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October 26, 2008

This Week’s Finds in Mathematical Physics (Week 271)

Posted by John Baez

In week271 of This Week’s Finds, see massive volcanic eruptions on Jupiter’s moon Io. Learn about allotropes of sulfur, 2d quasicrystals formed by slicing higher-dimensional A nA_n latices:

and a 4d quasicrystal formed by slicing the E 8E_8 lattice. Read about Jeffrey Morton’s wonderful extension of the "groupoidification" idea. And hear what Stephen Summers has to say about new work on constructive quantum field theory!

Posted at October 26, 2008 6:35 PM UTC

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Re: This Week’s Finds in Mathematical Physics (Week 271)

“During the journey we commonly forget its goal. Almost every profession is chosen as a means to an end but continued as an end in itself. Forgetting our objectives is the most frequent act of stupidity”

Nietzsche must have forgotten that Life itself is a journey without any goal except dying of course. What then would be best: Focusing ones attention towards the objective (dying) or being stupid and enjoying the journey??

Posted by: eric on October 26, 2008 7:42 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 271)

eric said

Nietzsche must have forgotten that Life itself is a journey without any goal except dying ofcourse.

Well, I think it’s pushing it a bit to deduce that much from a single aphorism. For instance, the following aphorism appears, in The Wanderer and his Shadow, a little before the one John quoted.

End and goal: Not every end is the goal. The end of a melody is not its goal; and yet: if a melody has not reached its end, it has not reached its goal. A parable.

Actually, I’m only writing this post so that I can get my favouritist Nietzsche aphorism in, and this also appears in the same place in The Wanderer and his Shadow.

Pleasure tourists.- They climb the hill like animals, stupid and perspiring: no one has told them there are beautiful views on the way.

I can’t climb a hill without it coming to mind.

Posted by: Simon Willerton on October 26, 2008 10:11 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 271)

Reminds me of a story I heard when I was going to or from the Teton Mtns

apparently tourists would arrive at Jackson Hole after traversing Yellowstone with all its `wonders’ and ask
anything to see around here?

the State of Michigan motto is something like

if you seek a beuatiful penninsula, look around you!

Posted by: jim stasheff on October 27, 2008 6:11 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 271)

Death is the end of life, not the goal. The goal, you get to choose.

Posted by: John Baez on October 26, 2008 11:04 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 271)

And if life is a journey, it is one of those journeys to which the saying It is better to travel than to arrive applies with particular force.

Posted by: Tim Silverman on October 26, 2008 11:34 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 271)

Tim wrote:

…the saying It is better to travel than to arrive

By the way, I’ve heard the related saying getting there is half the fun was invented by the Cunard Line only after airplanes rendered its original function — efficiently carrying travelers across the Atlantic by boat — obsolete.

I’m not sure there’s any grand metaphysical conclusion to be drawn from this, but it’s worth pondering.

Posted by: John Baez on October 27, 2008 4:13 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 271)

and i am not even talking about serendipity here. ;)

Posted by: eric on October 26, 2008 7:47 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 271)

equivalence classes of spans of spans finite groupoids as morphisms

should be 2-morphisms.

Posted by: David Corfield on October 27, 2008 12:02 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 271)

Thanks! Fixed.

Posted by: John Baez on October 27, 2008 4:24 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 271)

Thanks for mentioning “constructive AQFT”!

I believe this is an important subject to be investigated beyond the general “structural results” in AQFT, not the least for proving the relevance of the entire AQFT axiomatics in the first place (noticing that Summers argues that the previous approach, the “semicalssical one” was superceded by constructive AQFT as that failed to produce the desired examples).

To me it seems of particular relevance for the application of AQFT in the general scope of what people are doing (as opposed to just studying it for its own sake) to better understand the following two kinds of constructions

a) AQFT nets from continuum limits of lattice models.

b) AQFT nets from smearing of vertex operator algebras.

I know that people have thought about a), but no real discussion in the literature seems to exist.

I also know that people are currently thinking about b), but last time that I (almost) had the chance to hear a talk about this and speak with the person working on it, I had to leave on a vacation.

Posted by: Urs Schreiber on October 27, 2008 1:25 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 271)

For b) I guess you’re talking about 2d QFTs?

Constructive quantum field theory is vastly easier in 2 spacetime dimensions than in higher dimensions — for example, scalar fields with polynomial interactions were rigorously constructed by Segal, Glimm and Jaffe in the 1970s. But general nonlinear σ\sigma-models have long seemed out of reach. This has got to be fixed.

