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This is just one example of thosands that prove that that this particular patent stands in a framework defended by the 5 natural laws of interspecific interaction. Thes laws can be represented by digital logic forming a logical geometry of an iterated dodecahedron a "Bucky Ball". It also reinforces the fact that the 4 transmission types are deductive and must be utilized together within a control plane. IP will NOT work as a stand alone transmission system by mere definition. Cisco is pushing this concept and it will only make matters worse in a 2D telecommunications "cloud". A logical geometrical FRAME work MUST be created.

If systems are put in all IP random telecommunications outages will occur throughout the world. The reson they are trying to put all systems into IP is to create the illusion of true 3D convergence andkeep everything in overlay. Shame on them. The US nor any country wants to end up being subject to random telecommunication blackouts that entail everything from fax, radio, gps, phones, tv, satellites, Internet to high end broadcast TV stations. The FBI should look into what Cisco is trying to do by eliminatining ATM, FRAME RELAY AND LEGACY NETWORKS to create a face centered cubic axis pattern with all IP connections. it's blatant obfuscation and it must stop. If not, latency will exponentially increase in the 2Dtelecommunications "cloud" creating information cascades that will destroy financial institutions and businesses across the world. These cascades of triilions of latent signals hit the finacial market computer algorithms and produce false information and transactions are made on this data.

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- 1. Lost in spaceA multiconnecteddodecahedraluniverse gives theimpression of being120 times biggerthan it actually is,similar to a hallof mirrors. Jeff Weeks
- 2. physicsweb.org Feature: Cosmic topologyA cosmic hall of mirrorsMost astronomers think that the universe is tance apart as they do in the familiar Euclidean space; Jean-Pierre Luminet is at the Département negative curvature (Ω < 1), with parallel lines diverginginfinite, but recent measurements suggest as they do on the hyperbolic surface of a saddle; or pos- Univers et Théories, Observatoire dethat it could be finite and relatively small. itive curvature (Ω > 1), which means that parallel lines Paris, Meudon, eventually cross one another as they do on the surfaceIndeed, as Jean-Pierre Luminet describes, of a sphere. France, e-mail jean- pierre.luminet@we could be living in an exotic universe In the standard model of cosmology, space has been obspm.fr flat and infinite ever since the universe underwent ashaped rather like a football short period of extremely rapid expansion called infla- tion shortly after the Big Bang. Moreover, we now knowFor centuries the size and shape of space has intrigued that the expansion of the universe is actually accelerat-the human race. The Greek philosophers Plato and ing due to a mysterious repulsive force caused by “dark”Aristotle claimed that the universe was finite with a energy (see Physics World May 2004 pp37–42).clear boundary. Democritus and Epicurus, on the other In 2003 the Wilkinson Microwave Anisotropy Probehand, thought that we lived in an infinite universe filled (WMAP) produced a high-resolution map of the cos-with atoms and vacuum. Today, 2500 years later, cos- mic microwave background that provided clues aboutmologists and particle physicists can finally address the expansion rate of the universe and its curvature.these fundamental issues with some certainty. Combined with other astronomical observations, the Surprisingly, the latest astronomical data suggest that WMAP data suggest that Ω = 1.02 ± 0.02, whichthe correct answer could be a compromise between favours a spherical universe with positive curvature.these two ancient viewpoints: the universe is finite and The simplest such space is a “hypersphere”, which canexpanding but it does not have an edge or boundary. In be thought of as the 3D surface of a 4D ball, just as anparticular, accurate maps of the cosmic microwave ordinary sphere is the 2D surface of a 3D ball. Hyper-background – the radiation left over from the Big Bang spherical space is therefore finite but it does not have a– suggest that we live in a finite universe that is shaped problematic boundary (figure 1). However, as we willlike a football or dodecahedron, and which resembles see, many other spherical spaces can fit the data bettera video game in certain respects. than a hypersphere. In such a scenario, an object that travels away fromthe Earth in a straight line will eventually return from The topology of spacethe other side of the universe, having been rotated by Curvature is clearly central to the large-scale shape of36° in the process. Space might therefore act like a cos- space, but it is not the only factor. The global topolog-mic