From SciGuy comes a very interesting piece of news: for the next two months, The Royal Society will make available the complete archive of its scientific journals online. The archives, fully searchable, are available here. Over 340 years of research are awaiting for you!
Wondering how some famous physicist looks like? Chances are you’ll find her picture in this great gallery (collected by Carlos A Bertulani), featuring more that 160 portraits. There are also some group photos, like the one below: how many of these (very famous) guys can you recognise?
The IOP has just launched a podcast feed (hat tip Yan Feng). Their first post (mp3 here) is a 20 minutes interview with Michael Berry, of the eponymous phase fame, whose work on levitating frogs earned him an IgNobel Price in Physics in 2000. Professor Berry talks neither of his phase nor his frogs in the interview, but of his current research on optics (having to do with conical diffraction and the angular momentum of light), and the relationship between science and art (he is very fond of images representing optical phenomena, as you can see in this beautiful gallery). He has also a couple of things to say about the interplay between theoretical work and practical applications and the part played by science in our society. In this regard, his little piece Living with Physics (pdf) and his unpublished Night thoughts of a theoretical physicist (pdf) are very worth reading; for instance, here’s a quite inspiring musing on the unity of science:
From science come inspiring and magical connections between very different things. This observation counters one of our commonest criticisms: that by the reductionist disarticulation of the world into its parts, which are then studied separately, we lose the sense of the whole. My favourite example starts with the question: Why is matter hard? Atoms consist mostly of empty space, after all, so why doesn’t matter squash down, with all the electrons collapsing into their lowest quantum energy states near the nuclei? Because this is prevented by the Pauli exclusion principle: no two electrons can be in the same state. And where does that come from? It could well originate in a property of rotation in three-dimensional space4: holding a glass of wine, you can turn it completely twice (that is, through 720°) and find at the end of this contortion that your arm is untwisted (this does not work for a single turn). I find that `two into none’ connection, that totally unexpected association of microscopic hardness with geometry5, miraculous.
By the end of the interview, Berry returns to his research and mentions a quite curious recent result of his, the explanation of the workings of the oriental magic mirrors called Makyoh. This bit caught my eye (well, i guess it was my ear) because i had never hear of those magic mirrors before. They’re quite amusing. These cast and polished bronze mirrors, manufactured in China and Japan since at least 500 BC have a pattern embossed on the back that magically appears in a patch of light reflected by the mirror face (which to the naked eye looks as smooth and polished as one can get, except for a bit convexity). You can see a Makyoh in action in the figure on the right (more here): the pattern in the reflected light patch is nowhere to be seen in the mirror’s surface, which reflects images as a regular, slightly convex mirror would do. Credit for explaining the trick usually goes to Ayrthon and Perry, but, according to this article of the 1911 edition of the Encyclopedia Britannica,
The true explanation of the magic mirror was first suggested by the French physicist Charles Cleophas Person in 1847, who observed that the reflecting surface of the mirrors was not uniformly convex, the portions opposite relief surfaces being plane. Therefore, as he says, ‘ the rays reflected from the convex portion diverge and give but a feebly illuminated image,while, on the contrary, the rays reflected from the plane portions of the mirror preserve their parallelism, and appear on the screen as an image by reason of their contrast with the feebler illumination of the rest of the disk. Such differences of plane in the mirror surface are accidental, being due to the manner in which it is prepared, a process explained by W. E. Ayrton and J. Perry (Prot. Roy. Soc., 1878, vol. xxviii.), by whom ample details of the history, process of manufacture and composition of Oriental mirrors have been published.
I haven’t found these original papers on-line, but you can learn more about the history of Ayrton and Perry’s discoveries in this page on magic mirrors from Grand Illusions. A more in-depth treatment of the optics involved is given in Michael Berry’s article Oriental magic mirrors and the Laplace image (pdf), where he explains how the Laplacian of the relief height function gives rise to the image in the reflected patch (see also this article for comments on Berry’s and a bit more on the history of Makyoh).
I find Professor Berry’s willingness to investigate funny, every-day problems refreshing, not to mention his concern on making the physicist’s world closer to outsiders, like, say, taxi drivers. Or, as Berry himself puts it:
A source of delight is uncovering down-to-earth or dramatic and sometimes beautiful examples of abstract mathematical ideas: the arcane in the mundane.
