tag:blogger.com,1999:blog-10614348.post7639353989230665999..comments2024-01-22T11:26:37.599-08:00Comments on TGD diary: Scattering Amplitudes and the Positive GrassmannianMatti Pitkänenhttp://www.blogger.com/profile/13512912323574611883noreply@blogger.comBlogger16125tag:blogger.com,1999:blog-10614348.post-79308329644645300192013-01-05T13:17:36.449-08:002013-01-05T13:17:36.449-08:00http://www.dailygalaxy.com/my_weblog/2013/01/-eins...http://www.dailygalaxy.com/my_weblog/2013/01/-einsteins-emc2-may-breakdown-in-outer-space.htmlUllahttps://www.blogger.com/profile/16634036177244152897noreply@blogger.comtag:blogger.com,1999:blog-10614348.post-16885029953253628772013-01-01T08:40:00.460-08:002013-01-01T08:40:00.460-08:00Methionine is the starter of proteinsynthesis, and...Methionine is the starter of proteinsynthesis, and it seems it is started by blue light. Once we discussed this. I understand you have many other questions to think about.<br /><br />I understood that this mechanism is maybe the one that gives the dark matter or magnetic flux (blue light) regulation. <br /><br />Here a link to a PhD thesis about attention-awareness and how it is experienced. Usually there is no differentation, and then it is difficult to discuss this. Downloadable, in words <br /><br />http://denisewilbanks.com/?attachment_id=169<br /><br />An equally difficult thing to think of is what is a signal that cause the sensations. It comes down to what is the action principle, but then the qualias is just a different processing? Ramachandran uses brain areas for this interpretation http://www.youtube.com/watch?v=Rl2LwnaUA-k&feature=youtu.be<br /><br />Are there brain areas that interpret different meanings for the same thing, like reals and complex numbers? Left versus right? Then there are sensations of brainless organisms (microtubuli also have a left and right side of the body) and the superorganisms which make problems. Sensings are interpreted in many steps and the most important ones are in the sense organ themselves? We use brain mostly to selection and projection, and in that way we create maybe meaning?<br /><br />Happy new year. Still a few days left til the Chinese change, to the Snake. I feel this year will be the year of transformation :)Ullahttps://www.blogger.com/profile/16634036177244152897noreply@blogger.comtag:blogger.com,1999:blog-10614348.post-32087839720996399402013-01-01T03:01:47.640-08:002013-01-01T03:01:47.640-08:00To Ulla:
The finding is certainly interesting but...<br />To Ulla:<br /><br />The finding is certainly interesting but I must honestly confess that it would require a considerable effort to get even gist of what "adenine can replace methionine in redox reaction" implies. Here I sense my biochemical limitations strongly. Happy New Year in any case!Matti Pitkanenhttp://tgdtheory.com/noreply@blogger.comtag:blogger.com,1999:blog-10614348.post-55015840553579776482013-01-01T02:52:11.716-08:002013-01-01T02:52:11.716-08:00Continuation to the reply to Santeri:
[SS] I'...<br />Continuation to the reply to Santeri:<br /><br />[SS] I've also sometimes wondered, in connection to attention concentrating on "internal" body sense (not implying that it ends at skin level but can <br />be also sensual experience of the whole "magnetic body"), how many dimensions such "internal" sensual experiences contain and is such a question even meaningful?<br /><br />[MP] A model for OBEs and various sensory illusions leads to the view that various sensations could relate to the movement or imagined movement (say the nasty feeling in stomach when one imagines falling down to nearby cliff or moving train illusion) involve relative motion of physical body and magnetic body whose conformation provides a representation for it and maybe also for the environment. Motion of magnetic body alone gives rise to imagined sensory experience and motion of biological body alone to purely real sensory experience. OBEs would relate to the relative motion of magnetic body. The perceptions would result from the change of the hologram defined by the radiation exchanged between magnetic body and biological body.<br /><br />[SS] Assuming, based on phenomena of synaesthesia etc., that various senses are *filters* of holistic flow of information, rather than purely classical <br />mechanisms of data-receiving and neurological reproduction and representation, is for example the experience of orthogonality (that we sense most directly with our sense of balance (and gravity towards Earth's gravity center) necessary for experience of dimensionality?