- #1
John Baez
Also available as http://math.ucr.edu/home/baez/week241.html
November 18, 2006
This Week's Finds in Mathematical Physics (Week 241)
John Baez
I've been working too hard, and running around too much, to write
This Week's Finds for a while. A bunch of stuff has built up
that I want to explain. Luckily I've been running around explaining
stuff - higher gauge theory, and tales of the dodecahedron.
This weekend I went to Baton Rouge. I was invited to Louisiana
State University by Jorge Pullin of loop quantum gravity fame, and
I used the opportunity to get a look at LIGO - the Laser
Interferometry Gravitational-Wave Observatory! I took a bunch
of pictures, which you can see in the webpage version of this article.
I described this amazing experiment back in "week189", so I won't
rehash all that. Suffice it to say that there are two installations:
one in Hanford Washington, and one in Livingston Louisiana. Each
consists of two evacuated tubes 4 kilometers long, arranged in an L
shape. Laser beams bounce back and forth between mirrors suspended
at the ends of the tubes, looking for tiny changes in their distance
that would indicate a gravitational wave passing through, stretching
or squashing space. And when I say "tiny", I mean smaller than the
radius of a proton! This is serious stuff.
Jorge drove me in his SUV to Livingston, a tiny town about 20 minutes
from Baton Rouge. While he runs the gravity program at Louisiana
State University, which has links to LIGO, he isn't officially part
the LIGO team. His wife is. When I first met Gabriela Gonzalez, she
was studying the Brownian motion of torsion pendulums. The mirrors in
LIGO are hung on pendulums made of quartz wire, to minimize the effect
of vibrations. But, the random jittering of atoms due to thermal
noise still affects these pendulums. She was studying this noise to
see its effect on the accuracy of the experiment.
This was way back when LIGO was just being planned. Now that LIGO is
a reality, she's doing data analysis, helping search for gravitational
waves produced by pairs of neutron stars and/or black holes as they
spiral down towards a sudden merger. Together with an enormous
pageful of authors, she helped write this paper, based on data taken
from the "first science run" - the first real LIGO experiment, back
in 2002:
1) The LIGO Scientific Collaboration, Analysis of LIGO data for
gravitational waves from binary neutron stars, Phys. Rev. D69 (2004),
122001. Also available at gr-qc/0308069.
She's one of the folks with an intimate knowledge of the experimental
setup, who keeps the theorists' feet on the ground while they stare
up into the sky.
On the drive to Livingston, Jorge pointed out the forests that
surround the town. These forests are being logged. I asked him
about this - when I last checked, the vibrations from falling trees
were making it impossible to look for gravitational waves except at
night! He said they've added a "hydraulic external pre-isolator" to
shield the detector from these vibrations - basically a super-duper
shock absorber. Now they can operate LIGO day and night.
I also asked him how close LIGO had come to the sensitivity levels
they were seeking. When I wrote "week189", during the first
science run, they still had a long way to go. That's why the above
paper only sets upper limits on neutron star collisions within 180
kiloparsecs. This only reaches out to the corona of the Milky
Way - which includes the Small and Large Magellanic Clouds. We
don't expect many neutron star collisions in this vicinity: maybe one
every 3 years or so. The first science run didn't see any, and the
set an upper limit of about 170 per year: the best experimental upper
limit so far, but definitely worth improving, and nowhere near as fun
as actually *seeing* gravitational waves.
But Jorge said the LIGO team has now reached its goals: they should
be able to see collisions out to 15 megaparsecs! By comparison,
the Virgo cluster is about 20 megaparsecs away. They're on their
seventh science run, and they'll keep upping the sensitivity in
future projects called "Enhanced LIGO" and "Advanced LIGO". The
latter should see neutron star collisions out to 300 megaparsecs:
2) Advanced LIGO, http://www.ligo.caltech.edu/advLIGO/[/URL]
When we arrived at the gate, Jorge spoke into the intercom and got
us let in. Our guide, Joseph Giaimie, was running a bit a late,
so we walked over and looked at the interferometer's arms, each
of which stretched off beyond sight, 2.5 kilometers of concrete
tunnel surrounding the evacuated piping - the world's largest vacuum
facility.
One can tell this is the South. The massive construction caused
pools of water to form in the boggy land near the facility, and
these pools then attracted alligators. These have been dealt with firmly.
The game hunters who occasionally fired potshots at the facility were
treated more forgivingly: instead of feeding them to the alligators,
the LIGO folks threw a big party and invited everyone from the local
hunting club. Hospitality works wonders down here.
