Greg Egan has a new novel out, called "Incandescence" - so I want to talk about that. Then I'll talk about three of my favorite numbers: 5, 8, and 24. I'll show you how each regular polytope with 5-fold rotational symmetry has a secret link to a lattice living in twice as many dimensions. For example, the pentagon is a 2d projection of a beautiful shape that lives in 4 dimensions. Finally, I'll wrap up with a simple but surprising property of the number 12.
But first: another picture of Jupiter's moon Io! Now we'll zoom in much closer. This was taken in 2000 by the Galileo probe:
1) A continuous eruption on Jupiter's moon Io, Astronomy Picture of the Day, http://apod.nasa.gov/apod/ap000606.html
Here we see a vast plain of sulfur and silicate rock, 250 kilometers across - and on the left, glowing hot lava! The white dots are spots so hot that their infrared radiation oversaturated the detection equipment. This was the first photo of an active lava flow on another world.
If you like pictures like this, maybe you like science fiction. And if you like hard science fiction - "diamond-scratching hard", as one reviewer put it - Greg Egan is your man. His latest novel is one of the most realistic evocations of the distant future I've ever read. Check out the website:
2) Greg Egan, Incandescence, Night Shade Books, 2008. Website at http://www.gregegan.net/INCANDESCENCE/Incandescence.html
The story features two parallel plots. One is about a galaxy-spanning civilization called the Amalgam, and two of its members who go on a quest to our Galaxy's core, which is home to enigmatic beings that may be still more advanced: the Aloof. The other is about the inhabitants of a small world orbiting a black hole. This is where the serious physics comes in.
I might as well quote Egan himself:
"Incandescence" grew out of the notion that the theory of general relativity - widely regarded as one of the pinnacles of human intellectual achievement - could be discovered by a pre-industrial civilization with no steam engines, no electric lights, no radio transmitters, and absolutely no tradition of astronomy.As you can tell, this is a novel of ideas. You have to be willing to work through these ideas to enjoy it. It's also not what I'd call a feel-good novel. As with "Diaspora" and "Schild's Ladder", the main characters seem to become more and more isolated and focused on their work as they delve deeper into the mysteries they are pursuing. By the time the mysteries are unraveled, there's almost nobody to talk to. It's a problem many mathematicians will recognize. Indeed, near the end of "Diaspora" we read: "In the end, there was only mathematics".
At first glance, this premise might strike you as a little hard to believe. We humans came to a detailed understanding of gravity after centuries of painstaking astronomical observations, most crucially of the motions of the planets across the sky. Johannes Kepler found that these observations could be explained if the planets moved around the sun along elliptical orbits, with the square of the orbital period proportional to the cube of the length of the longest axis of the ellipse. Newton showed that just such a motion would arise from a universal attraction between bodies that was inversely proportional to the square of the distance between them. That hypothesis was a close enough approximation to the truth to survive for more than three centuries.
When Newton was finally overthrown by Einstein, the birth of the new theory owed much less to the astronomical facts it could explain - such as a puzzling drift in the point where Mercury made its closest approach to the sun - than to an elegant theory of electromagnetism that had arisen more or less independently of ideas about gravity. Electrostatic and magnetic effects had been unified by James Clerk Maxwell, but Maxwell's equations only offered one value for the speed of light, however you happened to be moving when you measured it. Making sense of this fact led Einstein first to special relativity, in which the geometry of space-time had the unvarying speed of light built into it, then general relativity, in which the curvature of the same geometry accounted for the motion of objects free-falling through space.
So for us, astronomy was crucial even to reach as far as Newton, and postulating Einstein's theory - let alone validating it to high precision, with atomic clocks on satellites and observations of pulsar orbits - depended on a wealth of other ideas and technologies.
How, then, could my alien civilization possibly reach the same conceptual heights, when they were armed with none of these apparent prerequisites? The short answer is that they would need to be living in just the right environment: the accretion disk of a large black hole.