Judging a paper’s quality may be hard for human referees, and people are looking for alternatives. For instance, this recent PhysicsWeb news gives an overview of P. Chen et al. article Finding Scientific Gems with Google, where the authors take advantage of Google’s page rating algorithm to assess the relative importance of all publications in the Physical Review family of journals from 1893 to 2003. Since the rating algorithm weights pages by number of referrers [0], there’s in principle no value added to traditional citation indexes: both popularity measures are linearly correlated. The catch is that there are exceptions: papers that are not widely cited but that, judging for the number of web pages linking to them, seem to be much more influential than one would think (the article mentions quite a few, Feynman, Murray and Gell-Mann’s one on Fermi interactions being an example). Amusing; although i must confess that this kind of democratic assessments of our scientific endeavours remind me somewhat of a well-known Planck dixit:
A new scientific truth does not triumph by convincing its opponents and making them see the light, but rather because its opponents eventually die, and a new generation grows up that is familiar with it.
Max Planck, 1858–1947
(I remember i jotted down this same quotation some twenty years ago, together with a note showing my skepticism… nowadays i think i’m much more of a planckian than i used to be.)
Returning to our electronic referees, over at PhysOrg there’s a story about how computer science may help us in detecting bogus papers, where by bogus i mean automatically generated ones (looks like our human referees do sometimes find their task really hard!). Probably the most popular case of such a prank was the article accepted at WMSCI 2005 whose author came out to be SCIgen, an automatic paper generator created by the guy in portrait on the right. And our field is not immune to similar problems, as exemplified by the amusing Bogdanoff Affair (besides, as you’ll see, most probably no computer program would be of much help in this case).
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[0] Of course, i’m oversimplifying: see here for the complete history behind Google’s PageRank.
Update: Andrew Jaffe, in his excellent blog (recommended, but you probably already knew it), has some interesting thoughts about peer review, and a recent initiative by Nature to open a debate on the issue and looking for ways of improvement.
From the Curiosities Department comes this news over at PhysicsWeb (see also a previous piece at nature.com) about recent advancements in our understanding of pebbles:
A question that has been around since the time of Aristotle — what shape is a pebble? — has now been solved by physicists in France and the US. Douglas Durian of the University of Pennsylvania and colleagues in Strasbourg say that a pebble is “a nearly round object with a near-Gaussian distribution of curvatures”. All pebbles, regardless of their original shape, end up with a similar shape that depends solely on how the pebble was eroded over time. The results could help geologists determine the history of a pebble simply by looking at its geometry (Phys. Rev. Lett. 97 028001).
You can also take a look at this nice presentation for more details, and even see some movies by the people of Strasbourg. For all the nitty-gritty details, the articles can be found in the arXiv, here and here.
This news reminded me of an older one, The Mistery of the Skipping Stone, where the physics of bouncing stones in water is (more or less) explained (unsurprisingly, the determining factor seems to be the initial velocity of the stone: see this very readable paper, to appear in AJP, by the same author, Lyderic Bocquet). A piece of amazing trivia included in the article: In 2002 an American called Kurt Steiner set a new world record when he threw a stone across a river in Pennsylvania and made it bounce… 40 times. Unbelievable? I thought so, but here’s the proof. By comparison, the team of physicists writing the article were using a specially designed catapult for their experiments, but they got just 20 bounces. What’s your mark?
Update: And, when it comes to talk about physics and throwing, nobody more apt (one would say) than a physicist let loose at the Baseball Major League: in A Magnus Force on the Mound, Major league pitcher Jeff Francis brings an educated insight to the physics of baseball (besides giving me an excellent excuse to publicize the excellent Symmetry Magazine, a joint SLAC/Fermilab publication about particle physics for the rest of us).
As part of the series of articles it is running on CERN’s LHC, Seed Magazine has just released Lords of the Ring, a very nice short film featuring some of the physicists (Brian Cox, Jon Butterworth and Albert de Roeck) working on the collider and, of course, the collider itself. It’s only five minutes, but worth viewing (with congratulations to SM for using a GNU/Linux/PPC compatible format! ;)).
Update: And a new premier, this time Seed’s first audio slideshow, where Luis Alvarez-Gaume and Ulrich Fuchs discuss the magnitude of the LHC, how it works and why future CERN experiments could revolutionize our understanding of how the universe works.