hall of mirrors by creating multiple images of far- ical properties of space are also important because theyaway light sources, which raises new questions about determine whether the universe is finite or infinite. Allthe physics of the early universe. However, this is just spherical spaces are finite, for instance, but not allone possibility and other proposals made by research- finite spaces are spherical. Indeed, flat and hyperbolicers in the expanding field of cosmic topology include spaces can have finite or infinite volumes dependingtetrahedral and octahedral spaces, flat doughnuts and on their topologies.an infinite “horn-shaped” universe. To illustrate this in two dimensions, think of a square and identify opposite sides as being the same, as hap-The curvature of space pens in video games where a spaceship disappearing toThe first testable predictions about the size and shape the right of the screen reappears on the left. In threeof the universe were made by Einstein in 1916 as part of dimensions, a spaceship or anything else (such as a par-his general theory of relativity. In general relativitymassive bodies such as stars change the shape of At a Glance: Cosmic topologyspace–time around them, much as a bowling ball wouldchange the shape of a trampoline. Indeed, it is this local G There are three possibilities for the curvature of the universe: space can be flat,deformation of space–time that is responsible for grav- spherical or hyperbolicity in Einstein’s theory. G The geometry of the universe depends on its curvature and also on its topology, The average curvature of space therefore depends which governs the way space is connected and so determines if it is finite or infiniteon the overall density of matter and energy in the uni- G Measurements of the cosmic microwave background constrain the curvature ofverse. This density is usually expressed in terms of the the universe and provide hints about its topologyratio Ω, which is defined as the actual density of the uni- G Recent data suggest that the universe might be multiply connected, like theverse divided by the critical density required for space left- and right-hand sides of the screen in a computer gameto be flat or Euclidean. Space can therefore have three G Since the early 1990s the number of cosmo-topologists around the world haspossible curvatures: zero curvature (Ω = 1), which grown to more than 50means that two parallel lines remain a constant dis-Physics World September 2005 3
- 3. Feature: Cosmic topology physicsweb.org 1 How flat is the universe? easily misinterpret them as distinct galaxies in an end- less space, much as a visitor to a mirrored room has the illusion of seeing a crowd. Could we, in fact, be living in such a cosmic hall of mirrors? Topologists have proved that in addition to the ordi- nary, simply connected Euclidean, spherical and hyper- bolic spaces, there are 17 other Euclidean spaces and an infinite number of spherical and hyperbolic spaces – all of which are multiply connected. These spaces differ horizon sphere in the shape of their fundamental blocks, which can take the form of a parallepiped or a hexagonal prism for a Euclidian space or more complicated polyhedrons for spherical and hyperbolic spaces. The way the faces of these blocks are glued together also differs between hyperspherical each space. The surprise from the WMAP data is that universe the topology of space seems indeed to be multiply con- nected and described by a special class of shapes that are called “well proportioned”. Cosmic harmonics The best way to determine the shape of our universe is The curvature of space and our horizon radius are determined by the to go back to its beginning, just after the Big Bang. The average density of the universe and its expansion rate. Cosmologists infant universe is thought to have been crossed by often say that space is nearly flat because the observed value of the acoustic waves that would have caused tiny density fluc- density is close to the critical value for a flat universe. However, if the tuations in the primordial plasma. After about 380 000 density was 2% more than the critical density, the horizon radius years, however, the universe had expanded and cooled (yellow line) would be about 46 billion light-years and the radius of enough to allow matter and antimatter to decouple. curvature of the corresponding hypersphere would be only 2.6 times This meant that photons could travel unhindered greater. We would therefore see a modest but non-trivial portion of the through space, carrying with them vital information hypersphere. If the density is exactly equal to the critical value, space about the primordial density fluctuations (which are is Euclidean, the radius of curvature is infinite and we can only see an now thought to have been the seeds for galaxies and infinitesimal portion of the universe. clusters of galaxies to form). Today, 13.7 billion years after the Big