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January 23, 2010

This Week’s Finds in Mathematical Physics (Week 291)

Posted by John Baez

In week291 of This Week’s Finds, learn about Carlos Simpson’s new book. Listen to a crab canon on a Möbius strip. See how the Mandelbrot set mimics infinitely many Julia sets:


Then, continue our exploration of analogies between electrical circuits and other physical systems. This time we’ll meet the most important 2-ports and 3-ports — and get a tiny taste of Poincaré duality for electrical circuits.

Posted at January 23, 2010 4:51 AM UTC

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Re: This Week’s Finds in Mathematical Physics (Week 291)

Did you see this question at Math Overflow, which asks for an interpretation of the claim that Mandelbrot sets are initial objects in the category of bifurcations?

If nothing else, I learned about the notion of multilimit from Mike Shulman’s comment as people there tried to interpret the claim.

Posted by: David Corfield on January 23, 2010 9:01 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 291)

Are you going to comment on Erik Verlinde’s provocative new paper some week soon? It’s starting to spawn interesting followups.

Posted by: Allen Knutson on January 23, 2010 1:35 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 291)

Quantum gravity: tired.

Electrical engineering: wired.

But I’m glad you made me look at Verlinde’s paper, because it starts with an example that meshes with my current interest in general systems theory: the ‘entropic force’ that makes a polymer want to coil up. In equation (2.3) we see that just as energy and inverse temperature are thermodynamically conjugate variables:

1T=SE \frac{1}{T} = \frac{\partial S}{\partial E}

so are the stretching xx of the polymer and this entropic force FF divided by temperature:

FT=Sx \frac{F}{T} = \frac{\partial S}{\partial x}

This makes me even more curious to understand better the relation between ‘canonically conjugate’ variables and ‘thermodynamically conjugate’ variables. A lot of the same math governs both — for example, Legendre transforms turning functions on tangent bundles into functions on cotangent bundles. But physically they seem very different.

If Verlinde is right and gravity is an entropic force, this issue could even be important for understanding gravity! But the issue is fascinating in its own right. It must have been studied for a long time. My plea from week289 still stands:

For example: in classical mechanics it’s really important that we can define “Poisson brackets” of smooth real-valued functions on the cotangent bundle. So: how about in thermodynamics? Does anyone talk about the Poisson bracket of temperature and entropy, for example?

And Poisson brackets are related to quantization - see week282 for more on that. So: does anyone try to quantize thermodynamics by taking seriously the analogies I’ve described? I’m not sure it makes physical sense, but it seems mathematically possible.

Posted by: John Baez on January 23, 2010 7:03 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 291)

[..]

So: does anyone try to quantize thermodynamics by taking seriously the analogies I’ve described? I’m not sure it makes physical sense, but it seems mathematically possible.

Well, further clarifying (after thinking about it) my previous comment in TWF 289, *if* S is conserved, then (T,S) are related to (p,V) by an equation of state. I’ll assume the ideal gas equation pV = nRT, but this can be generalised. In acoustics, you usually eliminate (T,S) and work with (p,V). v is Volume flow per area, better known as velocity:

dp/dt = - (Cp/Cv p_0) dv/dx
dv/dt = - (1/rho) dp/dx

But you could also write (s is entropy flow per area) :

dT/dt = - (T_0/C_v) ds/dx
ds/dt = - (R C_p) dT/dx

In both cases, the speed of sound squared = C_p/C_v R T_0
In the (p,V) case, the “capacitor” is an imaginary cylinder that you adiabatically compress, thereby slightly changing its volume. The “inductor” is a mass of air that you accelerate.

In the (S,T) case, the “capacitor” is an imaginary *constant volume* that you supply with matter and heat, thereby slightly changing its entropy at the cost of entropy of its neighbours. The “inductor” is work that you have to associate with change in reversible entropy flow.

