice Surfaces," mathematicians Jayadev Athreya (of the University of Washington), David Aulicino (of Brooklyn College), and W. Patrick Hooper (of the City University of New York) showed the astonishing result that alone of the Platonic solids, the answer for the dodecahedron is yes -- and in fact, there are 31 different classes of pathways that return you to your starting point without crossing another corner.

The way they did this started out by imagining taking the dodecahedron and opening it up and flattening it out.  You then have a flat surface made of twelve different pentagons, connected along their edges in some way (how depends on exactly how you did the cutting and unfolding).  You start at the vertex of one of the pentagons, and strike off in a random direction.  When you reach the edge of the flattened shape, you glue a second, identical flattened dodecahedron to that edge so you can continue to walk.  This new grid will always be a rotation of the original grid by some multiple of 36 degrees.  Reach another edge, repeat the process.  Athreya et al. showed that after ten iterations, the next flattened dodecahedron you glue on will have rotated 360 degrees -- in other words, it will be oriented exactly the same way the first one was.

Okay, that's kind of when my brain pooped out.  From there, they took the ten linked, flattened dodecahedrons and folded that back up to make a shape that is like a polygonal donut with eighty-one holes.  And that surface is related mathematically to a well-studied figure called a double pentagon, which allowed the researchers to prove that not only was a straight line returning to your origin without crossing another corner possible, there were 31 ways to do it.

"This was one of the most fun projects I've worked on in my entire career," lead author Jayadev Athreya said, in an interview with Quora.  "It's important to keep playing with things."

But it's also pretty critical to have a brain powerful enough to conceptualize the problem, and I'm afraid I'm not even within hailing distance.  I'm impressed, intrigued, and also convinced that I'd never survive in such rarified air.

So on the whole, it's good that I ended my pursuit of mathematics when I did.  Biology was probably the better choice.  I think I'm more suited to pursuits like ear-tagging fruit bats than calculating straight paths on Platonic solids, but I'm really glad there are people out there who are able to do that stuff, because it really is awfully cool.

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

This week's Skeptophilia book recommendation of the week should be in everyone's personal library.  It's the parting gift we received from the brilliant astrophysicist Stephen Hawking, who died two years ago after beating the odds against ALS's death sentence for over fifty years.

In Brief Answers to the Big Questions, Hawking looks at our future -- our chances at stopping anthropogenic climate change, preventing nuclear war, curbing overpopulation -- as well as addressing a number of the "big questions" he references in the title.  Does God exist?  Should we colonize space?  What would happen if the aliens came here?  Is it a good idea to develop artificial intelligence?

And finally, what is humanity's chance of surviving?

In a fascinating, engaging, and ultimately optimistic book, Hawking gives us his answers to the questions that occupy the minds of every intelligent human.  Published posthumously -- Hawking died in March of 2018, and Brief Answers hit the bookshelves in October of that year -- it's a final missive from one of the finest brains our species ever produced.  Anyone with more than a passing interest in science or philosophy should put this book on the to-read list.

[Note: if you purchase this book using the image/link below, part of the proceeds goes to support Skeptophilia!]

Falling in line

What amazes me about so many crazy claims is that you get the impression that the people making them didn't even try to find a natural explanation.

It's one thing to speculate wildly about a phenomenon for which science is still searching for explanations.  Déjà vu, for example, is one experience that virtually everyone shares, and for which no convincing explanation has yet been found.  It's no wonder that it's fertile ground for people who prefer to ascribe such occurrences to the paranormal.

But in other cases, there is such a simple, convincing natural explanation that you have to wonder why the claimant isn't going there.  Such, for example, is the suggestion over at the phenomenally bizarre quasi-religious site The Watchman's Cry that geographical locations on the Earth that have been the sites of disasters (natural or manmade) fall along connecting lines, making some sort of mystical, meaningful pattern.

The article starts out with a bang, with the phrase, "Several months ago, I had four prophetic dreams which took place on the same night."  Four precognitive dreams is pretty impressive, I have to say, especially since most skeptics don't think precognition occurs at all.  Be that as it may, these dreams involved train wrecks, which is ironic, because that is what the rest of the site turns out to be.

Both literally and figuratively.

The site goes into great detail about various train derailments, and how if you connect them by lines (great circles, to be more precise), those lines then go around the Earth and connect to other sites that have had bad things happen.  These then intersect other such great circles, which go other interesting places, and so on.

[Image is in the Public Domain]

It's just ley lines all over again, isn't it?  If your search parameters are wide enough -- basically, "anywhere that anything bad has happened in the past two centuries" -- you can find great circles that link them up.  Which is entirely unsurprising. I could draw a great circle anywhere on Earth and pretty much guarantee that I'll find three or more sites near it that had some kind of natural or manmade calamity in the past two centuries.  The Earth is a big place, and there are lots of calamities to choose from.

So this whole thing is an excellent example of the Texas Sharpshooter Fallacy, the choosing of data points favorable to your hypothesis after the fact.  The name comes from a folk story:

A traveler through Texas passed a barn that had a bullseye painted on the side, with three bullet holes near the dead center of the target.  There were two old-timers leaning on a fence nearby, and the visitor slowed down his car and said, "That's some pretty good shooting, right there."

One of the old-timers grins, and says, "Why, thank you."

The other one scowls.  "Don't pay any attention to him.  He just got drunk one night and shot the side of his barn, then the next morning painted a bullseye around the bullet holes."