<br /><br />I cannot really imagine a ball containing more than three orthogonal axis, but i can imagine - and internally feel! - a ball consisting of infinity of <br />directions emanating from the center and give that infinity of directions at least symbolic interpretation of n-dimensionality.<br /><br />[MP] Angle pi/2 is special. Hilbert space is one basic example. More I am unable to say.<br />I have a strong temptation to interpret imagined higher-dimensional structures as p-adic constructs. As I probably already mentioned, p-adics allow infinite number of algebraic extensions and also non-algebraic extensions which are infinite-D. These are higher-D structures in the same sense as complex numbers are a 2-D structure. That we experience the world as 3-D and are not able to sensorily imagine higher dimensions could be simply due to the fact that we ourselves are 3-dimensional in real sense! Our cognitive me's can have arbitrarily high dimension. We are what we perceive;-). Buddhists would be correct when they interpret also cognition as sensory perception if cognitive perception is p-adic sensory perception.<br />Matti Pitkanenhttp://tgdtheory.com/noreply@blogger.comtag:blogger.com,1999:blog-10614348.post-81758181717133511982013-01-01T02:51:06.697-08:002013-01-01T02:51:06.697-08:00To Santeri Satama:
Thank you for interesting ques...<br />To Santeri Satama:<br /><br />Thank you for interesting questions and Happy New Year.<br /><br />[SS] You bring up sensory experience. It's area where we run into problems of definition. What and how is sensing, and what experiences do we classify as <br />sensual experiences? For example, Buddhist philosophy consider thinking just another sense. Wiki article (http://en.wikipedia.org/wiki/Sense) gives a list of many "internal" senses in addition to the five classical "external" senses, and also what it considers non-human senses.<br /><br />[MP] Certainly the definition of sensory experience requires some assumptions. It is very difficult to separate pure sensory qualia from the cognitive representations constructed from them. The sensory input must be decomposed to objects and this requires a lot of processing. Essentially standardized mental images are generated using a virtual sensory input to sensory organs which would be seats of primary sensory qualia (phantom limb is one basic objection which can be circumvented in TGD Universe).<br /><br />Concerning qualia my basic assumptions are roughly the following. See http://tgdtheory.com/public_html/hologram/hologram.html#qualia .<br /><br />a) Sensory qualia are the building bricks of sensory experience and since moment of consciousness corresponds to quantum jump and quantum states are labelled by quantum numbers, sensory qualia are characterized by increments of quantum numbers. Different kind of quantum numbers wold correspond to different kinds of basic qualia. For instance, colors would correspond to increments of color quantum numbers of quarks and gluons so that the term QCD color would not be a mere algebraic joke. This of course makes sense only if one accepts the almost-prediction of TGD about hierarchy of QCD like theories in various scales: in particular in the scales of living cell. If the people at LHC discover M_89 hadron physics, we are rather close to asking whether this radically new vision really makes sense.<br /><br />b) Sensory experience contains also purely geometric information: vision, touch, and hearing do so.This information corresponds to 4-D geometry so that also dynamical information becomes geometric with this definition. I have talked about geometric qualia: angles, distances,... What is interesting, is that topologist Barbara Shipman found that honeybee dance has a mathematical description in terms of a flag manifolds associated with QCD color. Different choices of quantization axes (two kinds of them corresponding to the Cartan algebra of SU(2): color isospin and color hypercharge) for color form a flag manifold SU(3)/U(1)xU(1) and this space is involved with her model. <br /><br />To be continued...Matti Pitkanenhttp://tgdtheory.com/noreply@blogger.comtag:blogger.com,1999:blog-10614348.post-28130802771226135022012-12-31T09:02:53.503-08:002012-12-31T09:02:53.503-08:00I continue with my finding, and mean not to interr...I continue with my finding, and mean not to interrupt the discussion. Sorry for this. I am excited as usual :)<br /><br />http://www.ncbi.nlm.nih.gov/pubmed/20184295?dopt=AbstractPlus<br /><br />.. adenine acts as an electrostatic bouncer that shoves the charge flow from flavin toward the DNA lesion that photolyase repairs. This explanation is provided by an explicit time-dependent quantum mechanical approach...The electron wave function dynamics accurately accounts for the previously proposed mechanism of transfer via the terminal methyl group of the flavin moiety present in the catalytic electron-donor cofactor, FADH(-), which also contains adenine. <br /><br />adenine can replace methionine in a redox reaction? Can then also the mutagenesis be directed/controlled by pH-changes or wavelength changes of light?