(p,V) representation:
Voltage: pressure
Current: Volume velocity
Capacitance: (C_v V_0)/(C_p p_0), V_0 is the volume, and p_0 the ambiant pressure
Inductance: rho dx^2 /V_0, dx is the length of the chunk of air
speed of sound squared : (C_p/C_v) p_0/rho = (C_p/C_v) R T_0
LC = (dx/c)^2

(T,S) representation:
Voltage: Temperature
Current: Entropy flow
Capacitance: (R/Cv)(S_0/T_0), S_0 is the ambiant entropy of the control volume
Inductance: dx^2 (C_v/R)^2/(C_p S_0), dx is the length of the chunk of air
speed of sound squared : (C_p/C_v) R T_00
LC = (dx/c)^2

So in the (T,S) case, a capacitor is just a container with entropy, whose temperature gets increased if you squeeze in more entropy. It is connected to its neighbours by Carnot-engines (inductors). The heat that is displaced from container to container through these engines gets partly converted to work. The work is stored in the inductors. In an ideal gas, these Carnot engine inductors are simply masses that contain kinetic energy.

Check out ThermoAcoustic engines.
(I used to work on resonating flames, its cool stuff, and electric circuits are really useful in this field)
Check out Pyrophones!.

Comment on transformers:
Transformers are also useful as “dimension transformers”. On one side, you can have say electricity, the other side can for example be a mechanical vibration. The resulting resonators can be a mix of electric and mechanical oscillation.

Gerard

Posted by: Gerard Westendorp on January 27, 2010 10:29 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 291)

[…] Erik Verlinde’s provocative new paper […] interesting followups.

Reminds me of what a famous book on modern theoretical physics says in the last sentence of its preface:

With a large part of the theoretical physics community busy attacking grand structures with arguably insufficient tools, it doesn’t seem out of the question that when the next major progress does occur, it will have come out of the math departments.

;-)

Posted by: Urs Schreiber on January 23, 2010 8:31 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 291)

On Verlinde’s paper;

Having read it, it seems he’s got a heuristic argument for a statistical mechanics of geometry — however weird that may sound; it’s certainly got its own cleverness.

Never having myself looked into this holographic universe idea, I couldn’t say that this paper would encourage me to study it. Perhaps I’m just obtuse, but I don’t quite see where the local topology isn’t taken to be the 4-d thing we comfortably imagine ourselves to inhabit. Put another way: while I’m happy to suppose, when he writes a Laplace or Laplace-Beltrami operator, that it’s really a discrete thing, it doesn’t really seem to matter much to the heuristic. In a related way, when he talks of space as “emergent”, I have a great feeling along the lines of wanting to say “well, once you’ve made it abstract and with uncertain topology at tiny scales, what else could it be really?”.

I’m amused to see how Dr. Smolin has been quick to pipe up with a paper saying “we can do that”, and I’m curious how many other fine-structure schemes could be concocted to fit the hypotheses of Verlinde’s heuristic argument — I’m betting there’s lots, obviously. In any case, what seems to matter is the relative number of small-scale configurations fitting some large-scale outline; at some point, I’d expect either the large-scale outlines to be surprisingly rigid, or a central-limit effect would make distinguishing between well-tuned fine-structure theories impractical.

I must say I’m a bit annoyed with the way Verlinde writes of dismissing gravity as a “fundamental force”; because I rather got the impression that this sort of language was only current in a very narrow community, and hadn’t really been encouraged among classical physicists, e.g. of the Hawking and Penrose and Wheeler variety, for some time now. Am I missing something? (well, obviously; but anything pertinent?)

Oh, again, as it looks like a statistical mechanics of geometry, I’m curious: does the heuristic lend itself to a quantum mechanics/field theory of geometry? Is there an obvious “Wick rotation” or other trick to be done?

Posted by: some guy on the street on January 31, 2010 7:48 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 291)

Some guy on the street wrote:

Oh, again, as it looks like a statistical mechanics of geometry, I’m curious: does the heuristic lend itself to a quantum mechanics/field theory of geometry?

I don’t have any good direct answer to this question. But not surprisingly, some of the main people who think about a ‘quantum field theory of geometry’ are people working on… quantum gravity! And the connections between thermodynamics and quantum gravity have been tantalizingly visible, though still shrouded in mystery, for a long time. Verlinde’s new paper is just the lates of many clues.

For example, Hawking’s calculation of the temperature of a black hole made it clear that in situations where ‘Wick rotation’ makes sense, the Wick rotated version of time… is inverse temperature!

For anyone who wants to understand this with a minimum of fuss, I strongly recommend Hawking’s chapters of the little book by Hawking and Penrose, The Nature of Space and Time.