While a was a full-time physics and maths student, i seldom, if ever, thought of proving anything using a diagram, or any kind of non-algebraic method, for that matter. One could make a couple of drawings every now and then to help understanding, but that was all. Not even after learning differential geometry did my view change. As a matter of fact, with the emphasis on (and the beauty of) abstract representations (as in abstract tensor notations), using drawings of surfaces embedded in Euclidean space felt like cheating. To make things even worse, my first serious physics book had been Landau and Lifshitz’s Classical Field Theory, where even words are scarce, let alone drawings or diagrammatic reasoning [1]. In a nutshell, i would have felt at home reading Lagrange’s introduction to his Méchanique Analytic [2]:
No figures will be found in this work. The methods like i set forth require neither constructions nor geometrical or mechanical arguments, but only algebraic operations, subject to a regular and uniform procedure.
I’m stealing the quote above from a talk entitled Proofs and Pictures [3], which started me re-thinking about diagrams in physics (and maths) in the first place. It was given at the Perimeter Institute by James Brown, a professor of Philosophy of Science at the University of Toronto. In this fun talk, professor Brown explores the use of geometrical reasoning in maths and physics as a means of actually proving results. Some simple but instructive (and, to me, somewhat surprising and definitely amusing) examples of such “proving by diagrams” are given in the figure on the left (click to enlarge), which shows how getting general formulas for arithmetic and geometric sums may be as easy as counting squares. I’m giving away just two of them, so that you can try your hand with the other two and have a little fun (you can also try to invent your own, maybe going to 3- or even n-dimensional cubes, in which case, please, don’t forget to post your discoveries below! :)). Although elementary, these proofs are intriguing: would you accept them as such? Brown argues that they do, since they can be used to show the validity of the induction step in the usual algebraic proofs. I’m not sure i buy the argument, but it’s a very interesting one.
Turning our attention to physics, probably the most famous diagrams in the field are Feynman’s. As i’m sure you know, they offer a convenient notation for manipulating terms in QED’s perturbative expansions. Taken at face value, or, one might say, analytically, the represent just algebraic combinations of functions (propagators) entering a power series expansion in a small parameter (the interaction coupling constant, alpha). But they’re usually interpreted as providing the actual physical mechanism for the interaction of real particles by means of exchanges of virtual, unobservable photons. Albeit intuitive and appealing, this interpretation has always bothered me. After reading about it in popular science books, i expected QED being somehow based on photon exchanges from scratch. Instead, what one has is a principle of least action which leads to differential equations unsolvable in exact analytical form. Then, when calculating an approximate solution to a scattering problem using a power series, one obtains (the analytical equivalent) of Feynman diagrams and interprets them, so to speak, after the fact. I would somehow feel more comfortable if the process were the other way around: start with the (supposedly) physical underlying process (the photon exchange) and derive the scattering amplitude. Each Feynman diagram would then represent an actually possible scenario, in the same sense that an electron choosing one slit in the two-slit experiment is possible: one can break the superposition and observe the electron in its way through the slit. But this is of course impossible: virtual photons are unobservable, if only because they travel faster than light and violate energy conservation. To add to my uneasiness, a plain Feynman series leads to divergences to be cured, non-diagrammatically, by renormalisation. Yet, everyone since Feynman discusses this spooky photon ping-pong as the right interpretation [4], so probably i’m just showing off my lack of understanding! And, besides, one could arguably point to measurable vacuum polarisation effects like Casimir’s as an experimental proof of the existence of virtual particles (see for instance this recent, accesible account at PR Focus). Or one could even see the situation as a derivation of the interaction underlying mechanism from first principles, an stunning testament to their power [5]. At any rate, and specially if one accepts the mainstream interpretation, Feynman diagrams appear as a good example of how diagrammatic tools can be more than just a picture, and not only in mathematics. For more on Feynman diagrams and pointers to further reading, see their WikiPedia entry, or get Diagrammar a CERN report by ‘t Hooft and Veltman with all the gory details with a deliciously retro (as in written in 1973 using a typewritter) flavour.
Before leaving the subject of Feynman diagrams, let me mention two bits of diagrammatic folklore stolen from Peter Woit’s latest book. Naturally enough, recurring diagrams have got pet names over the years. The first one seems to have been the tadpole (for a diagram shaped, well, like a tadpole), coined by Sidney Coleman and resignedly accepted by the Physical Review editors after he proposed lollypop and spermion as alternatives. The second anecdote involves a diagram (depicted above) known as penguin since Melissa Franklin won a dart match over John Ellis: Tommaso Dorigo has recently recounted the story in his blog.