What really excites me, though I don’t understand the details yet, are Summers’ remarks on constructing higher-dimensional quantum field theories — starting not from field Lagrangians but from particle interactions!

Posted by: John Baez on October 27, 2008 4:31 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 271)

For b) I guess you’re talking about 2d QFTs?

Yes, that’s what I was thinking of, 2d CFTs.

(On the other hand, one can consider higher dimensional analogs of 2d vertex operator algebras #, and eventually I would like to see them smeared to AQFT nets, too, if possible.)

What really excites me, though I don’t understand the details yet, are Summers’ remarks on constructing higher-dimensional quantum field theories — starting not from field Lagrangians but from particle interactions!

So the claim is that there are now examples of higher-dimensional interacting AQFT nets? Could you point out which reference precisely you are thinking of?

In Summers’ html notes that you linked to # he mentiones, if I understand correctly, higher dimensional theories which are either not local or not Poincaré covariant. I am not sure if there is a higher dimensional, local, covariant and interacting example in his list (but I haven’t tried to follow all the references yet).

For instance the reference [GL] Summers discusses under (3) gives a net which is not local, he says.

That may not be a surprise as this net is supposed to be the net of fields on a noncommutative deformation of Minkowski space, but it means for me that this does not yield an example of a local net of interacting fields.

Below that he discusses his own [BS3] which builds on [GL]. I will have to read this in detail, since from the summary it is not quite clear to me whether this discusses local covariant nets or not. Do you know?

Maybe it sounds like I am nitpicking, but my impression from other examples is that dropping the locality axiom from the AQFT axioms means dropping too much.

And I am also a bit reluctant to consider AQFT on noncommutative spacetimes before trying to understand Yang-Mills in AQFT. That would probably require looking at lattice models. Which brings me back to my point a).

Posted by: Urs Schreiber on October 27, 2008 7:05 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 271)

Urs wrote:

So the claim is that there are now examples of higher-dimensional interacting AQFT nets? Could you point out which reference precisely you are thinking of?

You’ve already read the list of references more carefully than I have. I’m so darn busy…

In Summers’ html notes that you linked to # he mentions, if I understand correctly, higher dimensional theories which are either not local or not Poincaré covariant.

Really? That would be a bummer.

Hmm, maybe you’re right. I’ll try to get Summers to comment on this issue here.

Maybe it sounds like I am nitpicking, but my impression from other examples is that dropping the locality axiom from the AQFT axioms means dropping too much.

Yes: at least back when I was paying attention, the conventional wisdom was that locality is what makes the game difficult.

Posted by: John Baez on October 27, 2008 9:31 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 271)

John Baez suggested that I come to this site and respond. My desultory reading of some physics and mathematics blogs in the recent past has suggested that the, shall we say, occasionally undisciplined and incompletely informed comments made in such blogs are then repeated by others and become “common wisdom” among the readers of the blog. Of course, that can’t happen at this blog, right?…

In any case, I very carefully reduced the content of many papers to a few pages, not with the intent to replace reading the papers but to entice readers to their gates. I am not going to try to reduce the matter to a few sentences here - it is nontrivial, and I shall just make a few comments.

To begin, I did not argue “that the previous approach, the “semiclassical one” was superceded by constructive AQFT”.
I argued that new ideas have found completely new mathematical implementation and have resulted in the rigorous construction of quantum field models which could not be constructed by the previous approaches. I hope the distinction can be appreciated.

Further, even restricting one’s attention to those non-free models in four spacetime dimensions which have been constructed so far (and I emphasize here, as I did in my summary, that this program is in its infancy), there are different families of models with different properties. All admit local, Poincare invariant nets of observable algebras, even though, in some cases, they are obtained by algebraic methods from fields which are not local. In some cases, the resultant nontrivial two-body scattering amplitudes are not Lorentz invariant (true of the models obtained by deformations motivated by the desire to have rigorous models on noncommutative Minkowski space - these *also* provide models on normal Minkowski space), while in other cases these amplitudes are Lorentz invariant. But the underlying nets of algebras *are* local and Poincare invariant (however, in some models the algebras associated with bounded spacetime regions can be rather small, though they are nontrivial).

I would encourage those who feel that other approaches to constructing models of quantum field theory are, for whatever reason, more promising to get to work. Personally, I would welcome any advance in that direction.

