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This Week's Finds in Mathematical Physics (Week 241)

  1. Nov 20, 2006 #1
    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/

    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) John Baez, Higher gauge theory,
    http://math.ucr.edu/home/baez/highergauge

    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 Urs Schreiber'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 travelling last weekend - I went to Dartmouth and gave this
    talk:

    4) John Baez, Tales of the Dodecahedron: from Pythagoras through Plato
    to Poincare, http://math.ucr.edu/home/baez/dodecahedron/

    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 http://www.ams.org/notices/200406/fea-weeks.pdf

    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
    http://www.nature.com/nsmb/journal/v8/n1/pdf/nsb0101_77.pdf

    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, http://en.wikipedia.org/wiki/Dodecahedrane

    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
    http://ads.ahds.ac.uk/catalogue/library/psas/

    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
    |
    |
    o


    And, 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

    http://math.ucr.edu/home/baez/

    For a table of contents of all the issues of This Week's Finds, try

    http://math.ucr.edu/home/baez/twfcontents.html

    A simple jumping-off point to the old issues is available at

    http://math.ucr.edu/home/baez/twfshort.html

    If you just want the latest issue, go to

    http://math.ucr.edu/home/baez/this.week.html
     
  2. jcsd
  3. Nov 22, 2006 #2
    John Baez wrote:
    >
    > 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

    [snip erudition]

    > 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!

    [snip]

    > 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, http://en.wikipedia.org/wiki/Dodecahedrane
    >
    > 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.


    The pagodane
    (unadecacyclo[9.9.0.01,5.02,12.02,18.03,7.06,10.08,12.011,15.0
    13,17.016,20]eicosane) route to dodecahedrane is more interesting for
    its synthetic versatility,

    http://cat.inist.fr/?aModele=afficheN&cpsidt=10538067
    <www.rsc.org/delivery/_ArticleLinking/DisplayArticleForFree.cfm?doi=a709107i&JournalCode=P2>

    In principle one could bridge two cyclopentane rings with five
    acetylenes, reduce the acetylenes to cis-olefins, and the whole thing
    spontaneously pericyclizes to dodecahedrane. Start with two moles of
    all-cis-1,2,3,4,5-pentacyanocyclopentane and use Schrock's
    hexa-t-butoxyditungsten alkyne metathesis catalyst with fivefold
    stoichiometric cleverness and precipitation of (polymeric)
    tri-t-butoxytungsten nitride. Aside from the bridged intermediate
    probably being impossible (or curiously explosive), it's a very
    elegant route.

    Spontaneous pericylization of azo-bridges would be even more
    interesting. Diacetylene bridges might be reasonably accessible
    (all-cis-1,2,3,4,5-pentaethynyl cyclopentane oxidatively dimerized
    (Glaser oxidation) with everything lining up just so (riiiight!).
    Turn the crank and the diacetylene-bridged stuff would close to give a
    belly-expaned dodecahedrane derivative.

    Do the analogous dance with chair all-axial all-cis-1,3,5-triethynyl
    cyclohexane, dimerized, reduced, closed, and you get four cyclohexane
    decks as a tiny bit of hexagonal diamond, Lonsdaleite.

    [snip more erudition]

    > 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
    >
    > http://math.ucr.edu/home/baez/
    >
    > For a table of contents of all the issues of This Week's Finds, try
    >
    > http://math.ucr.edu/home/baez/twfcontents.html
    >
    > A simple jumping-off point to the old issues is available at
    >
    > http://math.ucr.edu/home/baez/twfshort.html
    >
    > If you just want the latest issue, go to
    >
    > http://math.ucr.edu/home/baez/this.week.html


    --
    Uncle Al
    http://www.mazepath.com/uncleal/
    (Toxic URL! Unsafe for children and most mammals)
    http://www.mazepath.com/uncleal/qz3.pdf
     
  4. Nov 22, 2006 #3
    John Baez wrote:

    [..]

    > 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.


    I am a bit puzzled by the topology of this.
    In a 2 dimensional ordinary dodecahedron, the Euler characteristic
    together with the (5,3) pattern seems to dictate the topology.
    Simply by fitting together 3 pentagons at each vertex, you automatically
    build a dodecahedron.
    (I am not completely sure there is no way out, but I can't think of
    one at the moment)
    The Euler characteristic of a (5,3) pattern is F(1-5/2+5/3) = F/6

    So if F=12, we get Euler=2. If F-24, we have 2 disjoint dodecahedra.
    It might be fun to think if we can build anything else, satisfying the
    (5,3) restriction.