Roger Penrose’s thought is all but geometrical, and it comes as no surprise that he has made many a contribution to the physics by drawing camp. Every decent course on General Relativity touches conformal diagrams [6], a nifty method envisioned by Penrose and Brandon Carter (back in the sixties) to bring infinity back into your drawing board. The trick consists on scaling your metric by a global function that vanishes quickly enough when your original coordinates go to infinite. Such scaling is known as a conformal transformation, and has the virtue of preserving angles; in particular, null geodesics are mapped into null geodesics and, therefore, the causal structure (represented by null cones) is untouched. While beautiful and handy, i think that conformal diagrams do not add anything really new from a computational standpoint (as Feynman diagrams do), let alone serving as the basis for actual proofs.
More interesting for our current musings is Penrose’s graphical tensor notation. Tensor indexes (specially in its abstract flavour, also introduced by Penrose) are a quite convenient housekeeping device, ensuring almost automatically the consistency of your equations and even (once one has a bit of practice with them) suggesting their form [7]. But, convenient as they are, indexes seem to be confusing for geometrical minds like Penrose’s, who some fifty years ago devised a pictorial representation for tensor equations [8]. As you can see in the figure, the idea is simple: choose a closed polygon to represent the kernel letter of each tensor, and add an upwards leg for each contravariant index, and a downwards one for each covariant index. Index contraction is represented by joining the respective legs. A wiggly horizontal line represents symmetrisation; a straight one anti-symmetrisation. One can cross legs to indicate index shuffling. The metric gets no kernel figure (it’s just an arch), so that contractions of indexes in the same tensor are easily depicted, and raising and lowering indexes amounts to twist the requisite leg up or down. To indicate covariant differentiation, circle the tensor being differentiated and add the corresponding downwards (covariant) leg. And so on and so forth. Note also that commutative and associative laws of tensor multiplication allow your using any two dimensional arrangement of symbols that fits you, which aids in compactifying expressions. Penrose explains the many details and twists of the notation in The Road to Reality and in his (and Rindler’s) Spinors and Space-time I, where you’ll find extensions to deal graphically also with spinors and twistors. According to the latter,
The notation has been found very useful in practice as it greatly simplifies the appearance of complicated tensor or spinor equations, the various interrelations expressed being discernable at a glance. Unfortunately the notation seems to be of value mainly for private calculations because it cannot be printed in the normal way.
Besides the (not so obvious nowadays) difficulty mentioned above, i guess that the main hurdle in adopting Penrose’s notation is habit. After many years using indexes, my algebraic mind seldom finds equations confusing because of their indexes. But after a little practice it becomes easier, and i’d say that people who see equations will find it quite natural after a very little while [9]. I don’t know how popular Penrose graphics are among physicists for private use, but there’s many an example of their application and extension to related fields. A few years after its introduction, the notation was rediscovered by Pedrag Cvitanovic, who used a variation of it in an article on group theory and Feynman diagrams. More concretely, Cvitanovic uses diagrams similar to Penrose’s to represent to represent the structure constants of simple groups in the context of non-abelian gauge theories, interestingly linking them with Feynman diagrams (and closing a loop in this article!). Later on, he would use the notation very extensively in his on-line book on Group Theory, where the diagrams go by the name of bird-tracks. In a nutshell, the book is devoted to answer, in Cvitanovic words, a simple question:
“On planet Z, mesons consist of quarks and antiquarks, but baryons contain 3 quarks in a symmetric color combination. What is the color group?” If you find the particle physics jargon distracting, here is another way to posing the same question: “Classical Lie groups preserve bilinear vector norms. What Lie groups preserve trilinear, quadrilinear, and higher order invariants?”
From here, an amazing journey through the theory of Lie groups and algebras ensues, a journey conducted almost exclusively by diagrams. For, notably, Cvitanovic uses his bird-tracks (as mentioned, a very evolved kind of Feynman diagrams) to actually derive his results. We have here physics (and maths) by diagrams for real, actually replacing algebraic reasoning (and, incidentally, a proof that Penrose’s reservations about his notation not being apt for publications are unfounded nowadays–i wonder how Cvitanovic draws his diagrams).