For a really deep treatment of how the second law of thermodynamics is related to general relativity, I recommend Jacobson’s paper. Verlinde cites it liberally, yet I’m still afraid it’s not getting enough attention compared to Verlinde’s paper! Perhaps this is because Verlinde derived Newtonian gravity from a thermodynamic argument, while Jacobson merely derived general relativity.

Posted by: John Baez on January 31, 2010 8:56 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 291)

Perhaps it is sociological reason, that is, most people feel comfortable to justify their personal interests if someone that is considered a major mainstream theorist, a highly cited string theorist, gives them the green-light to pursue their dreams. That is quite depressing…

What would happen if Ed Witten wrote some papers in a row positively addressing LQG? Hmm.

Posted by: Daniel de Franca MTd2 on January 31, 2010 10:56 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 291)

Are you going to comment on Erik Verlinde’s provocative new paper some week soon? It’s starting to spawn interesting followups.

This post by Robert Helling looks reasonable to me.

Posted by: Urs Schreiber on February 11, 2010 5:39 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 291)

I found a video where one can make IFS fractals, like Sierpinski or fern-like using 3 projector and a camera, WITHOUT the use of a computer. All this because of the feedback of images. There is a cool music going on while someone generates and changes the fractals.

Posted by: Daniel de Franša MTd2 on January 23, 2010 2:28 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 291)

ah, but what that *really* means is that you’ve got an ultra-parallelized system finding an approximate fixed-point measure using analog addition; similarly, you could find the continental divide by building a mock-up of Your Continent and gradually flooding it. I wonder if there are fractal-drawing applications written with gpu-based accelleration, as I think nvidia has been pushing?

Posted by: some guy on the street on January 25, 2010 6:16 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 291)

Mobius strips are cool, and the Crab Canon is cool, but they’re essentially different. Notice that in the video, the two players are still going around the Mobius strip in opposite directions, and each is keeping to its own side of the strip. Moreover, in spite of visually putting in a twist, the “backwards” player is really playing the sound in a mirror, not up-side-down. There’s a reason Bach calls it “crab”: it can be played forward and backward.

Thus, the correct visualization is not a Mobius strip at all, but the orbifold with boundary formed by reflecting the rectangle in half. Making this is easy: take a piece of paper with the music written on one side, and fold it so that the music is on the outside. In this way, each side of the orbifold has half the music on it. Now start at the non-mirror end, but play both sides, reflecting through the orbifold boundary and continuing until you’re back where you started.

Posted by: Theo on January 23, 2010 7:48 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 291)

Theo wrote:

Moreover, in spite of visually putting in a twist, the “backwards” player is really playing the sound in a mirror, not up-side-down.

I had noticed that defect, which is why I didn’t heavily tout the Möbius aspect in my writeup. I was happiest about their movie before they curled up the strip of paper. But I didn’t analyze the problem as carefully as you did. So thanks!

Maybe Jos Leys and Xantox can make some more videos where the images really fit the music.

A comment, and two questions.

First of all, you’re using the word ‘mirror’ to mean the operation of reflecting the musical score left to right, i.e. playing the music backwards in time. That’s fine, but we shouldn’t get confused: in music, a ‘mirror canon’ involves the operation of reflecting the musical score top to bottom, i.e., playing high pitches for low and vice versa.

This causes difficulties if we take it seriously, because an upside-down major scale is a minor scale — just as an upside-down smile is a frown. So, playing both a melody and its upside-down version at the same time forces us to leave the realm of conventional classical harmony. We can solve the problem in various ways:

1) We can tweak the upside-down melody so it says in the same key. Then we have a canon in contrary motion, sometimes called an ‘inverted canon’. This technique appears in the Musical Offering.

2) We could work in a mode like Dorian, which doesn’t change when you flip it upside down. Has anyone written a mirror canon in Dorian? The Beatles’ Eleanor Rigby is in Dorian — that might provide a good theme?

3) Or, we can just bite the bullet and write mirror canons that involve a blend of major and minor scales! Timothy Smith writes:

Ordinarily, canons in contrary motion freely inflect interval qualities in order to stay within the key. Composers with exceptional skill have constructed a rigorous sub-category, called “mirror” canon, in which followers mimic the precise quality of intervals stated by leaders (albeit in the opposite direction). As the technique is difficult, mirror canons are quite rare. The rule of qualitative correspondence between intervals implies that mirror canons invoke more than the usual number of chromatic pitches as No. 6, No. 8, and No. 11 from the Fourteen on the Goldberg Ground, Canon perpetuus and Canon a 2 Quaerendo invenietis from the Musical Offering demonstrate.