Posted by: Stephen J. Summers on October 29, 2008 2:00 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 271)

But the underlying nets of algebras *are* local and Poincare invariant

Thanks for the clarification! That wasn’t clear to me from what I had seen on your page. I’ll have a look at your articles.

I would encourage those who feel that other approaches to constructing models of quantum field theory are, for whatever reason, more promising to get to work.

What is currently the next kind of model, the next open question, that practicioners in constructive AQFT are trying to attack?

I read your above remark as implying that for instance the idea of constructing AQFTs from lattice models which I mentioned is not on the priority list of people currently active in constructive AQFT. I would find it interesting to know if there is a deeper reason for that, or if it just so happens. Can you say anything about that? Is it known that people have tried and run into difficult obstacles? Is it not regarded as a potentially promising strategy? I am just trying to understand the status of that idea in the field currently.

Posted by: Urs Schreiber on October 29, 2008 10:16 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 271)

A number of people have looked at lattice theories from the vantage point of AQFT (Rudolph, Szlachanyi, Baumgaertel, etc.) for the purpose of bringing developments from AQFT (like superselection theory) to bear upon lattice theories. But I am not aware of attempts to directly construct nets of observable algebras on Minkowski theory as some kind of limit of nets on lattices. Note that local algebras for lattice theories are typically type I, while those for relativistic theories are typically type III. This makes a huge difference. Of course, one can still try to control the continuum limit, construct the continuum fields, then construct the continuum observable algebras. But those few of us working on algebraic constructive QFT are exploring deformation procedures to arrive at new models…

Posted by: Stephen J. Summers on October 30, 2008 2:43 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 271)

I wrote #

I know that people have thought about [AQFT nets from continuum limits of lattice models], but no real discussion in the literature seems to exist.

I just found something:

there is

Dirk Schlingemann, Constructive Aspects of Algebraic Euclidean Field Theory.

This builds on

Dirk Schligemann, From Euclidean Field theory to Quantum Field Theory

which defies Euclidean AQFT nets (algebraic statistical field theory, that is) and shows how these may be Wick rotated to Haag-Kastler nets under certain conditions.

In Constructive Aspects of Algebraic Euclidean Field Theory the idea is to construct Euclidean nets from lattice models, in the hope that some of them would then admit a Wick rotation.

That last step is not discussed here, but Schlingemann sets up plenty of machinery to talk about Euclidean lattice models , block spin transformations, continuum limits etc. in the AQFT context, relating them to inductive limits over the relative C *C^*-alegrbas, etc.

I haven’t yet looked at this in detail, but this seems to be the kind of discussion I was hoping would exist somewhere.

Posted by: Urs Schreiber on November 13, 2008 9:36 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 271)

In D. Schlingemann, Short-distance analysis for algebraic euclidean field theory there is more on relating Euclidean nets to Minkowski nets, supposedly potentially helpful for carrying Euclidean AQFT lattice models to the Minkowski side.

Posted by: Urs Schreiber on November 13, 2008 9:51 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 271)

I am being told that the following thesis is a considerable achievement for construction of 2-d AQFTs:

Gandalf Lechner, On the Construction of Quantum Field Theories with Factorizing S-Matrices.

A summary is in hep-th/0502184.

This crucially uses wedge algebras (algebras localized in spacelike wedges left or right of a point in 2d Minkowski space) to construct from them the usual double-cone algebras. This makes use of an old insight by Bert Schroer on “polarization free generators”.

I still have to absorb this. But it was indicated to me that this might help also in constructing from a local net the corresponding parallel transport 2-functor. As indicated here.

Posted by: Urs Schreiber on November 13, 2008 10:35 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 271)

I would be remiss if I did not mention Tony Robbin in the context of Penrose tiling and quasi-crystals. The quasi-crystal image was from a sculpture that has since been destroyed. The concept is the 3-d version of deBruijn. Alas, Tony is recovering from some heavy-duty chemotherapy. For windows users, the executables on his page give really nice hypercubes and hypercubical tessellations for playing.

Posted by: Scott Carter on October 27, 2008 10:47 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 271)

Cool!

Posted by: John Baez on October 27, 2008 11:17 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 271)

Those at the Minnesota nn-categories workshop in 2004 may remember that Tony was there. Great to have an artist interested in nn-categories, just as, as Tony showed, Picasso made a close study of 4-dimensional geometry around the turn of the nineteenth century. You can read about all this in Shadows of Reality: The Fourth Dimension in Relativity, Cubism, and Modern Thought.

