{"id":39791,"date":"2018-08-03T08:24:23","date_gmt":"2018-08-03T08:24:23","guid":{"rendered":"http:\/\/www.labri.fr\/perso\/barla\/blog\/?p=39791"},"modified":"2018-08-03T08:29:10","modified_gmt":"2018-08-03T08:29:10","slug":"fearful-symmetry","status":"publish","type":"post","link":"https:\/\/www.labri.fr\/perso\/barla\/blog\/?p=39791","title":{"rendered":"Fearful symmetry"},"content":{"rendered":"<p id=\"top\" \/><em>by Ian Stewart and Martin Golubitsky<\/em><\/p>\n<h3>Geometer God<\/h3>\n<p>Nature behaves in ways that look mathematical, but nature is not the same as mathematics. Every mathematical model makes simplifying assumptions; its conclusions are only as valid as those assumptions. The assumption of perfect symmetry is excellent as a technique for deducing the conditions under which symmetry-breaking is going to occur, the general form of the result, and the range of possible behaviour. To deduce exactly which effect is selected from this range in a practical situation, we have to know which imperfections are present. &#8211; p.16 <em>Humans might be sensitive to the detection of symmetries for ecological reasons, and might even have their reasonning (and mathematics) built to make sense of it in ways that nature only partially exhibit!<\/em><\/p>\n<h3>Where did it go?<\/h3>\n<p>So now the dreadful truth is out: symmetries are not so much broken as shared around. The phrase should really be symmetrysharing, not symmetry-breaking. Despite this, we continue to talk of . broken symmetry, because that&#8217;s the conventional terminology of our subject. It&#8217;s a reasonable phrase to use, because in experiments you usually can observe only one member of the symmetrically related bunch of solutions that the mathematics guarantees. A buckling sphere can&#8217;t buckle into two shapes at the same time. So, while the full potentiality of possible states retains complete symmetry, what . we observe seems to break it. A coin has two symmetrically related sides, but when you toss it it has to end up either heads or tails: not both. Flipping the coin breaks its flip symmetry: the actual breaks the symmetry of the potential. &#8211; p.60 <em>So should we think in terms of probablilities here?<\/em><\/p>\n<p>Time translations, we&#8217;ve already seen, correspond to periodic motion. What about time reflections? Reflecting a line reverses its negative and positive directions; reflecting time interchanges past and future. A motion that is invariant under a time reflection will look the same if you make a film of it and then run the film backwards. The point about which it is reflected &#8211; the &#8216;time mirror&#8217; &#8211; will then be the unique instant of time when the real world is identical to that in the reversed film. &#8211; p.63 Not exactly the same as thinking in possible outcomes, as with reversibility.<\/p>\n<h3>Forever stones<\/h3>\n<p>Indeed the mathematician Fejes Toth proved that the hexagonal lattice is the most efficient method for packing together identical circles, in the sense that it gets the largest number of them into the smallest space. The same goes for fish &#8211; and for atoms. This is analogous to minimizing their total energy relative to attractive and repulsive forces, so again we have evidence for the stability of a crystal lattice. Why aren&#8217;t all crystal lattices hexagonal, then? The answer is that other lattices can occur if the population doesn&#8217;t consist of identical units, which is commonly the case, since most crystals involve several types of atom. Moreover, even identical atoms can exert different forces in different directions, which may cause them to pack in other ways. &#8211; p.101<\/p>\n<h3>Stripped water<\/h3>\n<p>When low-level instability allows convection currents which rise above the condensation level, each is capped by a cloud of the familiar cumulus variety. These are often beautiful, sometimes menacing, clouds with flat bases and cauliflower tops. They show up hard and white . when lit directly by the sun, dark and with the proverbial silver lining when the sun is behind them. The pattern of clouds in a cumulus sky renders visible the initial pattern of thermals rising from the surface, which again depends on the distribution of heat sources. Sometimes the cloud groupings seem quite haphazard, at others it is surprisingly regular. On occasions the cumulus are neatly arranged in rows &#8211; cloud streets &#8211; and this sometimes signifies the drifting of successive thermals downwind from the same source. &#8211; p.120<\/p>\n<p>As with warm air, &#8216;hot spots&#8217; deep in the Earth can give rise to a plume of rising molten rock. As the plates of the lithosphere drift slowly across the top of such a plume, a chain of volcanoes forms, analogous to cloud streets drifting on the wind above a hot spot on the ground. The Hawaiian islands are an example. You probably never realized that they had anything in common with clouds. &#8211; p.122<\/p>\n<p>So we ask: what are the symmetries of the system? What is the catalogue of possible types of pattern in a system with that symmetry? What must we calculate to determine which of these possible patterns actually occurs, and what its stability is? Only after answering these general questions do we sit down and calculate those numbers. Admittedly we don&#8217;t get as much detail as we might by a full-blooded numerical simulation: that&#8217;s one price we must pay. But in compensation we get a broad understanding of the mathematical features not just of his system, but of any other system with the same symmetries. We use model-independent concepts as far as possible, and only put in the detailed physics of the model at the end. Symmetry selects the general range of possible patterns; physics tells us which patterns actually occur. &#8211; p.125<\/p>\n<h3>The Universe and Everything<\/h3>\n<p>The aim of science is not just the manufacture of new toys: it&#8217;s the enrichment of the human spirit. &#8211; p.128<\/p>\n<h3>Turing&#8217;s tiger<\/h3>\n<p>Much of the frog&#8217;s DNA is there to specify alternative development paths for different environmental factors, such as temperature of the pond in which the frog is growing. A developing chimpanzee, however, is kept at a constant temperature, because it is still inside its mother. By putting instructions for maternal temperature-regulation into the chimpanzee genetics, nature has managed to eliminate a much more extensive set of instructions for dealing with a changing environment. &#8211; p.159<em> Shorter is smarter!