Before leaving the subject, let me mention a couple more works inspired by Penrose’s diagrammatic notation. Yves Lafont has greatly extended it and carefully analysed its application to mathematical problems in the context of category theory and term rewriting systems. If you’re privy in the field, or simply curious, take a look at his articles Algebra and Geometry of Rewriting (PS) and Equational Reasoning With 2-Dimensional Diagrams , where Yves explores two-dimensional diagrams a la Penrose with an eye to (possibly automatic and computer-aided) derivations much in the spirit of Cvitanovic. And, turning back to physics, if there’s a theory prone to diagrammatic reasoning it must be Loop Quantum Gravity, where the basic constituents are graphs and their transformations. Arguably, LQG is the most fundamental example discussed so far of graphical reasoning applied to physics, for here graphs (and their combinations in spin foams, an evolution of another Penrose invention, spin networks) do stand for themselves, as opposed to representing some underlying algebraic mathematical entity. Wandering into the marvels of LQG would carry us too far afield, so i’ll just point out that Rovelli, Smolin and friends use not only Penrose’s spin networks, but, on occasion, also the graphical tensor notation we’ve been reviewing; see for instance their seminal paper Spin Networks and Quantum Gravity, where Rovelli and Smolin presented their famous derivation of exact solutions to the Wheeler-DeWitt equation. The notable thing is, again, the fact that graphic notation is key in many a derivation, and cannot be seen as just an aid to represent some calculations.
Our final example of physics by diagrams comes from the category theory-inspired view of Quantum Mechanics invented by Samsom Abramsky, who has managed to do “quantum mechanics using only pictures of lines, squares, triangles and diamonds”. This beautiful notation (or picture language, as their authors call it) is nicely explained in Bob Coecke’s Kindergarten Quantum Mechanics, a very pedagogical set of lecture notes where it is applied to the problem of quantum teleportation. Bob’s thesis is that teleportation was not discovered until the 90’s (despite it’s being a relatively straightforward result in QM) due to the inadequacy of the commonly used, low-level mathematical language used to describe Hilbert spaces. Had lines, squares, triangles and diamonds been used from the beginning, teleportation would have followed almost immediately. Or so thinks Bob: go take a look at his article and see what’s your take. In any case, its more than sixty full-color diagrams, used instead of boring algebraic formulae, make for a fun reading (or, should i say, viewing). By the way, don’t let the mention to category theory put you off: only very basic ideas (explained in the lecture notes) are needed, if at all, in this case, and actually the author’s enthusiasm goes as far as making the bold claim that this new graphical formalism could be taught in kindergarten! Maybe that’s the gist, since i, for one, find the notation hard to follow, undoubtedly due to my old-school, algebraic upbringing. Just to give you an idea of how this preschool notation looks like and close this long post as it deserves (i.e., with a diagram), here you have how the teleportation protocol (including a correctness proof) looks like:
[1] My copy (Spanish translation) of the fifth edition of L&L’s book has 500 pages and just 22 figures!
[2] The link above points to Volume 11 of the collection at Oeuvres de Lagrange, a site that contains what seems to be the complete Lagrange corpus, conveniently scanned and downloadable too.
[3] I would give you a direct link, did it exist. Unfortunately, PI’s website is not up to the quality of their other activities. You’ll find it by browsing to their Public Lectures Series and from there to page 2 (or search for James Brown). Another very unfortunate circumstance is that the videos are only available for those of you not using weird as in freedom operating systems :-(.
[4] That’s at least my impression. Penrose, for instance, advocates for their reality in his road. The subject is however controversial enough to grant the existence of monographs like the recent Drawing theories apart, by David Kaiser (which i cannot comment on since i’ve just added it to my wish list).
[5] But i find this argument hard to swallow. Think for instance in the interpretation of antiparticles as particles travelling backwards in time: it also follows naturally (for some definition of natural) from perturbative series and/or their diagrams, but it is not as easily accepted as the existence of virtual photons. One wonders, where’s the limit?
[6] If you haven’t your favourite textbook at hand (Hawking and Ellis being mine when it comes to anything related to causal structure), you can find a pretty good introduction on-line in this chapter of Sean Carroll’s lecture notes.
[7] There is only so many ways of combining indexes, and if you know what are the free ones on, say, your LHS and the tensors entering the RHS and its general properties (e.g. symmetries), it’s often an easy task how their indexes should be combined. It reminds me, in a way, of dimensional reasoning, where knowing the target units and the ingredients gives an often quite accurate clue of how to combine them.