Posted by: David Corfield on October 28, 2008 10:06 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 271)

anyone able to do a comparative review with
Linda Henderson ( art historian) “The Fourth Dimension and Non-Euclidian Geometry in …

Posted by: jim stasheff on October 28, 2008 3:14 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 271)

Greg Egan is back from Iran! And, he brought back pictures, including this one of a tiling with approximate 5-fold symmetry, on the Friday Mosque in Isfahan:

Posted by: John Baez on October 30, 2008 9:59 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 271)

Any idea why the blue lines are there?

Posted by: jim stasheff on October 31, 2008 1:25 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 271)

Any idea why the blue lines are there?

The short answer is: they’re decorative. They’re known as strapwork or girih, and are part of the aesthetic tradition.

Exactly how these kinds of lines appear as markings on a certain set of tiles was figured out by Lu and Steinhardt in Decagonal and quasi-crystalline tilings in medieval Islamic architecture, but there’s a nice summary of their work on Wikipedia.

Posted by: Greg Egan on October 31, 2008 2:48 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 271)

You can also see those lines on the Darb-i Imam shrine in Isfahan:

Posted by: John Baez on October 31, 2008 5:37 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 271)

Doesn’t it have perfect 5-fold symmetry if you choose the right centre of rotation, apart from the fact that it’s cut off at the edges?

Posted by: Jamie Vicary on October 31, 2008 6:46 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 271)

Jamie wrote:

Doesn’t it have perfect 5-fold symmetry if you choose the right centre of rotation […]?

It probably does. Many Penrose tilings formed by inflation have this property, too.


What a 2d pattern can’t have is perfect 5-fold symmetry about one center and also two linearly independent translation symmetries.

(In fact, right now it seems to me that even perfect 5-fold symmetry about one center and one translation symmetry is impossible.)

Posted by: John Baez on October 31, 2008 11:53 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 271)

Yea, verily! for were there ONE translation symmetry and a five-fold center, then there WOULD be two, independent translation symmetries: the point group acts in the usual way on the lattice of translations!

Posted by: some guy on the street on December 4, 2008 1:26 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 271)

Doesn’t it have perfect 5-fold symmetry if you choose the right centre of rotation, apart from the fact that it’s cut off at the edges?

I think there are subsets with perfect 5-fold symmetry, but the whole thing is not a subset of any finite pattern with perfect 5-fold symmetry.

If you look at Lu and Steinhardt’s Figure S7A, the Darb-i Imam portal John shows above is constructed by subdividing a figure with four large decagons and three bowties. Each decagon is subdivided in a way that retains its perfect rotational symmetry, but the whole assembly doesn’t have that symmetry (and can’t be extended in a way that does).

Posted by: Greg Egan on October 31, 2008 11:59 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 271)

Clearly the picture in my reply is some sort of ‘finite pattern’ with perfect 5-fold symmetry. But maybe by ‘finite pattern’ you mean what I meant: a finite pattern that can be repeated to form a periodic tiling of the whole plane: i.e., one with two linearly independent translation symmetries.

Posted by: John Baez on November 1, 2008 12:06 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 271)

John, I’m a bit confused by what you’re saying here, and why you think we’re disagreeing about anything! (I’m afraid my comment appeared after one by you that I hadn’t seen yet, which might be part of the confusion.)

In this comment you certainly do give a finite figure with perfect 5-fold rotational symmetry. I’m not saying such things don’t exist! All I’m saying is that the Darb-i Imam portal you showed in your preceding comment is not, in its entirety, a subset of such a figure, but merely contains such figures. So Jamie’s conjecture, if I’ve understood him correctly, isn’t right.

Posted by: Greg Egan on November 1, 2008 12:19 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 271)

If Jamie was wondering about my picture of the Friday Mosque, rather than the Darb-i Imam, the same is true. Lu and Steinhardt show (PDF link, Figure S7B) how it’s a piece of a wonderfully symmetrical assembly of decagons and bowties, but that whole assembly doesn’t have 5-fold rotational symmetry.

Posted by: Greg Egan on November 1, 2008 1:35 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 271)

Thanks for that link, Greg. I was indeed talking about the Friday Mosque. I think I can see where rotational symmetry breaks down in the Darb-i Imam picture, but it seems to me that the excerpt of the Friday Mosque pattern visible in John’s post does admit 5-fold symmetry, about the obvious point on the left-hand side just above the centre. Maybe the larger pattern it’s taken from doesn’t, though.

