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.

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The celestial dance

It's interesting how the approach to science has changed in the last four centuries.

It's easy to have the (mistaken) impression that as long as we humans have been doing anything scientific, we've always done it the same way -- looked at the evidence and data, then tried to come up with an explanation.  But science in Europe before the eighteenth-century Enlightenment was largely done the other way around; you constructed your model from pure thought, based on a system of how you believed things should act, and once you had the model, you cast about for information supporting it.

It's why Aristotle's statement that the rate of speed of a falling object is directly proportional to its mass stood essentially unchallenged for over a millennium and a half despite the fact that it's something any second grader could figure out was wrong simply by dropping two different-sized rocks from the same height and observing they hit the ground at exactly the same time.  As odd as it is to our twenty-first century scientific mindset, the idea of figuring out if your claim is correct by testing it really didn't catch on until the 1700s.  Which is why the church fathers got so hugely pissed off at Galileo; using a simple experiment he showed that Aristotle got it wrong, and then followed that up by figuring out how things up in the sky moved (such as the moons of Jupiter, first observed by Galileo through the telescope he made).  And this didn't result in the church fathers saying, "Whoa, okay, I guess we need to rethink this," but their putting Galileo on trial and ultimately under permanent house arrest.

That "think first, observe later" approach to science plagued our attempts to understand the universe for a long time after Galileo; people first came up with how they thought things should work, often based on completely non-scientific reasons, then looked for data to support their guess.  That we've come as far as we have is a tribute to scientists who were able to break out of the straitjacket of what the Fourth Doctor in Doctor Who called "not altering their views to fit the facts, but altering the facts to fit their views."

One of the best examples of this was the seventeenth-century astronomer Johannes Kepler.  He was a deeply religious man, and lived in a time when superstition ruled pretty much everything -- in fact, Kepler's mother, Katharina (Guldenmann) Kepler, narrowly escaped being hanged for witchcraft.  Kepler, and most other European astronomers from his time and earlier, were as much astrologers as scientists; they expected the heavens to operate by some kind of law of divine celestial perfection, where objects moved in circles (anything else was viewed as imperfect) and their movements had a direct effect on life down here on Earth.

At the beginning, Kepler tried to extend his conviction of the mathematical perfection of the cosmos to the distances at which the planets revolved around the Sun.  He became convinced that the spacing of the planets' orbits was determined by conforming to the five Platonic solids -- cube, dodecahedron, tetrahedron, icosahedron, and octahedron -- convex polyhedra whose sides are made up only of identical equal-sided polygons.  He tried nesting them one inside the other to see if the ratios of their spacing could be made to match the estimated spacing of the planets, and got close, but not close enough.  One thing Kepler had going for him was he was firmly committed to the truth, and self-aware enough to know when he was fudging things to make them fit.  So he gave up on the Platonic solids, and went back to "we don't know why they're spaced as they are, but they still travel in perfect circles" -- until careful analysis of planetary position data by the Danish observational astronomer Tycho Brahe showed him again that he was close, but not quite close enough.

This was the moment that set Kepler apart from his contemporaries; because instead of shrugging off the discrepancy and sticking to his model that the heavens had to move in perfect circles, he jettisoned the whole thing and went back to the data to figure out what sort of orbits did make sense of the observations.  After considerable work, he came up with what we now call Kepler's Laws of Planetary Motion, including that planets move in "imperfect" elliptical, not circular, orbits, with the Sun at one focus.

Start with the data, and see where it drives you.  It's the basis of all good science.

[Image licensed under the Creative Commons Gonfer, Kepler-second-law, CC BY-SA 3.0]

What got me thinking about Kepler and his abandonment of the Platonic-solid-spacing idea was a paper this week in Astronomy & Astrophysics showing that even though Kepler initially was on the wrong track, there are sometimes odd mathematical regularities that pop up in the natural world.  (A well-known one is how often the Fibonacci series shows up in the organization of things like flower petals and the scales of pine cones.)  The paper, entitled "Six Transiting Planets and a Chain of Laplace Resonances in TOI-178," by a team led by Adrien Leleu of the Université de Genève, showed that even though hard data dashed Kepler's hope of the motion of the heavens being driven by some concept of mathematical perfection, there is a weird pattern to the spacing of planets in certain situations.  The patterns, though, are driven not by some abstract philosophy, but by physics.

In physics, resonance occurs when the physical constraints of a system make them oscillate at a rate called the "natural frequency."  A simple example is the swing of a pendulum; a pendulum of a given length and mass distribution only will swing back and forth at one fixed rate, which is why they can be used in timekeeping.  The motion of planets (or moons) is also an oscillating system, and a given set of objects of particular masses and distances from their center of gravity will tend to fall into resonance, the same as if you try to swing a pendulum at a different rate than the rate at which it "wants to go," then let it be, it'll pretty much immediately revert to swinging at its natural frequency.

The three largest moons of Jupiter exhibit resonance; they've locked into orbits that are the most stable for the system, which turns out to be a 4:2:1 resonance, meaning that the innermost (Io) makes two full orbits in the time the next one (Europa) makes a single orbit, and four full orbits in the time it takes for the farthest (Ganymede).

This week's paper found a more complex resonance pattern in five of the planets around TOI-178, a star two hundred light years away in the constellation Sculptor.  It's a 18:9:6:4:3 resonance chain -- the nearest planet orbits eighteen times as the farthest orbits once, the next farthest nine times as the farthest orbits once, and so on.  This pattern was locked in despite the fact that the planets are all quite different from each other; some are small, rocky planets like Earth, others low-density gaseous planets like Neptune.

"This contrast between the rhythmic harmony of the orbital motion and the disorderly densities certainly challenges our understanding of the formation and evolution of planetary systems," said study lead author Adrien Leleu, in an interview with Science Daily.

So the dance of the celestial bodies is orderly, and shows some really peculiar regularities that you wouldn't have guessed.  But unlike Kepler's favored (but ultimately abandoned) idea that the perfect heavens had to be arranged by perfect mathematics, the Leleu et al. paper shows us that those patterns only emerge by analysis of the data itself, rather than the faulty top-down attempt to force the data to conform to the way you think things should be.  Once you open your mind up to going where the hard evidence leads, that's when the true wonders of the universe begin to emerge.

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Just last week, I wrote about the internal voice most of us live with, babbling at us constantly -- sometimes with novel or creative ideas, but most of the time (at least in my experience) with inane nonsense.  The fact that this internal voice is nearly ubiquitous, and what purpose it may serve, is the subject of psychologist Ethan Kross's wonderful book Chatter: The Voice in our Head, Why it Matters, and How to Harness It, released this month and already winning accolades from all over.

Chatter not only analyzes the inner voice in general terms, but looks at specific case studies where the internal chatter brought spectacular insight -- or short-circuited the individual's ability to function entirely.  It's a brilliant analysis of something we all experience, and gives some guidance not only into how to quiet it when it gets out of hand, but to harness it for boosting our creativity and mental agility.

If you're a student of your own inner mental workings, Chatter is a must-read!

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

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.

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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!]