<br /><br />http://www.ncbi.nlm.nih.gov/pubmed/9657679?dopt=AbstractPlus<br /><br />Ullahttps://www.blogger.com/profile/16634036177244152897noreply@blogger.comtag:blogger.com,1999:blog-10614348.post-73991081525060600142012-12-31T08:22:23.270-08:002012-12-31T08:22:23.270-08:00Your bring up sensory experience. It's area wh...Your bring up sensory experience. It's area where we run into problems of definition. What and how is sensing, and what experiences do we classify as sensual experiences? For example, Buddhist philosophy consider thinking just another sense. Wiki article (http://en.wikipedia.org/wiki/Sense) gives a list of many "internal" senses in addition to the five classical "external" senses, and also what it considers non-human senses.<br /><br />I've also sometimes wondered, in connection to attention concentrating on "internal" body sense (not implying that it ends at skin level but can be also sensual experience of the whole "magnetic body"), how many dimensions such "internal" sensual experiences contain and is such a question even meaningful? <br /><br />Assuming, based on phenomena of synaisthesia etc., that various senses are *filters* of holistic flow of information, rather than purely classical mechanisms of data-receiving and neurological reproduction and representation, is for example the experience of orthogonality (that we sense most directly with our sense of balance (and gravity towards Earth's gravity center) necessary for experience of dimensionality? <br /><br />I cannot really imagine a ball containing more than three orthogonal axis, but i can imagine - and internally feel! - a ball consisting of infinity of directions emanating from the center and give that infinity of directions at least symbolic interpretation of n-dimensionality.<br /><br />Santeri Satamanoreply@blogger.comtag:blogger.com,1999:blog-10614348.post-90236011608774844222012-12-31T07:49:57.339-08:002012-12-31T07:49:57.339-08:00Look at this! http://www.hindawi.com/journals/jna/...Look at this! http://www.hindawi.com/journals/jna/2010/794782/<br /><br />The DNA repair enzyme has also a too high efficiency, like the photosynthesis electron tunnelling. Is DNA repair also activated from a quantum state? This uses 350-450 nm wavelength, which is the same as methionine or the start-signal for proteinsynthesis use. Can this be an indicium for the DNA-phantoms or 'dark' biology as you have stated, is in fact the mechanism?Ullahttps://www.blogger.com/profile/16634036177244152897noreply@blogger.comtag:blogger.com,1999:blog-10614348.post-25598952346965748152012-12-31T03:03:34.152-08:002012-12-31T03:03:34.152-08:00To die is also often experienced as a tunnel, and ...To die is also often experienced as a tunnel, and then the time forms a standing wave (?) allowing you to be in many times at the same time? It is like some kind of bosonic condensate?<br /><br />Time can also cheat in emergency situations, giving you much more energy than would be possible, and also those 'lightnings in mind' of past times.Ullahttps://www.blogger.com/profile/16634036177244152897noreply@blogger.comtag:blogger.com,1999:blog-10614348.post-67238727556008668152012-12-30T22:21:51.018-08:002012-12-30T22:21:51.018-08:00I would tend to assign introspection and extrospec...<br />I would tend to assign introspection and extrospective areas to conscious information about the space-time sheets of observer one one hand and on space-time sheets of external world. <br /><br />The recent vision discussed in the posting about blackholes and blackhole evaporation is that the spacetime sheet defining "me" as something separate from the external world defines the line of generalized Feynman diagram (rather thick as compared with line of ordinary Feynman diagram!) and therefore has Eucdlian(!) signature of the induced metric with time and space in the same position. External world would correspond to Minkowskian signature.