Posted by: Jamie Vicary on November 1, 2008 11:43 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 271)

Jamie, I think you’re right! It’s hard to get good photos of these long panels from ground level – and I didn’t even have a tripod with me – but it looks as if I serendipitously got a shot that’s roughly centred on one of the “big decagons” whose centre lies on the left edge, and which hides the fact that the decagon’s joined to things above and below it that break the symmetry with the thing it’s joined to on the right.

Posted by: Greg Egan on November 1, 2008 12:11 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 271)

Greg: I thought the Darb-i Imam portal design was a subset of a figure with perfect 5-fold symmetry, centered at the midpoint of the bottom of the picture in my comment. It still looks that way to me, but maybe I’m wrong.

Since I thought this was obvious, I thought you must have meant something else when you denied it.

Posted by: John Baez on November 1, 2008 4:34 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 271)

John wrote:

It still looks that way to me, but maybe I’m wrong.

OK! I found it very hard to judge this by eye, so I looked at Lu and Steinhardt, and I think you can tell from their figure S7A that there isn’t 5-fold rotational symmetry here. If you move “two sides around” their large decagon which lies at the centre of what you’re suggesting might be a point of 5-fold symmetry, there are different things glued to its sides.

It’s interesting, because the designers obviously could have chosen to use a subset of a single, very large, fully 5-fold rotationally symmetrical figure, using subdivision alone to make it intricate and complex. But they seem to have preferred to join up a few of the largest 5-fold symmetrical units (Lu and Steinhardt’s big decagons) – along with the smaller “bowties” – in a pattern with other kinds of symmetries.

Posted by: Greg Egan on November 1, 2008 5:07 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 271)

Does anyone have any idea whether this search for non-symmetry had a reason? I know only very little about this, and have not read the sources, so can anyone give me some idea and whether the idea is based on textual evidence. The subject is fascinating!

Posted by: Tim Porter on November 1, 2008 8:53 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 271)

Tim, I can’t quite discern what it is you’re asking. You might find it’s worth reading Lu and Steinhardt’s paper, which gives a little bit of background on the traditions and methods, then explains the claim they make for the use of “girih tiles” replacing compass and straight-edge techniques, and the evidence that backs this up.

Posted by: Greg Egan on November 1, 2008 10:42 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 271)

I think my quetion is something like the following more philsophical query:

What would be interesting to ask is whether the appeal of these patterns was simply that they look very beautiful or was there some symbolic sense behind them, in which case the contrast between the symmetric designs and the more complex ones that you are showing could be interpreted in some non-mathematical sense.

The question is vague, and may not be worth pursuing here, it is just that the use of mathematics in art is often linked to a deep appreciation for the symbolic nature of the relationships exhibited by it. Perhaps the symmetries in Islam Art might be an expression of a view of the perfection of the universe as created, but then what would be the hidden meaning of these broken symmetries. I suspect we will never know, but thought I would ask!

Posted by: Tim Porter on November 1, 2008 11:22 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 271)

I forgot to mention that similar questions arise in the interpretation of Irish knot work where many of the most beautiful examples show an appreciation for the rules governing interlacing and hence of a quite complex combinatorics (no knowledge of proof of those rules is implied). The question is then whether there is some way that we can find out if there was some ‘meaning’ behind those patterns or was it visual enjoyment alone that was the influence.

Posted by: Tim Porter on November 1, 2008 11:28 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 271)

url for Irish knot work?? thanks

Posted by: jim stasheff on November 1, 2008 1:49 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 271)

Jim,

Perhaps this is getting a bit far away from the main topic but there is no real URL for that since it was the result of conversations with Michael Brennan of the Waterford Institute, and was work he was doing for a PhD in History and Archaeology at Bangor. I do not know quite what has happened since I was ‘retired’ by Bangor. I had been one of his supervisors.

Michael has a web page:

http://wit.academia.edu/MichaelBrennan

but it does not help much. Interestingly enough his new half supervisor is a friend of mine in Galway, who has looked at patterns in music (and who is a very good tin whistle player!)

Michael’s main supervisor is Nancy Edwards, who did work of a non-mathematical nature classifying the motifs used in Celtic interlacing.

Posted by: Tim Porter on November 1, 2008 3:54 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 271)

Tim wrote:

What would be interesting to ask is whether the appeal of these patterns was simply that they look very beautiful or was there some symbolic sense behind them, in which case the contrast between the symmetric designs and the more complex ones that you are showing could be interpreted in some non-mathematical sense.