    The generalization of this in 4D is C-F+E-V. This is zero for the
    (5,3,3) pattern, regardless of the number of cells. On the one hand,
    this suggests that you could topologically stack 3-space with
    dodecahedra. But the "wriggle room" is a bit confusing, because in
    2D the Euler characteristic corresponds to the total curvature. In
    3D, things seem to work differently, but I don't understand how.

    Anyway, if I just imagine gluing together dodecahedra, I get a
    sphere that has an outer shell that is composed of an ever-increasing
    number of dodecahedra. They don't seem to come together to a close,
    like the pentagons do in a dodecahedron.

    Gerard
     
  5. Nov 23, 2006 #4
    Gerard Westendorp wrote:
    > John Baez wrote:
    >
    > [..]
    >
    >
    >>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.

    >
    >
    > I am a bit puzzled by the topology of this.
    > In a 2 dimensional ordinary dodecahedron, the Euler characteristic
    > together with the (5,3) pattern seems to dictate the topology.
    > Simply by fitting together 3 pentagons at each vertex, you automatically
    > build a dodecahedron.
    > (I am not completely sure there is no way out, but I can't think of
    > one at the moment)
    > The Euler characteristic of a (5,3) pattern is F(1-5/2+5/3) = F/6
    >
    > So if F=12, we get Euler=2. If F-24, we have 2 disjoint dodecahedra.
    > It might be fun to think if we can build anything else, satisfying the
    > (5,3) restriction.
    >
    > The generalization of this in 4D is C-F+E-V. This is zero for the
    > (5,3,3) pattern, regardless of the number of cells. On the one hand,
    > this suggests that you could topologically stack 3-space with
    > dodecahedra. But the "wriggle room" is a bit confusing, because in
    > 2D the Euler characteristic corresponds to the total curvature. In
    > 3D, things seem to work differently, but I don't understand how.
    >
    > Anyway, if I just imagine gluing together dodecahedra, I get a
    > sphere that has an outer shell that is composed of an ever-increasing
    > number of dodecahedra. They don't seem to come together to a close,
    > like the pentagons do in a dodecahedron.
    >
    > Gerard
    >


    Gerard

    As an approach to this problem of

    'gluing together dodecahedra'

    and observing the anticipated spherical result,

    I started with the standard ball and stick
    organic chemistry sp3 (4) vertices
    and connected them with springs (~.75 inch long)

    From a central dodecahedron, an increasing number
    of 'springy' dodecahedron were added.

    It did not take long (a couple of dodecahedra layers)
    before a divergence became obvious.

    The added dodecahedra had to be distorted by springs
    as distance from center increased
    until it became impossible to connect sp3 vertices by springs.

    Conclusion:

    An 3D array of close packed dodecahedra does not fill space.

    Richard
     
  6. Nov 25, 2006 #5
    John Baez wrote:
    > 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.


    [Moderator's note: Large amount of quoted text deleted. -P.H.]

    > 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".


    Another name for it is 'symmetric group of five elements'.
    The automorphism group of the dodecahedron ist Alt(5).
    the smallest nonabelian simple group. Its double cover is Sym(5).
    The latter is also the group PGL(2,5) of rational linear transformations
    over the field with 5 elements; the transformations with determinant 1
    give PSL(2,5) isomorphic to Alt(5).

    Vertices of the dodecahedron can be labelled by the 20 ordered pairs of
    5 symbols; adjacent vertices have a common symbol, but only half of the
    pairs of vertices with a common symbol form edges, which is why not
    the full symmetric group acts.

    Arnold Neumaier

    Arnold Neumaier
     
  7. Nov 25, 2006 #6
    A reply to Gerard Westendorp, but first some errata: I misspelled
    Joe Giaime's name, and the final E7 should have been an E6:

    the Dynkin diagram of E6:


    o----o----o----o----o
    |
    |
    o

    In article <4563AA9F.8080905@xs4all.nl>,
    Gerard Westendorp <westy31@xs4all.nl> wrote:

    >John Baez wrote:


    >> 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.


    >I am a bit puzzled by the topology of this.