<br /><br />This is just one possible proposal. Real space-time sheets correspond naturally to sensory experience and p-adic space-time sheets to cognition. You are thus saying that introspection-extrospection difference corresponds to sensory-cognitive dichotomy. I would argue that I have sensory experience from both external world and internal world so that I would disagree. <br /><br />Mathematical imagination would naturally correspond to p-adic sector. The non-determinsm of p-adic differential equations suggests an interpretation in terms of the non-determinism of imagination. <br /><br />p-Adic number fields have infinite number of algebraic extensions analogous to complex numbers and thus having arbitrarily large algebraic dimensions. Could it be that our ability to imagine higher dimensions corresponds to a geometric realization of these dimensions p-adically. Evolution would correspond to a gradual increase of the algebraic dimension. <br /><br /> We cannot imagine what it is to be sensorily higher-dimensional (at least I cannot;-)) and this could relate to the absence of higher algebraic extensions of reals.<br /><br />You certainly noticed that I tried to cheat here!;-) Complex numbers definea 2-D algebraic extension of reals so that one might argue that we are able to see 8-D dreams in 4-D space-time! <br /><br />My defense is following. The preferred extremals of Kaehler action define a generalization of 2-D complex structure to 4-D case in both Minkowskian and Eucdliian space-time regions. One can say that physical space-time is 2-D in "complex" sense just as complex plane is 1-D in complex sense. As also strong form of holography (saying the quantum states depend on partonic 2-surfaces and their tangent space data only) states.<br /><br />Maybe you mean with "spots" the flow that I jheave perceived by closing my eyes in peaceful state of mind, say after having worked for few hours. The flow is to and from a kind of tunnel leading somewhere. I have considered the possibility that this flow is along magnetic flux tube connecting me somewhere.<br /><br />Neuroscientists could certainly represent a trivializing and pathologizing explanation for this flow, and colleagues would list this experience as an additional strong support for the conjecture that I am completely mad as they have thought from the beginning!;-)<br /><br /> Matti Pitkanenhttp://tgdtheory.com/noreply@blogger.comtag:blogger.com,1999:blog-10614348.post-63653091882334219552012-12-30T18:34:56.700-08:002012-12-30T18:34:56.700-08:00Thanks for your response. Fully accepting my limit...Thanks for your response. Fully accepting my limitations in capacity of mathematical imagination, one can only hope that my simple reflections have some meaningful relation to your vision. And keep on questioning. <br /><br />The CD as "spotlight of attention" sounds promising approach. Not necessarily limited to such movements, but the "pendulum" or "oscillator" of attention shifting between introspection and extrospection seems common experience, common enough to raise the question, if the introspective areas are characterized by p-adic areas and extrospective by real areas, can that be applied to CD's? E.g. describing the lower or "introspective" part or "light-like boundary" of CD p-adically and upper "extrospective" part by reals? Assuming those areas can be mapped on each in dynamic fashion. Or are the theory dependent and/or mathematical reasons denying such possiblity? <br /><br />Or if the question above does not get even close, what is the TGD inspired vision of introspective and extrospective movements of attention? And in that vision, does mathematical contemplation and imagination belong to either category.<br /><br />PS: You mentioned somewhere seeing "spots" long time after your Great Experience. Though you are not supporter of multiverse theories at least in standard form, I have hunch that those might represent other universes, perhaps characterized by other transcendentals than those revealed to us, but dunno. Maybe this hunch can be relevant to your question about WCW, maybe not, just thought maybe this would be worth mentioning.Santeri Satamanoreply@blogger.comtag:blogger.com,1999:blog-10614348.post-41667712329820895212012-12-30T05:02:24.455-08:002012-12-30T05:02:24.455-08:00To Santeri Satama:
Thank you for a nice summary o...<br /><br /><br />To Santeri Satama:<br /><br />Thank you for a nice summary of Ekelands vision. Ekeland describes basic notion of mechanics: energy constant surface of phase space at which particle orbit in phase space resides by energy conservation. This notion is central in thermodynamics. <br /><br />Causal diamonds are *not dynamical systems* such as multi-oscillators. They are not energy constant surfaces in phase space. They *contain the dynamical systems* defined by space-time surfaces. At quantum level zero energy states have wave function in the moduli space of CDs (CD has position, it can be Lorentz boosted and rotated and translated). CD's size scale is assumed be quantized for number theoretic reasons and come as integer multiple of CP_2 size.