The question is vague, and may not be worth pursuing here, it is just that the use of mathematics in art is often linked to a deep appreciation for the symbolic nature of the relationships exhibited by it.

I think it’s a fascinating question. I often wonder: What were the people who made this art thinking? What did they say about it? How did they teach their students to do it? Did they leave behind any writings — training manuals, for example?

Unfortunately, I don’t know the answers to any of these questions.

Maybe I should start by looking at these books:

  • J. Bourgoin, Arabic Geometrical Pattern and Design.
  • R. Ettinghausen, Islamic Art and Architecture.
  • S. Blair, Islamic Art and Architecture 1250–1800.

There are also living Islamic traditions of tiling, like the zillij masters of Fez. It would be fun to interview them and ask them lots of questions. (The link is quite interesting, but I’d enjoy more detail.)

Unfortunately, I’ve never seen zillij in Fez based on pentagons and decagons — though I’ve heard hints that they exist. I love periodic tilings, but the flirtation with quasiperiodic tilings seems even more daring and sophisticated, so that’s what gets me the most excited.

I also recommend Eric Broug’s website, and his book. I bought an English translation of his book from Amazon. It shows how to make a bunch of tiling patterns using ruler and compass. It also recommends the 3 references above.

Posted by: John Baez on November 2, 2008 4:31 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 271)

Greg wrote:

I found it very hard to judge this by eye, so I looked at Lu and Steinhardt, and I think you can tell from their figure S7A that there isn’t 5-fold rotational symmetry here.

You’re right! The symmetry I’d imagined isn’t really there!

I get the feeling these guys had mastered symmetrical tilings and were having fun with tilings that come close to symmetry but then veer away from it… sort of like jazz musicians who know perfectly well how to play things straight, but enjoy surprising their fellow musicians with unexpected chord changes. Doing tilings with pentagons reminds me of playing music in 5/4 time — it seems inherently show-offy.

But, we may never know what they were thinking.

Posted by: John Baez on November 2, 2008 4:47 AM | Permalink | Reply to this

A Pentagonal Crystal, … Re: This Week’s Finds in Mathematical Physics (Week 271)

arXiv:0811.0336
Title: A Pentagonal Crystal, the Golden Section, alcove packing and aperiodic tilings
Authors: Anthony Joseph
Comments: 57 pages, 12 figures
Subjects: Representation Theory (math.RT)

A Lie theoretic interpretation is given to a pattern with five-fold symmetry occurring in aperiodic Penrose tiling based on isosceles triangles with length ratios equal to the Golden Section. Specifically a $B(\infty)$ crystal based on that of Kashiwara is constructed exhibiting this five-fold symmetry. It is shown that it can be represented as a Kashiwara $B(\infty)$ crystal in type $A_4$. Similar crystals with $(2n+1)$-fold symmetry are represented as Kashiwara crystals in type $A_{2n}$. The weight diagrams of the latter inspire higher aperiodic tiling. In another approach alcove packing is seen to give aperiodic tiling in type $A_4$. Finally $2m$-fold symmetry is related to type $B_m$.

Posted by: Jonathan Vos Post on November 4, 2008 7:28 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 271)

Here’s a quasicrystal with icosahedral symmetry, found by projecting points from the I 6I_6 lattice onto a three-dimensional subspace chosen so that the coordinate axes and their opposites project onto the vertices of an icosahedron.

The “minimal” lattice for icosahedral symmetry is D 6D_6, but the construction there is trickier because the Voronoi cells around the lattice points are more complicated.

Posted by: Greg Egan on November 6, 2008 4:30 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 271)

Oops, there was a glitch in this image where I omitted some tiles because I wasn’t taking account of a certain degeneracy.

That’s fixed now, and this cluster looks much more “icosahedrally symmetric” to the naked eye.

Of course real quasicrystals in nature will almost never be centred on the single point of perfect icosahedral symmetry like this, but it’s still nice to see what’s theoretically possible.

Posted by: Greg Egan on November 9, 2008 3:45 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 271)

This post has got absolutely nothing to do with the workshop, but from reading TWF i got the impression that you ‘are into’ new elements etc..

So I thought why not mention the following post about a discovery made by Amnon Marinov at the Hebrew University of Jerusalem: Element Unbibium discovered in nature. (Arxiv article can be found here.)

If it’s true it would be an astonishing discovery, some renowned journals however have not put the study up for peer-review.