    To get the Poincare homology sphere, take a dodecahedron, and
    identify each point on any face with a point on the opposite face,
    in the simplest possible way. More precisely, identify each face
    with the opposite face after giving it a clockwise 1/10 turn!
    (Or, if you prefer, a counterclockwise 1/10 turn - but be consistent.)
    If you look, you'll see that a 1/10 turn (36 degrees) is the
    smallest amount of turning that can work.

    When you're done, you'll see that four edges and four faces meet
    at each vertex.

    As for the 120-cell:

    >Anyway, if I just imagine gluing together dodecahedra, I get a
    >sphere that has an outer shell that is composed of an ever-increasing
    >number of dodecahedra. They don't seem to come together to a close,
    >like the pentagons do in a dodecahedron.


    Well, they don't close until you "fold it up" into the fourth
    dimension. Did you look at these pictures?

    http://www.weimholt.com/andrew/120_stage1.html

    You might also like these:

    http://www.ams.org/featurecolumn/archive/boole.html

    which show the successive layers more systematically:

    1 + 12 + 20 + 12 + 30 + 12 + 20 + 12 + 1 = 120

    although they actually just go to the halfway-point:

    1 + 12 + 20 + 12 + 30

    which gives approximately the "top half" of the 120-cell.

    Also look at this:

    http://www.georgehart.com/hyperspace/hart-120-cell.html

    Since I'm posting to sci.physics.research, I should also recommend
    Brett McInnes' paper on the instability of the Poincare
    3-sphere in the context of brane-world cosmology:

    http://arxiv.org/abs/hep-th/0401035

    As is well known, classical General Relativity does not constrain the
    topology of the spatial sections of our Universe. However, the Brane-
    World approach to cosmology might be expected to do so, since in general
    any modification of the topology of the brane must be reflected in some
    modification of that of the bulk. Assuming the truth of the Adams-
    Polchinski-Silverstein conjecture on the instability of non-supersymmetric
    AdS orbifolds, evidence for which has recently been accumulating, we
    argue that indeed many possible topologies for accelerating universes
    can be ruled out because they lead to non-perturbative instabilities.
     
  8. Nov 27, 2006 #7
    On 24-Nov-2006, Arnold Neumaier <Arnold.Neumaier@univie.ac.at>
    wrote in message <456437A1.2060908@univie.ac.at>:

    > John Baez wrote:
    >
    > > 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.

    >
    > [Moderator's note: Large amount of quoted text deleted. -P.H.]
    >
    > > 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".

    >
    > Another name for it is 'symmetric group of five elements'.
    > The automorphism group of the dodecahedron ist Alt(5).
    > the smallest nonabelian simple group. Its double cover is Sym(5).
    > The latter is also the group PGL(2,5) of rational linear transformations
    > over the field with 5 elements; the transformations with determinant 1
    > give PSL(2,5) isomorphic to Alt(5).


    Not quite. The rotational symmetry group of the dodecahedron is
    indeed isomorphic to Alt(5) ~= PSL(2,5), and Sym(5) is indeed
    isomorphic to PGL(2,5).

    But the binary dodecahedral group, the double cover of Alt(5) ~=
    PSL(2,5), is isomorphic to the perfect group SL(2,5), not Sym(5).
    As required of a double cover, SL(2,5) has a center Z of order 2
    such that SL(2,5)/Z ~= PSL(2,5) ~= Alt(5), so SL(2,5) has a
    surjective homomorphism to Alt(5), but it turns out that it doesn't
    contain Alt(5) as a subgroup. In fact, SL(2,5) has only one element
    of order 2, namely the generator of Z.

    Also, for what it's worth, the full symmetry group of the
    dodecahedron in O(3), including not only rotations but also
    reflections and all of their products, is isomorphic to the direct
    product Alt(5) x C_2, of Alt(5) and the cyclic group of order 2.
    Here C_2 is inversion through the origin, the negative of the
    identity element of O(3).

    Up to isomorphism, the three groups Sym(5), SL(2,5) and
    Alt(5) x C_2 are the only nonsolvable groups of order 120.

    > Vertices of the dodecahedron can be labelled by the 20 ordered pairs of
    > 5 symbols; adjacent vertices have a common symbol, but only half of the
    > pairs of vertices with a common symbol form edges, which is why not
    > the full symmetric group acts.