When SF readers think of the experience of being close to a black hole, the phenomena that most easily come to mind are those that are most exotic from our own perspective: time dilation, gravitational blue-shifts, and massive distortions of the view of the sky. But those are all a matter of making astronomical observations, or at least arranging some kind of comparison between the near-black-hole experience and the experience of other beings who have kept their distance. My aliens would probably need to be sheltering deep inside some rocky structure to protect them from the radiation of the accretion disk - and the glow of the disk itself would also render astronomy immensely difficult.
Blind to the heavens, how could they come to learn anything at all about gravity, let alone the subtleties of general relativity? After all, didn't Einstein tell us that if we're free-falling, weightless, in a windowless elevator, gravity itself becomes impossible to detect?
Not quite! To render its passenger completely oblivious to gravity, not only does the elevator need to be small, but the passenger's observations need to be curtailed in time just as surely as they're limited in space. Given time, gravity makes its mark. Forget about black holes for a moment: even inside a windowless space station orbiting the Earth, you could easily prove that you were not just drifting through interstellar space, light-years from the nearest planet. How? Put on your space suit, and pump out all the station's air. Then fill the station with small objects - paper clips, pens, whatever - being careful to place them initially at rest with respect to the walls.
Wait, and see what happens.
Most objects will eventually hit the walls; the exact proportion will depend on the station's spin. But however the station is or isn't spinning, some objects will undergo a cyclic motion, moving back and forth, all with the same period.
That period is the orbital period of the space station around the Earth. The paper clips and pens that are moving back and forth inside the station are following orbits that are inclined at a very small angle to the orbit of the station's center of mass. Twice in every orbit, the two paths cross, and the paper clip passes through the center of the space station. Then it moves away, reaches the point of greatest separation of the orbits, then turns around and comes back.
This minuscule difference in orbits is enough to reveal the fact that you're not drifting in interstellar space. A sufficiently delicate spring balance could reveal the tiny "tidal gravitational force" that is another way of thinking about exactly the same thing, but unless the orbital period was very long, you could stick with the technology-free approach and just watch and wait.
A range of simple experiments like this - none of them much harder than those conducted by Galileo and his contemporaries - were the solution to my aliens' need to catch up with Newton. But catching up with Einstein? Surely that was beyond hope?
I thought it might be, until I sat down and did some detailed calculations. It turned out that, close to a black hole, the differences between Newton's and Einstein's predictions would easily be big enough for anyone to spot without sophisticated instrumentation.
What about sophisticated mathematics? The geometry of general relativity isn't trivial, but much of its difficulty, for us, revolves around the need to dispose of our preconceptions. By putting my aliens in a world of curved and twisted tunnels, rather than the flat, almost Euclidean landscape of a patch of planetary surface, they came better prepared for the need to cope with a space-time geometry that also twisted and curved.
The result was an alternative, low-tech path into some of the most beautiful truths we've yet discovered about the universe. To add to the drama, though, there needed to be a sense of urgency; the intellectual progress of the aliens had to be a matter of life and death. But having already put them beside a black hole, danger was never going to be far behind.
So, this novel is not for everyone! But then, neither is This Week's Finds.
In fact, I was carrying "Incandescence" with me when in mid-September I left the scorched and smoggy sprawl of southern California for the cool, wet, beautiful old city of Glasgow. I spent a lovely week there talking math with Tom Leinster, Eugenia Cheng, Danny Stevenson, Bruce Bartlett and Simon Willerton. I'd been invited to the University of Glasgow to give a series of talks called the 2008 Rankin Lectures. I spoke about my three favorite numbers, and you can see the slides here:
3) John Baez, My favorite numbers, available at http://math.ucr.edu/home/baez/numbers/
I wanted to explain how different numbers have different personalities that radiate like force fields through diverse areas of mathematics and interact with each other in surprising ways. I've been exploring this theme for many years here. So, it was nice to polish some things I've written and present them in a more organized way. These lectures were sponsored by the trust that runs the Glasgow Mathematical Journal, so I'll eventually publish them there. I plan to add a lot of detail that didn't fit in the talks.