Two questions:

Are there any ‘mirror crab’ canons? Here the follower would play the leader’s part backwards and also upside-down.

What would be the right way to animate a mirror crab canon?

Posted by: John Baez on January 23, 2010 8:34 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 291)

“because an upside-down major scale is a minor scale”

More precisely: the upside-down version of the major scale (or Ionian mode) is the Phrygian mode (which is indeed a minor scale but not the usual natural minor scale aka Aeolian mode).

Posted by: Goncalo Marques on January 24, 2010 12:51 PM | Permalink | Reply to this

Carlos Simpson’s book

Thanks for pointing our Carlos Simpson’s book Homotopy Theory of Higher Categories!

Am reading…

Some random comments that came to mind during my first rough go through chapter I, “History and motivation”.

  • I was looking for mentioning of Charles Rezk’s nice stuff on Theta-spaces, which claims to achieve much of what this book is about. From the remark on p. 23 I gather this got in a bit too late to be treated more in detail in this book. (?)

  • page 24: wondering about the conjunction by which nn-stacks and nn-gerbes are mentioned here. Larry Breen’s work on 1- and 2-gerbes concerns 1-groupoid and 2-groupoid valued sheaves, both examples of (n,0)(n,0)-sheaves that are fully under control for all nn \leq \infty (and, in hindsight, have been since a couple of decades). Maybe I am misreading the notation. I think the next interesting application of (,2)(\infty,2)-sheaves is not nn-gerbes which are a special case of principal bundles\infty-bundles but of quasicoherent (,2)(\infty,2)-sheaves corresponding to \infty-vector bundles (as principal bundles form a groupoid, vector bundles a category, and so on).

  • page 25, am enjoying seeing nonabelian \infty-stack cohomology amplified.

    But I am wondering if we can’t make confidently stronger statements by now: as we recently discussed here, feeding Rezk’s model structures on (,n)(\infty,n)-categories into Barwick’s result on enriched Bousfield localization of the presheaf model categories at the set of Cech-cover-nerves should give the (,1)(\infty,1)-categories of non-hypercomplete (,n)(\infty,n)-stacks for all nn (and probably even the corresponding (,n+1)(\infty,n+1)-categories/Theta-spaces). What would be missing were “just” the intrinsic point of view, where we’d be able to justify the Bousfield localization step concretely as modelling a left exact localization intrinsic to (,n)(\infty,n)-categories, as Lurie did for n=1n=1 in his book for Dugger’s theory.

  • still page 25: now fundamental \infty-groupoids aka Poincaré-groupoids and their relation to and motivation from the vanKampen theorem. Readers here know that I spent some time thinking about this. In the end I felt that the construction of the path \infty-groupoid functor on lined \infty-stack \infty-toposes that is left Quillen and preserves weak equivalences between fibrant objects is pretty much the answer, as this respect for weak equivalences is the desired vanKampen codescent, really.

    So I am skipping ahead to section 3.6 “Poncaré nn-groupoids” but probably I need to read more later on. (?)

  • page 26: that “\mathcal{F}” is probably meant to read “AA” (?)

  • page 27 “gerbs” – one has to be careful with this typo, a student of Hitchin once introduced this (i.e. gerbe without one of the e-s) as a technical term! :-)

So much for the moment. Any comments on my comments?

Posted by: Urs Schreiber on January 24, 2010 12:11 AM | Permalink | Reply to this

Re: Carlos Simpson’s book

page 27 “gerbs” – one has to be careful with this typo, a student of Hitchin once introduced this (i.e. gerbe without one of the e-s) as a technical term! :-)

And in the seminal paper ‘Bundle gerbes’, the word appears in both spellings! (but there it is a typo :) - it predates the thesis of Hitchin’s student)

Posted by: David Roberts on January 24, 2010 2:31 AM | Permalink | Reply to this

Re: Carlos Simpson’s book

On p17 of Homotopy Theory of Higher Categories we read:

Thanks to a careful reading by Georges Maltsiniotis, we now know that [Pursing Stacks] in fact contained a definition of weakly associative nn-groupoid, and that his definition is very similar to Batanin’s definition of nn-category. Grothendieck enunciated the deceptively simple rule:

Intuitively, it means that whenever we have two ways of associating to a finite family (u i) iI(u_i)_{i\in I} of objects of an \infty-groupoid, u iF n(i)u_i \in F_{n(i)}, subjected to a standard set of relations on the u iu_i’s, an element of some F nF_n, in terms of the \infty-groupoid structure only, then we have automatically a “homotopy” between these built in in the very structure of the \infty-groupoid, provided it makes sense to ask for one…

The structure of this as a definition was not immediately evident upon any initial reading…

This does seem to point strongly in the direction of Batanin’s definition, but I must say that it is still not evident to me after any number of readings how to consider it as being a definition. Has anyone looked at Maltsiniotis’ preprints here and here? What is this definition like?

Posted by: Mike Shulman on February 3, 2010 6:25 AM | Permalink | PGP Sig | Reply to this

Re: Carlos Simpson’s book

I was looking for mentioning of Charles Rezk’s nice stuff on Theta-spaces, which claims to achieve much of what this book is about. From the remark on p. 23 I gather this got in a bit too late to be treated more in detail in this book.

I am also wondering whether he is conflating Barwick’s iterated complete Segal spaces, which use n-fold simplicial spaces, with Rezk’s Theta-spaces? Both generalize ordinary CSS and are defined by an iterative procedure, but the two procedures are different.

Posted by: Mike Shulman on February 3, 2010 6:27 AM | Permalink | PGP Sig | Reply to this

Re: Carlos Simpson’s book

I am also wondering whether he is conflating Barwick’s iterated complete Segal spaces, which use n-fold simplicial spaces, with Rezk’s Theta-spaces? Both generalize ordinary CSS and are defined by an iterative procedure, but the two procedures are different.

True.

I haven’t yet had a chance to see if there are any relevant changes in the arXiv-version of the book.

Posted by: Urs Schreiber on February 3, 2010 4:15 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 291)

It is a bit of an urban legend that Bach improvised a six-voice fugue. There is certainly a six-voice fugue in the musical offering, but someone once compared the feat of improvising one to playing many parallel games of blindfold chess. I think the concensus (talking from memory here - a reference would be nice) is that Bach improvised a thee-voice fugue for the king, and included it (or one like it) in the offering.

Posted by: David Roberts on January 24, 2010 2:29 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 291)

Playing many parallel games of blindfold chess is not difficult, even I can do that — although I don’t stand a chance to beat the world record of Miguel Najdorf (at least I think he still holds the world record, I don’t remember the exact number of games he played, something between 50 and 60).

With regard to the six-voice fugue: The legend has it that Bach was not able to improvise one — one of the reasons the whole story got famous is that Frederick even dared to ask Bach to do it. I don’t have a specific reference —- but I never heard the claim that Bach was able to do it, so I could refer to every biography and Wikipedia entry about Bach that I know of.

Posted by: Tim vB on January 25, 2010 11:17 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 291)

I got my information from Timothy Smith’s webpage 10 Canons of the Musical Offering, which I cited. This page seems to be down right now, but you can still read the cached version, and it says:

In 1746 Frederick the Great extended an invitation to Johann Sebastian to visit the Prussian court in Berlin. The invitation may have come at the behest of Carl Philip Emanuel, Sebastian’s son then in the employ of the king, or it may have been engineered by Count Keyserlingk, Russian ambassador and patron of Bach. In the spring of 1747 the elderly composer arrived in Potsdam where he was received graciously, if not deferentially, as shown by Frederick’s (or Carl Philip Emanuel’s) calling upon Sebastian as “Old Bach.”

The guest was immediately asked to test Frederick’s new Silbermann fortepianos and the ensuing display of technique was impressive enough for the emperor to propose a musical subject upon which Bach was requested to improvise a fugue. If contemporary accounts are to be believed, Johann Sebastian improvised at that time two fugues: one for three voices and one for six. Upon his return to Leipzig Bach added to the fugues a strict set of canons and a trio sonata featuring the “royal theme” in the flute part (Frederick’s own instrument) which he had engraved and sent to the emperor as a “Musical Offering.”