[8] It was introduced in a chapter of the book Combinatorial Mathematics and its Applications (Academic Press, London, 1971), entitled Application of Negative Dimensional Tensors. But Penrose have been using it (according to this letter to Cvitanovic (PDF) since 1952.
[9] An interesting (and not too far fetched) software project would be to write a Penrose diagram editor, possibly with support for tablet input devices. Such a tool would also probably solve the publication issue. In an ideal world, one would use a stylus to draw equations which would get automatically imported as nice diagrams, regular tensor equations with indexes or both. Any takers? 
If one were to choose a classic science video, that’d probably be Powers of Ten. Now, a kind soul has posted it online:
The NASA/ESA Hubble Space Telescope has captured the first-ever picture of a distant quasar lensed into five images. In addition the picture holds a treasure of lensed galaxies and even a supernova.
More details on this beauty over at ESA’s site. Almost as beautiful, in its own way, is the soon-to-be-operative Large Hadron Collider at CERN. Seed Magazine has a nice page giving some numbers to put LHC’s dimensions in perspective.
One of the many puzzles (a.k.a. Mysteries of Life) faced by modern theoretical physics is the so-called hierarchy problem: when one compares [1] the relative strength of the four fundamental forces, two widely separated scales are evident:
| Interaction | Coupling constant |
| Strong | 1 |
| Electromagnetic | 1/137 |
| Weak | 1/10^6 |
| Gravitational | 1/10^39 |
Or, as Lisa Randall puts it in this interview:
The gist of it is that the universe seems to have two entirely different mass scales, and we don’t understand why they are so different. There’s what’s called the Planck scale, which is associated with gravitational interactions. It’s a huge mass scale, but because gravitational forces are proportional to one over the mass squared, that means gravity is a very weak interaction. In units of GeV, which is how we measure masses, the Planck scale is 10 to the 19th GeV. Then there’s the electroweak scale, which sets the masses for the W and Z bosons. These are particles that are similar to the photons of electromagnetism and which we have observed and studied well. They have a mass of about 100 GeV. So the hierarchy problem, in its simplest manifestation, is how can you have these particles be so light when the other scale is so big.
As you probably know, Randall’s response to this conundrum implies a long detour through multiple dimensions, as recently reviewed over at Backreaction, which was predated by a proposal by Arkani-Hamed, Dimopoulos and Dvali, nicely explained for the rest of us in this Physics Today article. (As a warmup for higher-dimensional physics, you may find entertaining this recent pedagogical review of Kaluza-Klein theories.)
An alternative solution has been put forward by the supersymmetry proponents. As explained (hyped?) in this beatiful review of particle physics:
According to supersymmetry, every “ordinary” particle has a companion particle — differing in spin by half a unit, but with otherwise identical properties. Furthermore, the strengths of the interactions of the superpartners are identical to those of the corresponding ordinary particle. Supersymmetry so simplifies the mathematics of quantum field theory and String Theory that it allows theoriests to obtain solutions that would otherwise be far beyond their calculating ability.
For reasons too complex to explain here (even if i really understood them: see here and here for some of the nitty-gritty details), supersymmetry is claimed to lead to a unification of fundamental forces at very high energies (some 10^28K, or 10^{-39} seconds after the Big Bang), somehow making natural the wild differences in scale of the (energy-dependent) coupling constants in our current universe. As mentioned, the theory also makes easier to define renormalizable QFTs (due to some magical cancellations), and has become one of the main ingredients of String theory, although there is at least another extension to the standard model of particle physics that seem to share these magic cancellation virtues, solving in the process the hierarchy problem: this Physics World article gives an introduction to this so-called ‘little Higgs’ theory.
Personally, i find all these untestable super-theories and multiple dimensions rather unconvincing, and would prefer some old good 4-dimensional solution. Alas, no one seems to be avaialable… maybe it’s time to embrace Compactified Dementia and be done with that.
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[1] The excellent little comparison of coupling constants for the fundamental forces pointed to by the above link is part of a nifty site called HyperPhysics, an amusing experiment combining HyperCards, Javascript experiments and similar online tricks with well written contents. Visit it for fun, hierarchy problem or not.
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José Antonio Ortega Ruiz