I began with the number 5, since the golden ratio and the five-fold symmetry of the dodecahedron lead quickly to a wealth of easily enjoyed phenomena: from Penrose tilings and quasicrystals, to Hurwitz's theorem on approximating numbers by fractions, to the 120-cell and the Poincare homology sphere.
After giving the first talk I discovered the head of the math department, Peter Kropholler, is a big fan of Rubik's cubes. I'd never been attracted to them myself. But his enthusiasm was contagious, especially when he started pulling out the unusual variants that he collects, eagerly explaining their subtleties. My favorite was the Rubik's dodecahedron, or "Megaminx":
4) Wikipedia, Megaminx, http://en.wikipedia.org/wiki/Megaminx
Then I got to thinking: it would be even better to have a Rubik's icosahedron, since its symmetries would then include M12, the smallest Mathieu group. And it turns out that such a gadget exists! It's called "Dogic":
5) Wikipedia, Dogic, http://en.wikipedia.org/wiki/Dogic
The Mathieu group M12 is the smallest of the sporadic finite simple groups. Someday I'd like to understand the Monster, which is the biggest of the lot. But if the Monster is the Mount Everest of finite group theory, M12 is like a small foothill. A good place to start.
Way back in "week20", I gave a cute description of M12 lifted from Conway and Sloane's classic book. If you get 12 equal-sized balls to touch a central one of the same size, and arrange them to lie at the corners of a regular icosahedron, they don't touch their neighbors. There's even room to roll them around in interesting ways! For example, you can twist 5 of them around clockwise so that this arrangement:
1 5 2 6 4 3becomes this:
5 4 1 6 3 2We can generate lots of permutations of the 12 outer balls using twists of this sort - in fact, all even permutations. But suppose we only use moves where we first twist 5 balls around clockwise and then twist 5 others counterclockwise. These generate a smaller group: the Mathieu group M12.
Since we can do twists like this in the Dogic puzzle, I believe M12 sits inside the symmetry group of this puzzle! In a way it's not surprising: the Dogic puzzle has a vast group of symmetries, while M12 has a measly
8 × 9 × 10 × 11 × 12 = 95040
elements. But it'd still be cool to have a toy where you can explore the Mathieu group M12 with your own hands!
The math department lounge at the University of Glasgow has some old books in the shelves waiting for someone to pick them up and read them and love them. They're sort of like dogs at the pound, sadly waiting for somebody to take them home. I took one that explains how Mathieu groups arise as symmetries of "Steiner systems":
6) Thomas Beth, Dieter Jungnickel, and Hanfried Lenz, Design Theory, Cambridge U. Press, Cambridge, 1986.
Here's how they get M12. Take a 12-point set and think of it as the "projective line over F11" - in other words, the integers mod 11 together with a point called infinity. Among the integers mod 11, six are perfect squares:
Call this set a "block". From this, get a bunch more blocks by applying fractional linear transformations:
z |→ (az + b)/(cz + d)
where the matrix
has determinant 1. These blocks then form a "(5,6,12) Steiner system". In other words: there are 12 points, 6 points in each block, and any set of 5 points lies in a unique block.
The group M12 is then the group of all transformations of the projective line that map points to points and blocks to blocks!
If I make more progress on understanding this stuff I'll let you know. It would be fun to find deep mathematics lurking in mutant Rubik's cubes.
Anyway, in my second talk I turned to the number 8. This gave me a great excuse to tell the story of how Graves discovered the octonions, and then talk about sphere packings and the marvelous E8 lattice, whose points can also be seen as "integer octonions". I also sketched the basic ideas behind Bott periodicity, triality, and the role of division algebras in superstring theory.
If you look at my slides you'll also see an appendix that describes two ways to get the E8 lattice starting from the dodecahedron. This is a nice interaction between the magic powers of the number 5 and those of the number 8. After my talk, Christian Korff from the University of Glasgow showed me a paper that fits this relation into a bigger pattern:
7) Andreas Fring and Christian Korff, Non-crystallographic reduction of Calogero-Moser models, Jour. Phys. A 39 (2006), 1115-1131. Also available as arXiv:hep-th/0509152.