The place was pretty lonely. During the week lots of scientists work
there, but this was Saturday, and on weekends there's just a skeleton
crew of two. There's usually not much to do now that the experiment
is up and running. As Joseph later said, there have been no "Jodie
Foster moments" like in the movie Contact, where the scientists on
duty suddenly see a signal, turn on the suspenseful background music,
and phone the President. There's just too much data analysis
required to see any signal in real time: data from both Livingston and
Hanfordis get sent to Caltech, and then people grind away at it. So,
about the most exciting thing that happens is when the occaisional
earthquake throws the laser beam out of phase lock.
When Joseph showed up, I got to see the main control room, which
is dimly lit, full of screens indicating noise and sensitivity
levels of all sorts - and even some video monitors showing the view
down the laser tube. This is where the people on duty hang out.
One of them had brought his sons, in a feeble attempt to dispose of
the huge supply of Halloween candy that had somehow collected here.
I also got to see a sample of the 400 "optical baffles" which have
been installed to absorb light spreading out from the main beam
before it can bounce back in and screw things up. The interesting
thing is that these baffles and their placement were personally
designed by Kip Thorne and some other godlike LIGO figure. Moral:
unless they've gone soft, even bigshot physicists like to actually
think about physics now and then, not just manage enormous teams.
But overall, there was surprisingly little to see, since the innermost
workings are all sealed off, in vacuum. The optics are far more
complicated than my description - "a laser bouncing between two
suspended mirrors" - could possibly suggest. But, all I got to
see was a chart showing how they work. Oh well. I'm glad I don't
need to understand this stuff in detail. It was fun to get a peek.
By the way, I wasn't invited to Louisiana just to tour LIGO and eat
beignets and alligator sushi. My real reason for going there was to
talk about higher gauge theory - a generalization of gauge theory
which studies the parallel transport not just of point particles, but
also strings and higher-dimensional objects:
3) [URL='https://www.physicsforums.com/insights/author/john-baez/']John Baez[/URL], Higher gauge theory,
[url]http://math.ucr.edu/home/baez/highergauge[/url]
This is a gentler introduction to higher gauge theory than my previous
talks, some of which I inflicted on you in "week235". It explains
how BF theory can be seen as a higher gauge theory, and briefly
touches on [URL='https://www.physicsforums.com/insights/author/urs-schreiber/']Urs Schreiber[/URL]'s work towards exhibiting Chern-Simons theory
and 11-dimensional supergravity as higher gauge theories. The webpage
has links to more details.
I was also traveling last weekend - I went to Dartmouth and gave this
talk:
4) [URL='https://www.physicsforums.com/insights/author/john-baez/']John Baez[/URL], Tales of the Dodecahedron: from Pythagoras through Plato
to Poincare, [url]http://math.ucr.edu/home/baez/dodecahedron/[/url]
It's full of pictures and animations - fun for the whole family!
I started with the Pythagorean fascination with the pentagram, and how
you can use the pentagram to give a magical picture proof of the
irrationality of the golden ratio.
I then mentioned how Plato used four of the so-called Platonic solids
to serve as atoms of the four elements - earth, air, water and fire -
leaving the inconvenient fifth solid, the dodecahedron, to play the
role of the heavenly sphere. This is what computer scientists call
a "kludge" - an awkard solution to a pressing problem. Yes, there
are twelve constellations in the Zodiac - but unfortunately, they're
arranged quite differently than the faces of the dodecahedron.
This somehow led to the notion of the dodecahedron as an atom of
"aether" or "quintessence" - a fifth element constituting the heavenly
bodies. If you've ever seen the science fiction movie "The Fifth
Element", now you know where the title came from! But once upon a
time, this idea was quite respectable. It shows up as late as
Kepler's "Mysterium Cosmographicum", written in 1596.
I then went on to discuss the 120-cell, which gives a way of chopping
a spherical universe into 120 dodecahedra. This leads naturally to
the Poincare homology sphere, a closely related 3-dimensional manifold
made by gluing together opposite sides of *one* dodecahedron.
The Poincare homology sphere was briefly advocated as a model
of the universe that could explain the mysterious weakness of the
longest-wavelength ripples in the cosmic background radiation -
the ripples that only wiggle a few times as we scan all around the
sky:
5) J.-P. Luminet, J. Weeks, A. Riazuelo, R. Lehoucq, and J.-P.
Uzan, Dodecahedral space topology as an explanation for weak
wide-angle temperature correlations in the cosmic microwave background,
Nature 425 (2003), 593. Also available as astro-ph/0310253.