Posted by: John Baez on January 25, 2010 7:12 PM | Permalink | Reply to this

Bach and the fugue with six voices

I checked my sources and can offer a simple solution to this question: It would seem that Bach improvised a fugue with six voices, but did not use the theme that Frederick had given to him. Bach noticed that this would be too difficult, chose a theme on his own and improvised the fugue using that.

Bach himself confirms this in the dedication of the “musikalisches Opfer”, referring to his visit to Potsdam:

“Ich bemerkte aber gar bald, dass wegen Mangels noetiger Vorbereitung die Ausfuehrung nicht also geraten wollte, als es ein so treffliches Thema erforderte.”

Roughly: “I soon noticed that - lacking the necessary preparation - the execution would not succeed in a way that would be appropriate for such an excellent theme.”

(That’s why I thought to remember that Bach did not succeed at all, which is wrong).

Posted by: Tim van Beek on January 26, 2010 9:37 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 291)

Conc. Bach: Historians of music apparently suspect that C. Ph. E. Bach suppressed the last parts of J.S. Bach’s “Art of Fugue”, here is an interview with Ton Koopman mentioning that.

Posted by: Thomas on January 24, 2010 10:57 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 291)

The statement that Thomas refers to is this, I think:

Koopman: Und ob! Ich glaube, dass Carl Philipp Emanuel den fehlenden Rest der “Kunst der Fuge” unterschlagen hat. Und zwar gleichfalls, um den Mythos Bach groesser zu machen. Ich bin sicher, die “Kunst der Fuge” war komplett. Schliesslich war Bach nur damit beschaeftigt, das Werk abzuschreiben. Er hat spaeter noch an der Hohen Messe gearbeitet. Ich kenne Musikwissenschaftler, die - selbst wenn sie es nicht zugeben - heimlich in Archiven nach dem Schluss der “Kunst der Fuge” suchen. Und sie werden ihn eines Tages finden!

…I think that Carl Phillip Emanuel (Bach) suppressed the missing part of “Kunst der Fuge”, in order to enlarge the myth of Bach. I am sure “Kunst der Fuge” was complete…I know musicologists who - without admitting it - search for the ending, secretly, in the archives. Some day they will find it!

Posted by: Tim vB on January 25, 2010 11:28 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 291)

Particle physicists should look at the above diagram and think about how Feynman diagrams with closed loops in them lead to infinities. Category theorists should think about “traces” and how sometimes traces diverge. It is my job to make these analogies precise. But not today.

For some reason, this reminded me of St. Augustine. “Oh, Lord, grant me precision and rigour — but not yet!”

Posted by: Blake Stacey on January 24, 2010 4:22 PM | Permalink | Reply to this

Biological circuits; Re: This Week’s Finds in Mathematical Physics (Week 291)

Clever, Blake Stacey. I’d expected you to say something about Biological circuits (Ecology/population biology, genetic regulatory, metabolomic, …) to tie together your Complex Systems expertise with this Categorical unifiying approach of John Baez.

Posted by: Jonathan Vos Post on January 24, 2010 7:21 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 291)

Heh. The difference, Blake, between Augustine and me is that Augustine was cutting himself some slack, while I was cutting you — that is, the reader — some slack.

Posted by: John Baez on January 25, 2010 4:10 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 291)

Hi John! About your question on Mandelbrot and Julia set similarity:

The idea is the following: consider a parameter cc; then it belongs to the Mandelbrot set iff ((c 2+c) 2+c) 2+...((c^2+c)^2+c)^2 +... is bounded. Now consider the corresponding *filled in* Julia set, namely those zz such that the orbit of zz under f(z)=z 2+cf(z)=z^2+c is bounded. To any point cc in the Mandelbrot set corresponds the point z=cz=c such that cc is coloured “black” iff zz is coloured black. Then, with the help of lots of ϵ\epsilon’s and δ\delta’s, you can see that microscopically changing cc has roughly a comparable effect as microscopically changing zz.

I’m not aware of statements that are true everywhere, but mainly of conjectures as to how these things behave. The fundamental conjecture (from my perspective) is that the Julia and Mandelbrot sets are described by simple combinatorial models: the boundary of the Mandelbrot and connected Julia sets are quotients of a circle by a simple equivalence relation. Remember that the Julia set of z 2+cz^2+c is connected iff cc belongs to the Mandelbrot set.