<\/em><\/p>\n<p>Thus what is coded in the DNA is not a complete description of where every cell must go and what it must do, but a prescription of what must be done to the dynamical system to control its development in the appropriate manner. [&#8230;] In a similar manner, you can direct a stream of water down a complicated hillside into a particular valley just by letting it flow I but nudging it gently from time to time near places where its path might branch away from the desired one. This kind of image is due to C. H. Waddington, who called it the &#8216;epigenetic landscape&#8217;. &#8211; p.160<\/p>\n<p>The chemistry of seawater is analogous to the role of DNA in development; the fluid dynamics of seawater is analogous to the &#8216;free-running&#8217; dynamics of the cell. Genes aren&#8217;t for shapes, they&#8217;re for chemistry. &#8211; p.162<\/p>\n<p>Symmetries may be local things, not necessarily related to the overall form of the object concerned. Homogeneity is a kind of symmetry, even if it occurs in an object that doesn&#8217;t have a symmetric form. The kind of &#8216;symmetry operation&#8217; we have in mind is this: cut two small balls out of the mass of tissue, and swap them. If the tissue is homogeneous, this operation makes no difference. If it&#8217;s not homogeneous, then changing balls whose contents differ does make a difference, so homogeneity is equivalent to this &#8216;local symmetry&#8217;, Turing is saying that even in an irregularly shaped mass of tissue, initial homogeneity is a type of symmetry, and it too can be broken. Incidentally, the two balls that we cut out might be the same: that is, we cut out a ball and replace it. If we are allowed to rotate it before we replace it, then &#8216;symmetry&#8217; implies that the state of the tissue is the same in any direction, that is, it is isotropic. So isotropy is local rotational symmetry. &#8211; p.168<\/p>\n<p>So it may be that nature repeatedly amplifies the tiny asymmetry of the weak interaction between elementary particles &#8211; first in the formation of the molecules of amino acids, then in the proteins that they form, then in the process by which those proteins control the development of an embryo, and finally in the growth and development of the embryo into an adult creature. &#8211; p.182<\/p>\n<h3>Well, is She?<\/h3>\n<p>Crystals are a symbol of symmetry and yet their order breaks the complete macroscopic symmetry of a melt or gas. This is a general rule: by ordering finite parts it is impossible to reach the complete symmetry in the mean resulting from chaos. Consider a television set that has not been turned off after the end of the program. The screen then flickers chaotically and is therefore (except at the boundaries) in the mean completely motion symmetric. Suppose now the flickering stops, and j one of the typical translation and rotation symmetric drawings of M. C. Escher appears on the screen. This would reduce the previously prevailing symmetry under arbitrary rotations and translations to a symmetry under only certain translations and rotations. &#8211; p.252<\/p>\n<p>Like the lattice symmetry of crystals, it may be merely a human invention &#8211; not in the sense that Nature never uses those patterns, for she does; but in the sense that we find them a convenient way to deal with some aspects of nature, and we select those aspects that fit the pattern. To us, a crystal lattice is &#8216;perfect&#8217; and (except to specialists) a dislocation is an &#8216;imperfection&#8217;, a defect, somehow secondary, something to be denigrated. Nature may well not appreciate that kind of distinction at all, in the following sense. Mathematically, both lattices and dislocations are solutions to the equations that govern the state of a large quantity of atoms: why select one solution as being superior to another? Both lattice and dislocation are concepts imposed by our perception of reality, rather than being inherent features of the way reality itself operates. &#8211; p.259<\/p>\n<p>In short, the way humans think about the universe is that we select or invent patterns. They may not be as fundamental as we think they are: but we have very little choice. As a result, however, we may focus our attention on &#8216;basic laws&#8217; that are nothing of the kind. &#8211; p.260<\/p>\n","protected":false},"excerpt":{"rendered":"<p>by Ian Stewart and Martin Golubitsky Geometer God Nature behaves in ways that look mathematical, but nature is not the same as mathematics. Every mathematical model makes simplifying assumptions; its conclusions are only as valid as those assumptions. The assumption of perfect symmetry is excellent as a technique for deducing the conditions under which symmetry-breaking &#8230; <a title=\"Fearful symmetry\" class=\"read-more\" href=\"https:\/\/www.labri.fr\/perso\/barla\/blog\/?p=39791\" aria-label=\"Read more about Fearful symmetry\">Read more<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[621],"tags":[],"class_list":["post-39791","post","type-post","status-publish","format-standard","hentry","category-books"],"_links":{"self":[{"href":"https:\/\/www.labri.fr\/perso\/barla\/blog\/index.php?rest_route=\/wp\/v2\/posts\/39791","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.labri.fr\/perso\/barla\/blog\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.labri.fr\/perso\/barla\/blog\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.labri.fr\/perso\/barla\/blog\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.labri.fr\/perso\/barla\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=39791"}],"version-history":[{"count":2,"href":"https:\/\/www.labri.fr\/perso\/barla\/blog\/index.php?rest_route=\/wp\/v2\/posts\/39791\/revisions"}],"predecessor-version":[{"id":39793,"href":"https:\/\/www.labri.fr\/perso\/barla\/blog\/index.php?rest_route=\/wp\/v2\/posts\/39791\/revisions\/39793"}],"wp:attachment":[{"href":"https:\/\/www.labri.fr\/perso\/barla\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=39791"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.labri.fr\/perso\/barla\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=39791"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.labri.fr\/perso\/barla\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=39791"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}