They set up a nice correspondence between some non-crystallographic Coxeter groups and some crystallographic ones:
the H2 Coxeter group and the A4 Coxeter group,
the H3 Coxeter group and the D6 Coxeter group,
the H4 Coxeter group and the E8 Coxeter group.
A Coxeter group is a finite group of linear transformations of Rn that's generated by reflections. We say such a group is "non-crystallographic" if it's not the symmetries of any lattice. The ones listed above are closely tied to the number 5:
H2 is the symmetry group of a regular
H3 is the symmetry group of a regular dodecahedron.
H4 is the symmetry group of a regular 120-cell.
Note these live in 2d, 3d and 4d space. Only in these dimensions are there regular polytopes with 5-fold rotational symmetry! Their symmetry groups are non-crystallographic, because no lattice can have 5-fold rotational symmetry.
A Coxeter group is "crystallographic", or a "Weyl group", if it is symmetries of a lattice. In particular:
A4 is the symmetry group of a 4-dimensional lattice also
D6 is the symmetry group of a 6-dimensional lattice also called D6.
E8 is the symmetry group of an 8-dimensional lattice also called E8.
You can see precise descriptions of these lattices in "week65" - they're pretty simple.
Both crystallographic and noncrystallographic Coxeter groups are described by Coxeter diagrams, as explained back in "week62". The H2, H3 and H4 Coxeter diagrams look like this:
5 o---o 5 o---o---o 5 o---o---o---oThe A4, D6 and E8 Coxeter diagrams (usually called Dynkin diagrams) have twice as many dots as their smaller partners H2, H3 and H4:
o---o---o---o o | o---o---o---o---o o | o | o---o---o---o---o---oI've drawn these in a slightly unorthodox way to show how they "grow".
In every case, each dot in the diagram corresponds to one of the reflections that generates the Coxeter group. The edges in the diagram describe relations - you can read how in "week62".
All this is well-known stuff. But Fring and Korff investigate something more esoteric. Each dot in the big diagram corresponds to 2 dots in its smaller partner:
5 o---o o---o---o---o A B B' A" B" A' o C" 5 | o---o---o o---o---o---o---o A B C C' B' A" B" A' o D" | o C" 5 | o---o---o---o o---o---o---o---o---o A B C D D' C' B' A" B" A'If we map each generator of the smaller group (say, the generator D in H4) to the product of the two corresponding generators in the bigger one (say, D'D" in E8), we get a group homomorphism.
In fact, we get an inclusion of the smaller group in the bigger one!
This is just the starting point of Fring and Korff's work. Their real goal is to show how certain exactly solvable physics problems associated to crystallographic Coxeter groups can be generalized to these three noncrystallographic ones. For this, they must develop more detailed connections than those I've described. But I'm already happy just pondering this small piece of their paper.
For example, what does the inclusion of H2 in A4 really look like?
It's actually quite beautiful. H2 is the symmetry group of a regular pentagon, including rotations and reflections. A4 happens to be the symmetry group of a 4-simplex. If you draw a 4-simplex in the plane, it looks like a pentagram:
So, any symmetry of the pentagon gives a symmetry of the 4-simplex. So, we get an inclusion of H2 in A4.
People often say that Penrose tilings arise from lattices in 4d space. Maybe I'm finally starting to understand how! The A4 lattice has a bunch of 4-simplices in it - but when we project these onto the plane correctly, they give pentagrams. I'd be very happy if this were the key.
What about the inclusion of H3 in D6?
Here James Dolan helped me make a guess. H3 is the symmetry group of a regular dodecahedron, including rotations and reflections. D6 consists of all linear transformations of R6 generated by permuting the 6 coordinate axes and switching the signs of an even number of coordinates. But a dodecahedron has 6 "axes" going between opposite pentagons! If we arbitrarily orient all these axes, I believe any rotation or reflection of the dodecahedron gives an element of D6. So, we get an inclusion of H3 in D6.