The idea is that if we lived in a Poincare homology sphere, we'd
see several images of each very distant point in the universe. So,
any ripple in the background radiation would repeat some minimum
number of times: the lowest-frequency ripples would be suppressed.
Alas, this charming idea turns out not to fit other data. We just
don't see the same distant galaxies in several different directions:
6) Neil J. Cornish, David N. Spergel, Glenn D. Starkman and
Eiichiro Komatsu, Constraining the topology of the universe,
Phys. Rev. Lett. 92 (2004) 201302. Also available as astro-ph/0310233.
For a good review of this stuff, see:
7) Jeffrey Weeks, The Poincare dodecahedral space and the mystery
of the missing fluctuations, Notices of the AMS 51 (2004), 610-619.
Also available at [url]http://www.ams.org/notices/200406/fea-weeks.pdf[/url]
In the abstract of my talk, I made the mistake of saying that
the regular dodecahedron doesn't appear in nature - that instead,
it was invented by the Pythagoreans. You should never say things
like this unless you want to get corrected!
Dan Piponi pointed out this dodecahedral virus:
8) Liang Tang et al, The structure of Pariacoto virus reveals a
dodecahedral cage of duplex RNA, Nature Structural Biology 8
(2001), 77-83. Also available at
[url]http://www.nature.com/nsmb/journal/v8/n1/pdf/nsb0101_77.pdf[/url]
Garett Leskowitz pointed out the molecule "dodecahedrane", with
20 carbons at the vertices of a dodecahedron and 20 hydrogens bonded
to these:
9) Wikipedia, Dodecahedrane, [url]http://en.wikipedia.org/wiki/Dodecahedrane[/url]
This molecule hasn't been found in nature yet, but chemists can
synthesize it using reactions like these:
10) Robert J. Ternansky, Douglas W. Balogh and Leo A. Paquette,
Dodecahedrane, J. Am. Chem. Soc. 104 (1982), 4503-4504.
11) Leo A. Paquette, Dodecahedrane - the chemical transliteration of
Plato's universe (a review), Proc. Nat. Acad. Sci. USA 14 part 2
(1982), 4495-4500. Also available at
http://www.pubmedcentral.nih.gov/articlerender.fcgi?artid=346698
So, there's probably a bit somewhere in our galaxy.
Of course, what I *meant* was that people didn't come up with
regular dodecahedra after seeing them in nature - that instead,
the Pythagoreans dreamt them up, possibly after seeing pyrite
crystals that look sort of similar.
These crystals are called "pyritohedra". Since pyrite is
fundamentally a cubic crystal, the pyritohedron is basically made out
of little cubic cells, as shown here:
12) Steven Dutch, Building isometric crystals with unit cells,
http://www.uwgb.edu/dutchs/symmetry/isometuc.htm
It has 12 pentagonal faces, orthogonal to these vectors:
(0,1,2) (0,2,1) (1,0,2) (1,2,0) (2,0,1) (2,1,0)
(0,-1,-2) (0,-2,-1) (-1,0,-2) (-1,-2,0) (-2,0,-1) (-2,-1,0)
formed by permuting and/or negating the entries of (0,1,2).
But, even here I made a mistake. The Pythagoreans seem not to have been
the first to discover the dodecahedron. John McKay told me that
stone spheres with Platonic solids carved on them have been found
in Scotland, dating back to around 2000 BC! There are even some in
the Ashmolean at Oxford:
13) Michael Atiyah and Paul Sutcliffe, Polyhedra in physics, chemistry
and geometry, available as math-ph/0303071.
14) Dorothy N. Marshall, Carved stone balls, Proc. Soc. Antiq.
Scotland, 108 (1976/77), 40-72. Available at
[url]http://ads.ahds.ac.uk/catalogue/library/psas/[/url]
Indeed, stone balls with geometric patterns on them have been found
throughout Scotland, and occasionally Ireland and northern England.
They date from the Late Neolithic to the Early Bronze age: 2500 BC to
1500 BC. For comparison, the megaliths at Stonehenge go back to
2500-2100 BC.
Nobody knows what these stone balls were used for, though the
article by Marshall presents a number of interesting speculations.
Let me wrap up by mentioning a fancier aspect of the dodecahedron
which has been intriguing lately. I already mentioned it in
"week230", but in such a general setting that it may have whizzed
by too fast. Let's slow down a bit and enjoy it.
The rotational symmetries of the dodecahedron form a 60-element
subgroup of the rotation group SO(3). So, the "double cover" of
the rotational symmetry group of the dodecahedron is a 120-element
subgroup of SU(2). This is called the "binary dodecahedral group".