To understand all of that, note that z 2+cz^2+c is conjugate, in the neighbourhood of \infty, to the map z 2z^2; i.e. there exists a map B:DC^B:D\to\widehat C where DD is the open unit disk, such that B(0)=B(0)=\infty and B(z 2)=f(B(z))B(z^2)=f(B(z)); and B(0)=1B'(0)=1 for normalization. The boundary of the Julia set “should be” the image of the boundary of DD, but it’s only known for some values of cc. Consider radial lines in DD; their images are rays R θR_\theta in the complement of the Julia set, starting at \infty, and conjecturally converging. Points in the boundary of the Julia set are therefore parameterized by an angle, with different angles possibly describing the same point.

Now the complement of the Mandelbrot set may also be parameterized by a disk; namely, for c¬c\not\in the Mandelbrot set, set f(z)=z 2+cf(z)=z^2+c and consider G(c)=lim2 nlogf n(0)G(c)=\lim 2^{-n}\log f^n(0). Then this defines a holomorphic map GG (careful with the log!), and again points on the boundary of the Mandelbrot set get an angle-valued parameter. GG-preimages of radial lines are called “parameter rays” P θP_\theta.

Another link between what happens in Mandelbrot and Julia space can now be described. Consider pairs of rays in Julia space that land at the same boundary point. A pair is called “characteristic” if it separates 00 from cc and none of its forward images separates cc from the initial pair. Then if the rays R θ,R ϕR_\theta,R_\phi form a characteristic pair, then P θP_\theta and P ϕP_\phi land at the same point in the boundary of the Mandelbrot set.

It follows that the “pinching” that describes the boundary of the Mandelbrot set as a quotient of a circle is well related to the “pinching” that describes Julia sets.

Posted by: Laurent on January 25, 2010 12:42 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 291)

Laurent wrote in part:

Then, with the help of lots of ϵ\epsilon’s and δ\delta’s, you can see that microscopically changing cc has roughly a comparable effect as microscopically changing zz.

This reminds me of a fairly elementary fact (no reference, I may have thought of it on my own, but I was a teenager and I don't remember): the Mandelbrot and Julia sets are cross sections of a single 44-dimensional fractal: the pair (z,c)(z,c) (of complex numbers) belongs to this fractal if, starting with z 0zz_0 \coloneqq z and applying z n +z n 2+cz_{n^+} \coloneqq z_n^2 + c, the orbit is bounded. (The Julia sets correspond to fixing cc; the Mandelbrot set corresponds to fixing z0z \coloneqq 0. Here I am ignoring the filled/boundary technicality.)

So the Mandelbrot set near cc and the Julia set for cc near 00 are both near (0,c)(0,c) in the hyper-fractal, but rotated from each other. I have not tried to chase down Laurent's ϵ\epsilons and δ\deltas, but I can hand-wavingly convince myself that the hyper-fractal satsifies an approximate symmetry under rotations which explains why these should look similar.

Posted by: Toby Bartels on January 25, 2010 9:39 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 291)

Hey, that is cool.
So you could take other sections too, and create fractals that are half Julia/ half Manderbrot!

As you say, In (z,c)-space, Julia is in a c=c0 plane, while Mandelbrot is in the z=z0 plane.
So what about the general plane Pz+Qc=R?

This is equivalent to setting:
c= c0-A z0
Again, if A=0, you get a Julia set. If c0=0, and A = 1, you get a Mandelbrot. But now we can make a completely new fractal. (That is, I hope it will be something new)

I’ll make one right away.

Posted by: Gerard Westendorp on January 25, 2010 11:42 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 291)

I have to quit now, but the first picture:

Indeed half Julia - half Mandelbrot (“JuliaBrot”)!

Click here for the source.

Posted by: Gerard Westendorp on January 26, 2010 1:00 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 291)

Surfing the web, I found others have already been doing this. You can also take 3D sections of the 4D fractal, that they call “Julia-Mandelbrot set”.

Did you see the Mandelbulb?

Gerard

Posted by: Gerard Westendorp on January 27, 2010 10:47 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 291)

Did you see the Mandelbulb?

Yes, someone reported on that in Week 286.