Posted by: ericv on November 25, 2008 3:39 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 271)

Eric V. wrote:

This post has got absolutely nothing to do with the workshop

Yeah, it doesn’t — so I moved it here, where it fits in a bit better. Please try to post comments in vaguely appropriate places.

I’ve heard about Marinov’s claims, but I also heard they’ve been heavily criticized, so I’ll wait until something new happens before getting very interested:

It would be cool to see the legendary island of stability — I’ve been waiting for it ever since high school! However, I sort of doubt Marinov has found it.



Posted by: John Baez on November 25, 2008 3:49 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 271)

Here’s a triangular tiling derived from A 6A_6. With these lattices, you have a choice between projecting down faces of the Voronoi cells and projecting down faces of the Delaunay cells, which are their duals. The former give rhombic tilings, the latter triangular tilings. In either case, the selection process is really cool: you select the faces whose duals intersect the plane you’re projecting to!

Tiling from 2-boundaries of Delaunay cells in A_6

Posted by: Greg Egan on November 25, 2008 1:27 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 271)

Beautiful! I see a lot of lines that extend across the whole picture you’ve shown us — a clear sign of long-range order. But I also see some lines that go on for quite a while but then fizzle out. Do some lines go on forever?

(Just an idle question.)

Posted by: John Baez on November 25, 2008 8:52 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 271)

If the plane you project onto passes through a point of the lattice (and is also oriented in a nice symmetrical way, which is taken for granted if you want to produce tilings with edges that belong to a nice symmetrical set of directions) then there are reflection symmetries of the whole setup that require there to be infinite lines passing through the lattice point that lies on the plane.

If you give the plane a generic offset, as in the picture above, then I’m not sure if there are any truly infinite lines. I suspect not, but I don’t know an easy proof of that.

Of course, the actual edges in the lattice that project onto these long lines are really zigzagging wildly in orthogonal dimensions. Unlike the rhombic tilings, where all the possible edge directions have distinct projections, the triangular tilings are degenerate in that respect.

Posted by: Greg Egan on November 26, 2008 1:38 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 271)

I’ve written an applet now that draws quasiperiodic tilings by projecting triangles from A nA_n.

Posted by: Greg Egan on December 14, 2008 8:04 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 271)

At the risk of stating the obvious, I wanted to spell out the precise connection between the rhombic tilings based on hypercubic lattices, and the rhombic tilings based on A nA_n.

In the hypercubic lattice I n+1I_{n+1}, the Voronoi cells are unit hypercubes centred on the lattice points, and the Delaunay cells are unit hypercubes whose vertices are lattice points. Dual to each mm-face of a Voronoi cell, there is an (n+1m)(n+1-m)-face of a Delaunay cell, whose vertices consists of the set of lattice points whose Voronoi cells share our chosen mm-face. For example, in I 3I_3, dual to any 22-face F of a Voronoi cell there is a Delaunay 11-face, or edge, joining the two lattice points whose Voronoi cells share the 22-face F.

The original deBruijn method involved projecting onto a suitable plane P every Voronoi 22-face in I n+1I_{n+1} that was dual to a Delaunay (n1)(n-1)-face that intersected P. So how can we get exactly the same result using A nA_n, rather than I n+1I_{n+1}?

If S is the subspace of n+1\mathbb{R}^{n+1} satisfying x 1+x 2+...+x n+1=0x_1 + x_2 + ... + x_{n+1} = 0, the Voronoi cells of A nA_n turn out to be the projections onto S of unit hypercubes in n+1\mathbb{R}^{n+1} centred on the A nA_n lattice points: in other words, projections of those Voronoi cells in I n+1I_{n+1} whose centres lie in A nA_n. Each A nA_n Voronoi cell has two less vertices than the original hypercube, because those two vertices project into the interior. What’s more, each mm-boundary of a Voronoi cell in A nA_n is the projection onto S of a particular mm-face of the hypercube in n+1\mathbb{R}^{n+1}.

But the Delaunay cells of A nA_n turn out to be intersections of S with the Delaunay cells in I n+1I_{n+1}, not projections.

Given any mm-boundary, M, of a Voronoi cell in A nA_n, the Delaunay (nm)(n-m)-boundary D dual to M is the convex hull of all the lattice points in A nA_n whose Voronoi cells share M.

But if you want, you can also find D via I n+1I_{n+1}, as follows:

  1. find the mm-face M’ of the I n+1I_{n+1} Voronoi cell that projects to M
  2. find the dual Delaunay (n+1m)(n+1-m)-face D’ of M’ in I n+1I_{n+1}
  3. intersect D’ with S, to get the Delaunay (nm)(n-m)-boundary D back down in A nA_n.