Let's call it G.
The group SU(2) is topologically a 3-sphere, so G acts as left
translations on this 3-sphere, and we can use a dodecahedron sitting
in the 3-sphere as a fundamental domain for this action. This gives
the 120-cell. The quotient SU(2)/G is the Poincare homology sphere!
But, we can also think of G as acting on C^2. The quotient C^2/G
is not smooth: it has an isolated singular coming from the origin
in C^2. But as I mentioned in "week230", we can form a "minimal
resolution" of this singularity. This gives a holomorphic map
p: M -> C^2/G
where M is a complex manifold. If we look at the points in M
that map to the origin in C^2/G, we get a union of 8 Riemann spheres,
which intersect each other in this pattern:
/\ /\ /\ /\ /\ /\ /\
/ \ / \ / \ / \ / \ / \ / \
/ \ \ \ \ \ \ \
/ / \ / \ / \ / \ / \ / \ \
\ \ / \ / \ / \ / \ / \ / /
\ \ \ \ \ /\ \ \ /
\ / \ / \ / \ / \ \ \ / \ /
\/ \/ \/ \/ / \/ \ \/ \/
/ \
\ /
\ /
\ /
\/
Here I've drawn linked circles to stand for these intersecting
spheres, for a reason soon to be clear. But, already you can
see that we've got 8 spheres corresponding to the dots in this
diagram:o----o----o----o----o----o----o
|
|
o
where the spheres intersect when there's an edge between the
corresponding dots. And, this diagram is the Dynkin diagram for
the exceptional Lie group E8!
I already mentioned the relation between the E8 Dynkin diagram and
the Poincare homology sphere in "week164", but now maybe it fits
better into a big framework. First, we see that if we take the
unit ball in C^2, and see what points it gives in C^2/G, and then
take the inverse image of these under
p: M -> C^2/G,
we get a 4-manifold whose boundary is the Poincare homology
3-sphere. So, we have a cobordism from the empty set to the
Poincare homology 3-sphere! Cobordisms can be described using
"surgery on links", and the link that describes this particular
cobordism is:
/\ /\ /\ /\ /\ /\ /\
/ \ / \ / \ / \ / \ / \ / \
/ \ \ \ \ \ \ \
/ / \ / \ / \ / \ / \ / \ \
\ \ / \ / \ / \ / \ / \ / /
\ \ \ \ \ /\ \ \ /
\ / \ / \ / \ / \ \ \ / \ /
\/ \/ \/ \/ / \/ \ \/ \/
/ \
\ /
\ /
\ /
\/
Second, by the "McKay correspondence" described in "week230", all
this stuff also works for other Platonic solids! Namely:
If G is the "binary octahedral group" - the double cover of the
rotational symmetry group of the octahedron - then we get a minimal
resolution
p: M -> C^2/G
which yields, by the same procedure as above, a cobordism from the
empty set to the 3-manifold SU(2)/G.
This cobordism can be described using surgery on this link:/\ /\ /\ /\ /\ /\
/ \ / \ / \ / \ / \ / \
/ \ \ \ \ \ \
/ / \ / \ / \ / \ / \ \
\ \ / \ / \ / \ / \ / /
\ \ \ \ /\ \ \ /
\ / \ / \ / \ \ \ / \ /
\/ \/ \/ / \/ \ \/ \/
/ \
\ /
\ /
\ /
\/which encodes the Dynkin diagram of E7:o----o----o----o----o----o
|
|
oAnd, if G is the "binary tetrahedral group" - the double cover of the
rotational symmetry group of the tetrahedron - then a minimal
resolution
p: M -> C^2/G
yields, by the same procedure as above, a cobordism from the
empty set to the 3-manifold SU(2)/G. This cobordism can be
described using surgery on this link:/\ /\ /\ /\ /\
/ \ / \ / \ / \ / \
/ \ \ \ \ \
/ / \ / \ / \ / \ \
\ \ / \ / \ / \ / /
\ \ \ /\ \ \ /
\ / \ / \ \ \ / \ /
\/ \/ / \/ \ \/ \/
/ \
\ /
\ /
\ /
\/which encodes the Dynkin diagram of E6:o----o----o----o----o
|
|
o
I don't fully understand this stuff, that's for sure. But, I
want to. The Platonic solids are still full of mysteries.