Bang, this radiation has cooled to a tem- ticle or a photon) that leaves the “fundamental” cube perature of about 2.7 K, which is in the microwave through one face re-enters it from the opposite face. In region. And the fluctuations are imprinted as hot and this case one can imagine a cubic block of space whose cold spots in this cosmic microwave background. opposite faces have been “glued” together to produce A good way to understand the connection between what is effectively a 3D torus. acoustics and topology is to sprinkle fine sand uni- At first glance all the familiar rules of Euclidean formly over a drumhead and then make it vibrate. The geometry hold in both of these examples, and the grains of sand will collect in characteristic areas and spaces look infinite to those who live in them. However, patterns that reveal information about the local geom- unless the spaceship keeps encountering the same etry of the drum and about the elasticity of its mem- objects on its travels, there is no way that its crew could brane. But the distribution of these spots also depends tell if it is moving through an infinite space or through on the global shape – i.e. the topology – of the drum. the same finite space again and again. For example, the waves will be reflected differently Furthermore, general relativity does not distinguish according to whether the drumhead is infinite or finite, between these possibilities because each of the three and whether it is shaped like a circle, an ellipse or some plausible cosmic geometries – flat, hyperbolic and other shape. spherical – is consistent with many different topolo- Just as the vibration of a drumhead may be expressed gies. For example, a 3D torus and ordinary flat Eu- as a combination of its harmonics, fluctuations in the clidean space are described by the same equations in temperature of the cosmic background radiation may general relativity, even though the former is finite and be expressed as combinations of the vibrational modes the latter infinite. Determining the topology of the cos- of space itself. When the level of fluctuations is plotted mos therefore requires some physical understanding as a function of angle, we therefore find a series of beyond relativity, in particular concerning the way dif- peaks that provides a signature of the geometry of ferent regions of space–time are connected. space 13.7 billion years ago (figure 2). For example, theFinite well- Cosmologists usually assume that the universe is sim- position and amplitude of the first peak – i.e. the peakproportioned ply connected like a plane, which means there is only at the largest angle – in this “angular power spectrum”spaces open a one direct path for light to travel from a source to an gives the radius of curvature of space.Pandora’s observer. A simply connected Euclidean or hyperbolic Different cosmological models predict different universe would indeed be infinite, but if the universe is power spectra, and high-resolution measurements ofbox for the multiply connected, like a torus, there would be many the cosmic microwave background from instrumentsphysics of the different possible paths. This means that an observer such as WMAP now allow us to compare different the-early universe would see multiple images of each galaxy and could ories against real data. However, when WMAP re-4 Physics World September 2005
- 4. physicsweb.org Feature: Cosmic topologyleased its first data in 2003, advocates of the standard 2 Angular power spectrumcosmological model were faced with several surprises. The position of peaks in the angular spectrum is usu- modeally described by their wavenumber or mode 2 90 360 900 = 180°/θ, where θ is the angular distance in the sky.In fact, the lowest mode – the dipole or = 1 mode – isswamped by the far stronger dipole induced by themotion of the solar system relative to the cosmic back- temperature fluctuationground, which means that it cannot be measured. Butwhen researchers determined the first observablemode – the = 2 or quadrupole mode – they found thatit was seven times weaker than the predictions for a flat,infinite universe. Furthermore, the octopole or = 3mode was also less than the expected value by a factorof about two-thirds. For higher modes up to = 900, which correspondto angular scales of just 0.2°, the WMAP data werefairly consistent with the standard model. But a morecareful analysis of the power spectrum also reveals that 90° 2° 0.5° 0.2°the distribution of temperature fluctuations is not fully angular scaleisotropic and that the fluctuations are distributed dif- modeferently on different angular scales. 