<br /><br />What matters in zero energy ontology is that causal diamonds have two *light-like* boundaries at which positive and negative energy parts of zero energy states are localized.The "upper" and "lower" ends of space-time surfaces (restaurants at the ends of the Universe are possible in TGD Universe!;-)) at the two light-like boundaries of CD: the analogs of initial and final state of a physical event. CDs contain the space-time surfaces in the superposition of preferred extremals (analogs of Bohr orbits) defining zero energy state. <br /><br />The interpretation of CD suggested by TGD inspired theory of consciousness is as an imbedding space correlate for self, kind of spotlight of attention. <br /><br /><br />Light-likeness of the boundaries of CD implies a generalization of conformal symmetry at the boundaries: it is essential that one has 4-D Minkowski space x CP_2. These conformal symmetries form only half of the conformal symmetries of TGD. Also light-like 3-surfaces defining parton orbits possess generalized conformal symmetries. The 4-dimensionality of both M^4 and space-time surface are crucial [I would be happy if this would finally induce a "click" in the learned heads of colleagues ;-) although I know from bitter experience that although I can bring a horse to the fountain, I cannot force it to drink;-)]. <br /><br />Number theoretic vision inspires a speculative picture about a connection between the hierarchy of infinite primes and hierarchical structure of space-time sheets of many-sheeted space-time. Why I take this vision seriously is that the hierarchy of infinite primes is a number theoretical correlate for a hierarchy of second quantizations whereas the many-sheeted hierarchy is classical space-time correlate for it. <br /><br />You mention frequencies. In the hierarchy of infinite primes frequencies labeling fermionic and bosonic oscillator operators are replaced by logarithms of primes. I remember that also in chaos theory frequencies comings as logarithms of primes appear: I do not however not the deeper mathematical reason.<br />Matti Pitkanenhttp://tgdtheory.com/noreply@blogger.comtag:blogger.com,1999:blog-10614348.post-40115853250865847122012-12-29T09:17:58.516-08:002012-12-29T09:17:58.516-08:00Correction to previous post: non-probable -> no...Correction to previous post: non-probable -> non-provable<br /><br />So, when you generalize from a simple harmonic oscillator to multioscillators, you still get only two countable results, either duality or infinity. Which I supposes also presupposes and requires infinite primes which can among other things describe various layers of the hypersurface I previously visioned? <br /><br />Another question about causal diamonds: aren they multioscillators that Ekeland discusses? Also I expect that both points of the arrows must touch infinity; what is the relation of real and p-adic infinities here, if any?Santeri Satamanoreply@blogger.comtag:blogger.com,1999:blog-10614348.post-30749789242825285142012-12-29T09:09:39.375-08:002012-12-29T09:09:39.375-08:00In the article discussed in above: "While a g...<br /><br />In the article discussed in above: "While a global picture is still missing, a huge amount of data has been generated..."<br /><br />Here's my current attempt for a global vision. Matti has said that all p-adic areas get in some sense the value of one, and this to me sounds highly untrivial, though I'd like to comprehend better how exactly, maybe and hopefully Matti can explain. Any case, I vision that p-adic unity as the Center or Source of a ball-like "phase space", first surrounded by p-adic regions or layers and then the "hypersurface" consisting of rational areas shared by p-adics and reals, which binds together the space-time sheets of how Matti visions the book or books. This hypersurface divides the "phase space" of p-adics and reals (etc.?) into inner and outer regions, inner areas (at first glance!?) converging in p-adically infinite "singularity" and outer areas scattering as infinite extensions of reals. This view could also help to explain the interest in non-probable "imaginary" black hole -singularity objects presumably "out there" as an attempt and act of gnothi seauton.<br /><br />Santeri Satamanoreply@blogger.comtag:blogger.com,1999:blog-10614348.post-85440328132932663922012-12-29T08:45:23.442-08:002012-12-29T08:45:23.442-08:00Cf. With Matti:
"The mathematical aspects of ...Cf. With Matti:<br />"The mathematical aspects of p-adicization of quantum TGD are discussed. In a well-<br />de ned sense Nature itself performs the p-adicization and p-adic physics can be regarded as<br />physics of cognitive regions of space-time which in turn provide representations of real space-<br />time regions. Cognitive representations presumably involve the p-adicization of the geometry<br />at the level of the space-time and imbedding space by a mapping of a real space time region to a p-adic one. One can di fferentiate between two kinds of maps: the identifi cation induced by the common rationals of real and p-adic space time region and the representations of the external real world to internal p-adic world induced by a canonical identi cation type maps.<br />Only the identi cation by common rationals respects general coordinate invariance, and<br />it leads to a generalization of the number concept. Di fferent number fields form a book like<br />structure with number fi elds and their extensions representing the pages of the book glued<br />together along common rationals representing the rim of the book. This generalization forces<br />also the generalization of the manifold concept: both imbedding space and con figuration space<br />are obtained as union of copies corresponding to various number fields glued together along<br />common points, in particular rational ones. Space-time surfaces decompose naturally to real<br />and p-adic space-time sheets. In this framework the fusion of real and various p-adic physics<br />reduces more or less to to an algebraic continuation of rational number based physics to various<br />number fields and their extensions."<br />http://www.scienceoflife.nl/MPitkanen-TGD-Number-06-Piadic3.pdf<br /><br />Ekeland continues by discussing multioscillators (cf. pendulums) and finding periodic solutions to Lagrange-Euler equations in convex hypersurfaces of two degrees of freedom and states resent proofs showing that they can can have only either 2 or infinity of periodic solutions of closed circuits. <br /><br />So both approaches end up with book metaphor, where as Matti's approach goes much further and combines the space-time sheets into global book where they can communicate at the level of p-adic areas.Santeri Satamanoreply@blogger.comtag:blogger.com,1999:blog-10614348.post-60261300369719836642012-12-29T08:44:07.615-08:002012-12-29T08:44:07.615-08:00To continue attempts to understand TGD, first a qu...To continue attempts to understand TGD, first a quote from Ekeland's 'Best of All Possible Worlds', with interesting parallels to Matti's vision; sorry it's in Finnish as I don't have English copy available:<br /><br /><br />"Tällä energian säilymisen ominaisuudella on geometrinen tulkinta, joka mahdollistaa liikkeiden määrittely ongelman yksinkertaistamisen. Ajatellaan suurta järjestelmää, jolla on lukuisia vapausasteita, sanokaamme n kpl, mikä tarkoittaa, että sen faasiavaruuden ulottuvuuksia on 2n. Kootaan tässä 2n-ulotteisessa avaruudessa kaikki tilat, joissa energia saa tietyn arvon, esimerkiksi 1: ne muodostavat eräänlaisen lehden, jossa on (2n - 1) ulottuvuutta, samanlaisen kuin tavallinen kaksiulotteinen pinta tavallisessa kolmiulotteiseessa tilassa. Tämä lehti voi sulkeutua kuin pallo tai torus, tai levitä äärettömiin kuten taso. Konstruoidaan sitten lehti, joka muodostuu kaikista järjestelmän tiloista, joissa energiataso on 2. Kyse on toisesta, joka on erillään ensimmäisestä ja joka ei leikkaa sitä, sillä energia ei voisi olla samalla kertaa 1 ja 2 samassa tilassa. Katsotaan sitten välissä olvevia energiatasoja 2,1:stä 2,9:ään, sitten s,01:stä 2,09:ään ja näin eteenpäin halki desimaalien: niitä on äärettömästi, ja jokainen niistä vastaa erillistä lehteä. Nämä kaikki lehdet täyttävät täysin faasiavaruuden, joka on näin kerrostunut eli stratifioitu: jokaisessa pisteessään faasiavaruus kohta yhden javain yhden rakennelehden (kyseisen kohdan energiatason). Tätä matemaatikot kutsuvat nimellä faasiavaruuden foliaatio eli lehditys, asian ymmärtämiseksi voi ajatella kirjan sivuja, tai proosallisemmin hienoikdsi siivuiksi leikattua ja sitten uudelleen koottua kinkkua.<br />(...)<br />Oletetaan faasiavaruudessa yksi rajattu alue, jonka reuna on faasiavaruuden kahteen osaan - sisäiseen ja ulkoiseen - jakava lehti: tätä kutsutaan matematiikassa hyperpinnaksi, joka on samanlainen kuin (kaksiulotteinen) pinta tavallisessa (kolmiulotteisessa) tilassa. Voidaan kirjoittaa puhtaasti geometrisella menetelmällä Eulerin-Lagrangen tyyppisiä yhtälöitä, jotka ovat hyperpinnalle ominaisia ja joita vastaavat kulkuradat eivät poistu siltä. Nämä yhtälöt eivät edellytä energian funktion määrittelemistä faasiavaruudessa, mutta jos toisaalta sellainen funktio on olemassa ja sillä on annettu hyperpinta energiatasona, silloin tähän energiatasoon kuuluvat liikeradat ovat yhtneväiset geometristä tietä löydettyjen ratojen kanssa." <br />Ekeland: Paras mahdollisista maailmoista - matematiikka ja kohtalo, pp 135-136.<br /><br />Santeri Satamanoreply@blogger.com