----------------------------------------------------------------------
Previous issues of "This Week's Finds" and other expository articles on
mathematics and physics, as well as some of my research papers, can be
obtained at
[url]http://math.ucr.edu/home/baez/[/url]
For a table of contents of all the issues of This Week's Finds, try
[url]http://math.ucr.edu/home/baez/twfcontents.html[/url]
A simple jumping-off point to the old issues is available at
[url]http://math.ucr.edu/home/baez/twfshort.html[/url]
If you just want the latest issue, go to
[url]http://math.ucr.edu/home/baez/this.week.html[/url]
November 18, 2006
This Week's Finds in Mathematical Physics (Week 241)
John Baez
I've been working too hard, and running around too much, to write
This Week's Finds for a while. A bunch of stuff has built up
that I want to explain. Luckily I've been running around explaining
stuff - higher gauge theory, and tales of the dodecahedron.
This weekend I went to Baton Rouge. I was invited to Louisiana
State University by Jorge Pullin of loop quantum gravity fame, and
I used the opportunity to get a look at LIGO - the Laser
Interferometry Gravitational-Wave Observatory! I took a bunch
of pictures, which you can see in the webpage version of this article.
I described this amazing experiment back in "week189", so I won't
rehash all that. Suffice it to say that there are two installations:
one in Hanford Washington, and one in Livingston Louisiana. Each
consists of two evacuated tubes 4 kilometers long, arranged in an L
shape. Laser beams bounce back and forth between mirrors suspended
at the ends of the tubes, looking for tiny changes in their distance
that would indicate a gravitational wave passing through, stretching
or squashing space. And when I say "tiny", I mean smaller than the
radius of a proton! This is serious stuff.
Jorge drove me in his SUV to Livingston, a tiny town about 20 minutes
from Baton Rouge. While he runs the gravity program at Louisiana
State University, which has links to LIGO, he isn't officially part
the LIGO team. His wife is. When I first met Gabriela Gonzalez, she
was studying the Brownian motion of torsion pendulums. The mirrors in
LIGO are hung on pendulums made of quartz wire, to minimize the effect
of vibrations. But, the random jittering of atoms due to thermal
noise still affects these pendulums. She was studying this noise to
see its effect on the accuracy of the experiment.
This was way back when LIGO was just being planned. Now that LIGO is
a reality, she's doing data analysis, helping search for gravitational
waves produced by pairs of neutron stars and/or black holes as they
spiral down towards a sudden merger. Together with an enormous
pageful of authors, she helped write this paper, based on data taken
from the "first science run" - the first real LIGO experiment, back
in 2002:
1) The LIGO Scientific Collaboration, Analysis of LIGO data for
gravitational waves from binary neutron stars, Phys. Rev. D69 (2004),
122001. Also available at gr-qc/0308069.
She's one of the folks with an intimate knowledge of the experimental
setup, who keeps the theorists' feet on the ground while they stare
up into the sky.
On the drive to Livingston, Jorge pointed out the forests that
surround the town. These forests are being logged. I asked him
about this - when I last checked, the vibrations from falling trees
were making it impossible to look for gravitational waves except at
night! He said they've added a "hydraulic external pre-isolator" to
shield the detector from these vibrations - basically a super-duper
shock absorber. Now they can operate LIGO day and night.
I also asked him how close LIGO had come to the sensitivity levels
they were seeking. When I wrote "week189", during the first
science run, they still had a long way to go. That's why the above
paper only sets upper limits on neutron star collisions within 180
kiloparsecs. This only reaches out to the corona of the Milky
Way - which includes the Small and Large Magellanic Clouds. We
don't expect many neutron star collisions in this vicinity: maybe one
every 3 years or so. The first science run didn't see any, and the
set an upper limit of about 170 per year: the best experimental upper
limit so far, but definitely worth improving, and nowhere near as fun
as actually *seeing* gravitational waves.
But Jorge said the LIGO team has now reached its goals: they should
be able to see collisions out to 15 megaparsecs! By comparison,
the Virgo cluster is about 20 megaparsecs away. They're on their
seventh science run, and they'll keep upping the sensitivity in
future projects called "Enhanced LIGO" and "Advanced LIGO". The
latter should see neutron star collisions out to 300 megaparsecs:
2) Advanced LIGO, http://www.ligo.caltech.edu/advLIGO/[/URL]
When we arrived at the gate, Jorge spoke into the intercom and got
us let in. Our guide, Joseph Giaimie, was running a bit a late,
so we walked over and looked at the interferometer's arms, each
of which stretched off beyond sight, 2.5 kilometers of concrete
tunnel surrounding the evacuated piping - the world's largest vacuum
facility.
One can tell this is the South. The massive construction caused
pools of water to form in the boggy land near the facility, and
these pools then attracted alligators. These have been dealt with firmly.