So the requirement in the original deBruijn method of selecting Delaunay (n1)(n-1)-faces in I n+1I_{n+1} that intersect P is really the same as selecting Delaunay (n2)(n-2)-faces in A nA_n that intersect P, so long as P lies in S. And of course projecting a Voronoi 22-face first from n+1\mathbb{R}^{n+1} into S and then down to P is exactly the same as projecting straight from n+1\mathbb{R}^{n+1} to P.

Posted by: Greg Egan on December 22, 2008 3:26 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 271)

Just to check that I’m understanding the simplest aspects of this, and maybe help other people:

The Voronoi cells for the A 2A_2 lattice are regular hexagons. A hexagon is what you see when you view a transparent 3d cube corner-on, so that two of its 8 corners line up in your sight, with the remaining 6 forming the corners of the hexagon.

I guess the A 3A_3 Voronoi cells are rhombic dodecahedra:

So, a rhombic dodecahedron must be what you get when you project a 4d cube into 3d space ‘corner-on’, so that two of its 16 corners map to the center of the rhombic dodecahedron, with the remaining 14 forming the corners of the rhombic dodecahedron.

So, a rhombic dodecahedron had better have 14 corners, or I made a mistake. But it’s damned hard to count them when they’re rotating. This makes it easier:

Posted by: John Baez on December 23, 2008 4:14 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 271)

John wrote:

I guess the A 3A_3 Voronoi cells are rhombic dodecahedra

Yes! And the A 3A_3 Delaunay cells dual to its 14 vertices are 8 tetrahedra and 6 octahedra. There’s a picture showing how this fits together (with the bases sliced off the tetrahedra, and only half of each octahedron portrayed) about 2/3 of the way down this page, but I won’t insert the picture here, in case people prefer to try imagining it for themselves first.

Posted by: Greg Egan on December 23, 2008 9:58 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 271)

I should just add a remark as to where the tetrahedral and octahedral Delaunay cells of A 3A_3 come from.

In A 2A_2, the Delaunay cells are triangles, dual to the hexagonal Voronoi cells:

A_2 lattice Voronoi and Delaunay cells

If you take a cube in three dimensions, and slice it orthogonally to one of its diagonals in a plane that intersects at least two vertices, the slice will be one of two triangles, which are upside down compared to each other. These are just the two kinds of triangles in the picture above.

And if you take a cube in four dimensions and slice it orthogonally to one of its diagonals in a 3-plane that intersects at least two vertices, there are three possibilities: two different tetrahedra, or an octahedron. We can check that the vertices add up: 4 + 6 + 4 = 14, plus the two vertices that lie on the diagonal, makes a total of 16.

There are some nice simple patterns in all the coordinates here, as you might expect when there’s an integer lattice involved. I won’t go into all the details, which are spelled out in this page, but for any dimension the vertices of the Voronoi cell of the origin in A nA_n, thought of as lying in a subspace of n+1\mathbb{R}^{n+1}, take the form (1s,1s,1s,...,s,s,s,...)(1-s,1-s,1-s,...,-s,-s,-s,...) or permutations thereof, and where there are qq coordinates of the form 1s1-s we have s=qn+1s=\frac{q}{n+1} in order that the coordinates sum to zero. These correspond to the projections of the (n+1)(n+1)-cube vertices with the form (12,12,12,...,12,12,12,...)(\frac{1}{2},\frac{1}{2},\frac{1}{2},...,-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},...) with qq positive coordinates.

And dual to a Voronoi cell vertex of that form, there is a Delaunay cell whose vertices are the lattice points that have an equal number of coordinates with the value 11 and the value 1-1 (the rest being zero), with all the values of 11 lining up with a 1s1-s in the Voronoi vertex, and all the values of 1-1 lining up with a s-s in the Voronoi vertex. This is simply because, relative to such a lattice point, the shift of origin gives you back another Voronoi vertex of the same form (merely swapping values of 1s1-s and s-s for the non-zero coordinates); in other words, the Voronoi vertex lies not just on the Voronoi cell of the origin, but also on the congruent Voronoi cell of the lattice point we’ve cooked up this way. And that’s the definition of the lattice point being a vertex of the dual Delaunay cell!

Posted by: Greg Egan on December 23, 2008 11:27 AM | Permalink | Reply to this

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