2 3 90 All these anomalies contradict the standard pictureof the universe, which has led some more conservativecosmologists to claim that they are due to errors in thedata analysis. Furthermore, the second round of WMAPdata – originally expected in February 2004 – has been temperature fluctuationdelayed for more than a year, which may hint at addi-tional trouble to come! Meanwhile, other cosmologistshave taken the problem seriously and proposed newlaws to explain the early universe, some of which haveexotic names such as “vanilla” and “racetrack” inflation. Cosmo-topologists, on the other hand, have tried tofind a more natural, geometrical explanation for theobserved power spectrum. Put simply, the unusuallylow amplitudes of the quadrupole and octopole modesmeans that long wavelengths (i.e. temperature fluctu-ations over large angular scales) are missing, possibly 90° 60° 2°because space is not big enough to sustain them. This angular scalecan be likened to oscillation of a string fixed at both The evidence that space might have an unusual shape is obvious when we plot the level ofends, where the maximum wavelength of an oscillation temperature fluctuations in the cosmic microwave background as a function of angle, θ (top).is twice the string length. The geometrical explanation The WMAP data are shown in green (purple depicts earlier data) and the modes of theof the power spectrum thus implies that we live in a spectrum are defined as = 180°/θ. The largest angular scales (or lowest modes) are very sensitive to the global shape or topology of space. In this region (bottom), data from WMAP dofinite, multiply connected space that is smaller than the not agree with theoretical predictions for an infinite Euclidean space (red curve), whereasobservable universe. predictions based on a Poincaré dodecahedral space (blue curve) do agree. This suggests that we live in a multiply connected spherical space.Dodecahedral spaceSurprisingly, not all small-volume universes suppressthe large-scale fluctuations. In 2003 the present author, space is essentially a multiply connected variant of aJeff Weeks and co-workers proved that the long-wave- simply connected hypersphere, although its volume islength modes tend to be relatively lowered only in a 120 times smaller.special family of finite, multiconnected spaces that are A rocket leaving the dodecahedron through a givencalled “well-proportioned spaces” because they have face immediately re-enters through the opposite face,a similar extent in all three dimensions. More specifi- and light propagates such that any observer whosecally, we discovered that the best candidate to fit the line-of-sight intercepts one face has the illusion of see-observed power spectrum is a well-proportioned space ing a slightly rotated copy of their own dodecahedron.called the Poincaré dodecahedral space. This means that some photons from the cosmic This space may be represented by a polyhedron with microwave background, for example, would appear12 pentagonal faces, with opposite faces being “glued” twice in the sky.together after a twist of 36° (figure 3). This is the only The power spectrum associated with the Poincaréconsistent way to obtain a spherical (i.e. positively dodecahedral space is different from that of a flat spacecurved) space from a dodecahedron: if the twist was because the fluctuations in the cosmic microwave back-108°, for example, we would end up with a radically dif- ground will change as a function of their wavelengths.ferent hyperbolic space. The Poincaré dodecahedral In other words, due to a cut-off in space correspondingPhysics World September 2005 5
- 5. Feature: Cosmic topology physicsweb.org3 The many shapes of the universe Jeff WeeksThe Poincaré dodecahedral space (left) can be described as the interior of a “sphere” made from 12 slightly curved pentagons. However, there is one big differencebetween this shape and a football because when one goes out from a pentagonal face, one immediately comes back inside the ball from the opposite face after a 36°rotation. Such a multiply connected space can therefore generate multiple images of the same object, such as a planet or a photon. Other such well-proportioned,spherical spaces that fit the WMAP data are the tetrahedron (middle) and octahedron (right). to the size of the dodecahedron, one expects fewer fluc- had to cross a much larger distance due to the expan- tuations at large angular scales than in an infinite flat sion of the universe.) space, but at small angular scales one must recover the If physical space is indeed smaller than the observ- same pattern as in the flat infinite space. In order to cal- able universe, some points on the map of the cosmic culate the power spectrum we varied the mass–energy microwave background will have several copies. As first density of the dodecahedral universe and computed shown by Neil Cornish of Montana State University the quadrupole and the octopole modes relative to the and co-workers in 1998, these ghost images would WMAP data. To our delight, we found a small interval appear as pairs of so-called matched circles in the cos- of values over which both these modes matched the mic microwave background where the temperature observations perfectly. Moreover, the best fit occurred fluctuations