The game hunters who occasionally fired potshots at the facility were
treated more forgivingly: instead of feeding them to the alligators,
the LIGO folks threw a big party and invited everyone from the local
hunting club. Hospitality works wonders down here.
The place was pretty lonely. During the week lots of scientists work
there, but this was Saturday, and on weekends there's just a skeleton
crew of two. There's usually not much to do now that the experiment
is up and running. As Joseph later said, there have been no "Jodie
Foster moments" like in the movie Contact, where the scientists on
duty suddenly see a signal, turn on the suspenseful background music,
and phone the President. There's just too much data analysis
required to see any signal in real time: data from both Livingston and
Hanfordis get sent to Caltech, and then people grind away at it. So,
about the most exciting thing that happens is when the occaisional
earthquake throws the laser beam out of phase lock.
When Joseph showed up, I got to see the main control room, which
is dimly lit, full of screens indicating noise and sensitivity
levels of all sorts - and even some video monitors showing the view
down the laser tube. This is where the people on duty hang out.
One of them had brought his sons, in a feeble attempt to dispose of
the huge supply of Halloween candy that had somehow collected here.
I also got to see a sample of the 400 "optical baffles" which have
been installed to absorb light spreading out from the main beam
before it can bounce back in and screw things up. The interesting
thing is that these baffles and their placement were personally
designed by Kip Thorne and some other godlike LIGO figure. Moral:
unless they've gone soft, even bigshot physicists like to actually
think about physics now and then, not just manage enormous teams.
But overall, there was surprisingly little to see, since the innermost
workings are all sealed off, in vacuum. The optics are far more
complicated than my description - "a laser bouncing between two
suspended mirrors" - could possibly suggest. But, all I got to
see was a chart showing how they work. Oh well. I'm glad I don't
need to understand this stuff in detail. It was fun to get a peek.
By the way, I wasn't invited to Louisiana just to tour LIGO and eat
beignets and alligator sushi. My real reason for going there was to
talk about higher gauge theory - a generalization of gauge theory
which studies the parallel transport not just of point particles, but
also strings and higher-dimensional objects:
3) [URL='https://www.physicsforums.com/insights/author/john-baez/']John Baez[/URL], Higher gauge theory,
[url]http://math.ucr.edu/home/baez/highergauge[/url]
This is a gentler introduction to higher gauge theory than my previous
talks, some of which I inflicted on you in "week235". It explains
how BF theory can be seen as a higher gauge theory, and briefly
touches on [URL='https://www.physicsforums.com/insights/author/urs-schreiber/']Urs Schreiber[/URL]'s work towards exhibiting Chern-Simons theory
and 11-dimensional supergravity as higher gauge theories. The webpage
has links to more details.
I was also traveling last weekend - I went to Dartmouth and gave this
talk:
4) [URL='https://www.physicsforums.com/insights/author/john-baez/']John Baez[/URL], Tales of the Dodecahedron: from Pythagoras through Plato
to Poincare, [url]http://math.ucr.edu/home/baez/dodecahedron/[/url]
It's full of pictures and animations - fun for the whole family!
I started with the Pythagorean fascination with the pentagram, and how
you can use the pentagram to give a magical picture proof of the
irrationality of the golden ratio.
I then mentioned how Plato used four of the so-called Platonic solids
to serve as atoms of the four elements - earth, air, water and fire -
leaving the inconvenient fifth solid, the dodecahedron, to play the
role of the heavenly sphere. This is what computer scientists call
a "kludge" - an awkard solution to a pressing problem. Yes, there
are twelve constellations in the Zodiac - but unfortunately, they're
arranged quite differently than the faces of the dodecahedron.
This somehow led to the notion of the dodecahedron as an atom of
"aether" or "quintessence" - a fifth element constituting the heavenly
bodies. If you've ever seen the science fiction movie "The Fifth
Element", now you know where the title came from! But once upon a
time, this idea was quite respectable. It shows up as late as
Kepler's "Mysterium Cosmographicum", written in 1596.
I then went on to discuss the 120-cell, which gives a way of chopping
a spherical universe into 120 dodecahedra. This leads naturally to
the Poincare homology sphere, a closely related 3-dimensional manifold
made by gluing together opposite sides of *one* dodecahedron.
The Poincare homology sphere was briefly advocated as a model
of the universe that could explain the mysterious weakness of the
longest-wavelength ripples in the cosmic background radiation -
the ripples that only wiggle a few times as we scan all around the
sky:
5) J.-P. Luminet, J. Weeks, A. Riazuelo, R. Lehoucq, and J.-P.