And finally, what about the inclusion of H4 in E8?
H4 is the symmetry group of the 120-cell, including rotations and reflections. In 8 dimensions, you can get 240 equal-sized balls to touch a central ball of the same size. E8 acts as symmetries of this arrangement. There's a clever trick for grouping the 240 balls into 120 ordered pairs, which is explained by Fring and Korff and also by Conway's "icosian" construction of E8 described at the end of my talk on the number 8. Each element of H4 gives a permutation of the 120 faces of the 120-cell - and thanks to that clever trick, this gives a permutation of the 240 balls. This permutation actually comes from an element of E8. So, we get an inclusion of H4 in E8.
My last talk was on the number 24. Here I explained Euler's crazy "proof" that
1 + 2 + 3 + ... = -1/12
and how this makes bosonic strings happy when they have 24 transverse directions to wiggle around in. I also touched on the 24-dimensional Leech lattice and how this gives a version of bosonic string theory whose symmetry group is the Monster: the largest sporadic finite simple group.
A lot of the special properties of the number 24 are really properties of the number 12 - and most of these come from the period-12 behavior of modular forms. I explained this back in "week125". I recently ran into these papers describing yet another curious property of the number 12, also related to modular forms, but very easy to state:
8) Bjorn Poonen and Fernando Rodriguez-Villegas, Lattice polygons and the number 12. Available at http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.43.2555
9) John M. Burns and David O'Keeffe, Lattice polygons in the plane and the number 12, Irish Math. Soc. Bulletin 57 (2006), 65-68. Also available at http://www.maths.tcd.ie/pub/ims/bull57/M5700.pdf
Consider the lattice in the plane consisting of points with integer coordinates. Draw a convex polygon whose vertices lie on this lattice. Obviously, the differences of successive vertices also lie on the lattice. We can create a new convex polygon with these differences as vertices. This is called the "dual" polygon.
Say our original polygon is so small that the only lattice point in its interior is (0,0). Then the same is true of its dual! Furthermore, the dual of the dual is the original polygon!
But now for the cool part. Take a polygon of this sort, and add up the number of lattice points on its boundary and the number of lattice points on the boundary of its dual. The total is 12.
You can see an example in Figure 1 of the paper by Poonen and Rodriguez-Villegas:
I like how Poonen and Rodriguez-Villegas' paper uses this theorem as a springboard for discussing a big question: what does it mean to "explain" the appearance of the number 12 here? They write:
Our reason for selecting this particular statement, besides the intriguing appearance of the number 12, is that its proofs display a surprisingly rich variety of methods, and at least some of them are symptomatic of connections between branches of mathematics that on the surface appear to have little to do with one another. The theorem (implicitly) and proofs 2 and 3 sketched below appear in Fulton's book on toric varieties. We will give our new proof 4, which uses modular forms instead, in full.
Addenda: I thank Adam Glesser and David Speyer for catching mistakes.
The only noncrystallographic Coxeter groups are the symmetry groups of the 120-cell (H4), the dodecahedron (H3), and the regular n-gons where n = 5,7,8,9,... The last list of groups is usually called In - or better, I2(n), so that the subscript denotes the number of dots in the Dynkin diagram, as usual. But Fring and Korff use "H2" as a special name for I2(5), and that's nice if you're focused on 5-fold symmetry, because then H2 forms a little series together with H3 and H4.
If you examine Poonen and Rodriguez-Villegas' picture carefully, you'll see a subtlety concerning the claim that the dual of the dual is the original polygon. Apparently you need to count every boundary point as a vertex! Read the papers for more precise details.
For more discussion visit the n-Category Café.
When the blind beetle crawls over the surface of a globe, he doesn't realize that the track he has covered is curved. I was lucky enough to have spotted it. - Albert Einstein
© 2008 John Baez