Posted by: Toby Bartels on January 27, 2010 11:16 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 291)

I was a bit bemused to see that particular Julia set so neatly approximated by a speck of the M-set MM, particularly because it’s so disconnected — recent reading calls it a species of “Fatou Dust” — which means it corresponds to a point very much NOT in the M-set; so I asked gnofract4d to show me the Julia for the point identified, and it looked nothing like that! Trying to zoom in the Mandelbrot disappointingly pixelated before I found anything similar, but I also suspect a little distortion due to rounding — I’m not good at estimating these sorts of error, you know. But then reading the wikipedia article seemed to clear things up for me (that article’s captions will have to be edited, because they really are misleading!)

These “embedded Julia sets” aparently aren’t part of the classic “easy” renormalization; instead they seem to show up somewhat in analogy to the satelite sub-mandelbrot structures;

Disclaming honestly, this is for myself some mildly wild conjecture: for a sub-mandelbrot, it looks like there’s a neighborhood UU of MM and an injective conformal map φ:UU\varphi:U\rightarrow U which is in some sense locally maximal such that φ|MM\varphi|M\rightarrow M — and furthermore, M\φ(M)M\backslash \varphi(M) accumulates at φ(M)\partial \varphi(M). The fine structure of this accumulation is quite intriguing: suppose the local filigree looks like X; then approaching the satelite mandelbrot, there’s a progression to finer spacing of parallel shorter branches of rough shape X. It’s something akin to the fuzzy appearance of the logistic Feigenbaum tree just prior to the emergence of an attracting odd cycle (which surely isn’t a coicidence). However, approaching the boundary of φ(M)\varphi(M), experimentally there are other regions than second-order satelites where X-shaped branches can accumulate in this manner, and if there is such an exotic accumulation near a point φ(c)\varphi(c), then it looks like the Julia set for cc.

It should then also follow that these embedded Julias are internally Cantor, because of necessity they correspond (via φ\varphi) to points cc outside MM.

Posted by: Jesse McKeown on January 25, 2010 12:58 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 291)

when I said

These “embedded Julia sets” apparently aren’t part of the classic “easy” renormalization …

I spoke too soon!

Posted by: some guy on the street on January 25, 2010 9:54 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 291)

Mixo Lydian sent me an email answering my question about why Japan has currents of two different frequencies — 50 and 60 cycles per second. As expected, there’s some history involved:

The 50Hz/60Hz divide in Japan is due to historic reasons. Towards the end of the Meiji era, Japan made the switch from DC to AC. Tokyo Dento (Japan’s first electric power company) adopted 50Hz German AEG generators while its rival Osaka Dento decided to adopt 60Hz American GE generators to power their respective electric grids.
Neighboring regions built their electric infrastructure adopting either Tokyo or Osaka standards which has led to a east-west / Tokyo-Osaka divide which continues to the present day, the exact border being the Fuji river which runs thru Shizuoka prefecture: east of the river the frequency is 50Hz, west of river the frequency is 60Hz.

This has hilarious consequences for the town of Shibakawa-cho, Shizuoka. The Fuji river runs directly thru Shibakawa-cho: some parts of town use 50Hz while others use 60Hz! All you have to do is cross a bridge to alternate between (intentional pun)!

Hope this has been helpful.

Posted by: John Baez on January 25, 2010 7:00 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 291)

From Fractal Groups to Fractal Sets
Authors: Laurent Bartholdi, Rostislav I. Grigorchuk, Volodymyr V. Nekrashevych

“In its survey, the paper discusses finite-state transducers, growth of groups and languages, limit spaces of groups, hyperbolic spaces and groups, dynamical systems,Hecke-type operators, C^*-algebras, random matrices, ergodic theorems and entropy of non-commuting transformations. Self-similar groups appear then as a natural weaving thread through these seemingly different topics.

The aim of this paper is to present a survey of ideas, notions and results that are connected to self-similarity of groups, semigroups and their actions; and to relate them to the above-mentioned classical objects. Besides that, our aim is to exhibit new connections of groups and semigroups with fractals objects, in particular with Julia sets.”


Posted by: Stephen Harris on January 26, 2010 10:47 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 291)

Wikipedia has an article on the Newton fractal which may relate J- to M-sets.

Posted by: Doug on January 31, 2010 2:31 PM | Permalink | Reply to this

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