Uzan, Dodecahedral space topology as an explanation for weak
wide-angle temperature correlations in the cosmic microwave background,
Nature 425 (2003), 593. Also available as astro-ph/0310253.
The idea is that if we lived in a Poincare homology sphere, we'd
see several images of each very distant point in the universe. So,
any ripple in the background radiation would repeat some minimum
number of times: the lowest-frequency ripples would be suppressed.
Alas, this charming idea turns out not to fit other data. We just
don't see the same distant galaxies in several different directions:
6) Neil J. Cornish, David N. Spergel, Glenn D. Starkman and
Eiichiro Komatsu, Constraining the topology of the universe,
Phys. Rev. Lett. 92 (2004) 201302. Also available as astro-ph/0310233.
For a good review of this stuff, see:
7) Jeffrey Weeks, The Poincare dodecahedral space and the mystery
of the missing fluctuations, Notices of the AMS 51 (2004), 610-619.
Also available at [url]http://www.ams.org/notices/200406/fea-weeks.pdf[/url]
In the abstract of my talk, I made the mistake of saying that
the regular dodecahedron doesn't appear in nature - that instead,
it was invented by the Pythagoreans. You should never say things
like this unless you want to get corrected!
Dan Piponi pointed out this dodecahedral virus:
8) Liang Tang et al, The structure of Pariacoto virus reveals a
dodecahedral cage of duplex RNA, Nature Structural Biology 8
(2001), 77-83. Also available at
[url]http://www.nature.com/nsmb/journal/v8/n1/pdf/nsb0101_77.pdf[/url]
Garett Leskowitz pointed out the molecule "dodecahedrane", with
20 carbons at the vertices of a dodecahedron and 20 hydrogens bonded
to these:
9) Wikipedia, Dodecahedrane, [url]http://en.wikipedia.org/wiki/Dodecahedrane[/url]
This molecule hasn't been found in nature yet, but chemists can
synthesize it using reactions like these:
10) Robert J. Ternansky, Douglas W. Balogh and Leo A. Paquette,
Dodecahedrane, J. Am. Chem. Soc. 104 (1982), 4503-4504.
11) Leo A. Paquette, Dodecahedrane - the chemical transliteration of
Plato's universe (a review), Proc. Nat. Acad. Sci. USA 14 part 2
(1982), 4495-4500. Also available at
http://www.pubmedcentral.nih.gov/articlerender.fcgi?artid=346698
So, there's probably a bit somewhere in our galaxy.
Of course, what I *meant* was that people didn't come up with
regular dodecahedra after seeing them in nature - that instead,
the Pythagoreans dreamt them up, possibly after seeing pyrite
crystals that look sort of similar.
These crystals are called "pyritohedra". Since pyrite is
fundamentally a cubic crystal, the pyritohedron is basically made out
of little cubic cells, as shown here:
12) Steven Dutch, Building isometric crystals with unit cells,
http://www.uwgb.edu/dutchs/symmetry/isometuc.htm
It has 12 pentagonal faces, orthogonal to these vectors:
(0,1,2) (0,2,1) (1,0,2) (1,2,0) (2,0,1) (2,1,0)
(0,-1,-2) (0,-2,-1) (-1,0,-2) (-1,-2,0) (-2,0,-1) (-2,-1,0)
formed by permuting and/or negating the entries of (0,1,2).
But, even here I made a mistake. The Pythagoreans seem not to have been
the first to discover the dodecahedron. John McKay told me that
stone spheres with Platonic solids carved on them have been found
in Scotland, dating back to around 2000 BC! There are even some in
the Ashmolean at Oxford:
13) Michael Atiyah and Paul Sutcliffe, Polyhedra in physics, chemistry
and geometry, available as math-ph/0303071.
14) Dorothy N. Marshall, Carved stone balls, Proc. Soc. Antiq.
Scotland, 108 (1976/77), 40-72. Available at
[url]http://ads.ahds.ac.uk/catalogue/library/psas/[/url]
Indeed, stone balls with geometric patterns on them have been found
throughout Scotland, and occasionally Ireland and northern England.
They date from the Late Neolithic to the Early Bronze age: 2500 BC to
1500 BC. For comparison, the megaliths at Stonehenge go back to
2500-2100 BC.
Nobody knows what these stone balls were used for, though the
article by Marshall presents a number of interesting speculations.
Let me wrap up by mentioning a fancier aspect of the dodecahedron
which has been intriguing lately. I already mentioned it in
"week230", but in such a general setting that it may have whizzed
by too fast. Let's slow down a bit and enjoy it.