Anyhow, what gets me most about the claim in The Watchman's Cry is that they don't even seem to understand that given the fact that the Earth is a sphere (an oblate spheroid, to be precise, but let's not get technical), a given point on Earth has an infinite number of great circles passing through it.  Just as two points on a plane define a line, two points on a sphere define a great circle.  And his lack of grasp of simple geometry becomes apparent when he tells us that it's amazing that two intersecting great circles (ones connecting Houston, Texas to train derailment sites in Rosedale, Maryland and Bear Creek, Alabama, respectively) were "only 900 feet apart."

How can you say that two intersecting lines are any specific distance apart?  If they intersect, they are (at that point) zero feet apart.  Farther from the intersection, they are farther apart.  Because that's how intersection works.

But the author of this site trumpets this statement as if it were some kind of epiphany.  It's like being excited because you found a triangle that had three sides.

I'll leave you to explore the site on your own, if you're curious to see more of this false-pattern malarkey, but suffice it to say that there's nothing at all mystical going on here.  He's adding geometry to coincidence and finding meaning, and it's no great surprise that it turns out to be the meaning he already believed going into it.

So like the ley lines people, this guy doesn't seem to be trying very hard to see if there's a natural explanation that sufficiently accounts for all of the facts, a tendency I have a hard time comprehending.  Why are people attracted to this kind of hokum?  Science itself is a grand, soaring vision, telling us that we are capable of understanding how the universe works, from the realm of the enormous to the realm of the unimaginably small.  With a little work, you can find out the rules that govern everything from galaxies to quarks.

But that, apparently, isn't enough for some people.

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

This week's Skeptophilia book recommendation of the week is a fun one -- George Zaidan's Ingredients: The Strange Chemistry of What We Put In Us and On Us.  Springboarding off the loony recommendations that have been rampant in the last few years -- fad diets, alarmist warnings about everything from vaccines to sunscreen, the pros and cons of processed food, substances that seem to be good for us one week and bad for us the next, Zaidan goes through the reality behind the hype, taking apart the claims in a way that is both factually accurate and laugh-out-loud funny.

And high time.  Bogus health claims, fueled by such sites as Natural News, are potentially dangerous.  Zaidan's book holds a lens up to the chemicals we ingest, inhale, and put on our skin -- and will help you sort the fact from the fiction.

[Note: if you purchase this book using the image/link below, part of the proceeds goes to support Skeptophilia!]

Lines of sight

What amazes me about so many crazy claims is that you get the impression that the people making them didn't even try to find a natural explanation.

It's one thing to speculate wildly about a phenomenon for which science is still searching for explanations.  Déjà vu, for example, is one experience that virtually everyone shares, and for which no convincing explanation has yet been found.  It's no wonder that it's fertile ground for people who prefer to ascribe such occurrences to the paranormal.

But in other cases, there is such a simple, convincing natural explanation that you have to wonder why the claimant isn't going there.  Such, for example, is the suggestion over at the phenomenally bizarre quasi-religious site The Watchman's Cry that geographical locations on the Earth that have been the sites of disasters (natural or manmade) fall along connecting lines, making some sort of mystical, meaningful pattern.

The article starts out with a bang, with the phrase, "Several months ago, I had four prophetic dreams which took place on the same night."  Four precognitive dreams is pretty impressive, I have to say, especially since most skeptics don't think precognition occurs at all.  Be that as it may, these dreams involved train wrecks, which is ironic, because that is what the rest of the site turns out to be.

Both literally and figuratively.

The site goes into great detail about various train derailments, and how if you connect them by lines (great circles, to be more precise), those lines then go around the Earth and connect to other sites that have had bad things happen.  These then intersect other such great circles, which go other interesting places, and so on.

[image courtesy of the Wikimedia Commons]

It's just ley lines all over again, isn't it?  If your search parameters are wide enough -- basically, "anywhere that anything bad has happened in the past two centuries" -- you can find great circles that link them up.  Which is entirely unsurprising.  I could draw a great circle anywhere on Earth and pretty much guarantee that I'll find three or more sites near it that had some kind of natural or manmade calamity in the past two centuries.  The Earth is a big place, and there are lots of calamities to choose from.

But what gets me most about this guy is that he doesn't even seem to understand that given the fact that the Earth is a sphere (an oblate spheroid, to be precise, but let's not get technical), a given point on Earth has an infinite number of great circles passing through it.  Just as two points on a plane define a line, two points on a sphere define a great circle.  And his lack of grasp of simple geometry becomes apparent when he tells us that it's amazing that two intersecting great circles (ones connecting Houston, Texas to train derailment sites in Rosedale, Maryland and Bear Creek, Alabama, respectively) were "only 900 feet apart."

How can you say that two intersecting lines are any specific distance apart?  If they intersect, they are (at that point) zero feet apart.  Further from the intersection, they are further apart.  Because that's how intersection works.

But the author of this site trumpets this statement as if it were some kind of epiphany.  It's like being excited because you found a triangle that had three sides.

I'll leave you to explore the site on your own, if you're curious to see more of this false-pattern malarkey, but suffices to say that there's nothing at all mystical going on here.  He's adding geometry to coincidence and finding meaning, and it's no great surprise that it turns out to be the meaning he already believed going into it.

So like the ley lines people, this guy doesn't seem to be trying very hard to see if there's a natural explanation that sufficiently accounts for all of the facts, a tendency I have a hard time comprehending.  Why are people attracted to this kind of hokum?  Science itself is a grand, soaring vision, telling us that we are capable of understanding how the universe works, from the realm of the enormous to the realm of the unimaginably small.  With a little work, you can find out the rules that govern everything from galaxies to quarks.

But that, apparently, isn't enough for some people.