    --
    Jim Heckman
     
  9. Nov 27, 2006 #8
    Some addenda:

    Someone with the handle "Dileffante" has found another nice
    example of the dodecahedron in nature - and even in Nature:

    While perusing a Nature issue I found this short notice on a paper,
    and I remembered that in your talk (which I saw online) you mentioned
    that the dodecahedron was not found in nature. Now I see in "week241"
    that there are some things dodecahedral after all, but nevertheless,
    I send this further dodecahedron which was missing there.

    Nature commented in issue 7075:

    15) The complete Plato, Nature 439 (26 January 2006), 372-373.

    According to Plato, the heavenly ether and the classical elements -
    earth, air, fire and water - were composed of atoms shaped like
    polyhedra whose faces are identical, regular polygons. Such shapes
    are now known as the Platonic solids, of which there are five: the
    tetrahedron, cube, octahedron, icosahedron and dodecahedron.
    Microscopic clusters of atoms have already been identified with
    all of these shapes except the last.

    Now, researchers led by Jose Luis Rodriguez-Lopez of the Institute
    for Scientific and Technological Research of San Luis Potose in Mexico
    and Miguel Jose-Yacaman of the University of Texas, Austin, complete
    the set. They find that clusters of a gold-palladium alloy about two
    nanometres across can adopt a dodecahedral shape.

    The article is in:

    16) Juan Martin Montejano-Carrizales, Jose Luis Rodriguez-Lopez,
    Umapada Pal, Mario Miki-Yoshida and Miguel Jose-Yacaman, The
    completion of the Platonic atomic polyhedra: the dodecahedron,
    Small, 2 (2006), 351-355.

    Here's the abstract:

    Binary AuPd nanoparticles in the 1-2 nm size range are
    synthesized. Through HREM imaging, a dodecahedral atomic
    growth pattern of five fold axis is identified in the
    round shaped (85%) particles. Our results demonstrate the
    first experimental evidence of this Platonic atomic solid
    at this size range of metallic nanoparticles. Stability of
    such Platonic structures are validated through theoretical
    calculations.

    Either there is some additional value in the construction, or
    the authors (and Nature editors) were unaware of dodecahedrane.

    Dodecahedrane is a molecule built from carbon and hydrogen - a bit
    different from an "atomic cluster" of the sort discussed here.
    It's a matter of taste whether that's important, but I bet these
    gold-palladium nanoparticles occur in nature, while dodecahedrane
    seems to be unstable.

    My friend Geoffrey Dixon contributed these pictures of Platonic
    life forms:

    http://math.ucr.edu/home/baez/platonic_lifeforms.jpg

    They look a bit like Ernst Haeckel's pictures from his book
    "Kunstformen der Natur" (artforms of nature).

    ..................................................................

    Two puzzles:

    # Which job are only blind people allowed to do in Korea?

    # Who owns all the unmarked mute swans on the River Thames?

    If you get stuck, try

    http://math.ucr.edu/home/baez/puzzles/32.html

    and

    http://math.ucr.edu/home/baez/puzzles/33.html
     
  10. Nov 30, 2006 #9
    John Baez wrote:
    > Some addenda:
    >
    > Someone with the handle "Dileffante" has found another nice
    > example of the dodecahedron in nature - and even in Nature:
    >
    > While perusing a Nature issue I found this short notice on a paper,
    > and I remembered that in your talk (which I saw online) you mentioned
    > that the dodecahedron was not found in nature. Now I see in "week241"
    > that there are some things dodecahedral after all, but nevertheless,
    > I send this further dodecahedron which was missing there.
    >
    > Nature commented in issue 7075:
    >
    > 15) The complete Plato, Nature 439 (26 January 2006), 372-373.
    >
    > According to Plato, the heavenly ether and the classical elements -
    > earth, air, fire and water - were composed of atoms shaped like
    > polyhedra whose faces are identical, regular polygons. Such shapes
    > are now known as the Platonic solids, of which there are five: the
    > tetrahedron, cube, octahedron, icosahedron and dodecahedron.
    > Microscopic clusters of atoms have already been identified with
    > all of these shapes except the last.


    and of the tetrahedron, cube, octahedron, icosahedron and dodecahedron

    only the 3D array (tessellation) of the cube fills space (no intervening gaps).

    http://mathworld.wolfram.com/Space-FillingPolyhedron.html

    Aristotle erroneously thought that the tetrahedron did fill space.

    For information, the parity forms in

    http://arxiv.org/abs/physics/9905007
    Figure 2.2.1

    do fill space.

    Richard
     
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