The rotational symmetries of the dodecahedron form a 60-element
subgroup of the rotation group SO(3). So, the "double cover" of
the rotational symmetry group of the dodecahedron is a 120-element
subgroup of SU(2). This is called the "binary dodecahedral group".
Let's call it G.
The group SU(2) is topologically a 3-sphere, so G acts as left
translations on this 3-sphere, and we can use a dodecahedron sitting
in the 3-sphere as a fundamental domain for this action. This gives
the 120-cell. The quotient SU(2)/G is the Poincare homology sphere!
But, we can also think of G as acting on C^2. The quotient C^2/G
is not smooth: it has an isolated singular coming from the origin
in C^2. But as I mentioned in "week230", we can form a "minimal
resolution" of this singularity. This gives a holomorphic map
p: M -> C^2/G
where M is a complex manifold. If we look at the points in M
that map to the origin in C^2/G, we get a union of 8 Riemann spheres,
which intersect each other in this pattern:
/\ /\ /\ /\ /\ /\ /\
/ \ / \ / \ / \ / \ / \ / \
/ \ \ \ \ \ \ \
/ / \ / \ / \ / \ / \ / \ \
\ \ / \ / \ / \ / \ / \ / /
\ \ \ \ \ /\ \ \ /
\ / \ / \ / \ / \ \ \ / \ /
\/ \/ \/ \/ / \/ \ \/ \/
/ \
\ /
\ /
\ /
\/
Here I've drawn linked circles to stand for these intersecting
spheres, for a reason soon to be clear. But, already you can
see that we've got 8 spheres corresponding to the dots in this
diagram:o----o----o----o----o----o----o
|
|
o
where the spheres intersect when there's an edge between the
corresponding dots. And, this diagram is the Dynkin diagram for
the exceptional Lie group E8!
I already mentioned the relation between the E8 Dynkin diagram and
the Poincare homology sphere in "week164", but now maybe it fits
better into a big framework. First, we see that if we take the
unit ball in C^2, and see what points it gives in C^2/G, and then
take the inverse image of these under
p: M -> C^2/G,
we get a 4-manifold whose boundary is the Poincare homology
3-sphere. So, we have a cobordism from the empty set to the
Poincare homology 3-sphere! Cobordisms can be described using
"surgery on links", and the link that describes this particular
cobordism is:
/\ /\ /\ /\ /\ /\ /\
/ \ / \ / \ / \ / \ / \ / \
/ \ \ \ \ \ \ \
/ / \ / \ / \ / \ / \ / \ \
\ \ / \ / \ / \ / \ / \ / /
\ \ \ \ \ /\ \ \ /
\ / \ / \ / \ / \ \ \ / \ /
\/ \/ \/ \/ / \/ \ \/ \/
/ \
\ /
\ /
\ /
\/
Second, by the "McKay correspondence" described in "week230", all
this stuff also works for other Platonic solids! Namely:
If G is the "binary octahedral group" - the double cover of the
rotational symmetry group of the octahedron - then we get a minimal
resolution
p: M -> C^2/G
which yields, by the same procedure as above, a cobordism from the
empty set to the 3-manifold SU(2)/G.
This cobordism can be described using surgery on this link:/\ /\ /\ /\ /\ /\
/ \ / \ / \ / \ / \ / \
/ \ \ \ \ \ \
/ / \ / \ / \ / \ / \ \
\ \ / \ / \ / \ / \ / /
\ \ \ \ /\ \ \ /
\ / \ / \ / \ \ \ / \ /
\/ \/ \/ / \/ \ \/ \/
/ \
\ /
\ /
\ /
\/which encodes the Dynkin diagram of E7:o----o----o----o----o----o
|
|
oAnd, if G is the "binary tetrahedral group" - the double cover of the
rotational symmetry group of the tetrahedron - then a minimal
resolution
p: M -> C^2/G
yields, by the same procedure as above, a cobordism from the
empty set to the 3-manifold SU(2)/G. This cobordism can be
described using surgery on this link:/\ /\ /\ /\ /\
/ \ / \ / \ / \ / \
/ \ \ \ \ \
/ / \ / \ / \ / \ \
\ \ / \ / \ / \ / /
\ \ \ /\ \ \ /
\ / \ / \ \ \ / \ /
\/ \/ / \/ \ \/ \/
/ \
\ /
\ /
\ /
\/which encodes the Dynkin diagram of E6:o----o----o----o----o
|
|
o
I don't fully understand this stuff, that's for sure. But, I
want to. The Platonic solids are still full of mysteries.
----------------------------------------------------------------------
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