should be the same (figure 4). This “lens- in the range 1.01 < Ω < 1.02, which sits comfortably ing” effect, which can be precisely calculated, is thus with the observed value. purely attributable to the topology of the universe. The Poincaré dodecahedral space therefore ac- Due to its 12-sided regular shape, the Poincaré dodec- counts for the lack of large-scale fluctuations in the ahedral model actually predicts six pairs of diametri- microwave background and also for the slight positive cally opposite matched circles with an angular radius of curvature of space inferred from WMAP and other 10–50°, depending on the precise values of cosmologi- observations. Moreover, given the observed values of cal parameters such as the mass–energy density.The most fundamental issue Circles in the sky When news of our dodecahedral model appeared in Nature in October 2003, it was not long before the pressis to link the topology of space to started running headlines based on what was being hailed the new “football-shaped” model of the uni-a quantum origin verse. However, since cosmo-topology is a very com- petitive field, the initial response from other groups the mass–energy densities and of the expansion rate of was not always favourable. the universe, the size of the dodecahedral universe can For instance, the New York Times ran the headline be calculated. We found that the smallest dimension of “Cosmic soccer ball? Theory already takes sharp kicks”, the Poincaré dodecahedron space is 43 billion light- based on an apparently negative search for matched cir- years, compared with 53 billion light-years for the cles in the WMAP data performed by Cornish and co- “horizon radius” of the observable universe. Moreover, workers. Using massive computer simulations, they the volume of this universe is about 20% smaller than claimed to have found no evidence of matching on the volume of the observable universe. (There is a com- angular sizes greater than 25° and thus rejected the mon misconception that the horizon radius of a flat uni- Poincaré hypothesis the same day it appeared. verse is 13.7 billion light-years, since that is the age of In fact, their rejection was rather premature because the universe multiplied by the speed of light. However, they had only looked for non-rotated matched circles the horizon radius is actually much larger because pho- that were diametrically opposite one another – a case tons from the horizon that are reaching us now have that did not test the dodecahedron model at all. After6 Physics World September 2005
- 6. physicsweb.org Feature: Cosmic topologythe initial excitement, Cornish and co-workers went 4 Simulated circle matchingback and reassessed the data. Taking account of the ad- A Riazueloditional 36° twist took a few additional months of com-puter time, but the matched circles remained elusive.This led them to conclude that there was no reasonabletopology for the universe that had a characteristiclength smaller than the observable horizon. However, it turned out that the researchers hadtaken a short cut to save computer time. While theycorrectly took into account the possible rotationsbetween matched circles that are implied by mostmulti-connected topologies, they only searched formatched circles that were back-to-back or very nearlyback-to-back. This led them to exclude all likely multi-ply connected spaces. In the mean time, however, wehad proved that in most multiply connected, well-pro-portioned topologies space is not homogeneous. Thismeans that the position of matched circles in the skydepends on the location of the observer, and they arenot, therefore, back-to-back. Only in the simplest of The topology of the universe describes how different regions are connected and could thereforetopologies, such as the hypertorus in flat space and the leave its imprint on the cosmic microwave background. For example, if our physical space isPoincaré dodecahedron in a spherical space, is space smaller than the observable universe (as recent data suggest it is) then the horizon spherehomogeneous and the circles back-to-back. wraps around the universe and intersects itself. As a result, duplicated images of the cosmic This violates one of the most basic principles of cos- microwave background (in which the colours represent temperature fluctuations) will intersect along a circle and we would observe this circle on different sides of the sky.mology, that there is no privileged position in the uni-verse. But this principle could be illusory, like the antin the desert that is convinced the whole world is filled curved space, it does not fit the experimental con-with sand and dunes. For instance, in a flat-torus uni- straints we already have on the curvature of space.verse, any gluing together of the opposite faces com- After studying the horn-shaped topology further,bined with a screw motion produces pair of circles that Steiner and co-workers realized that well-proportionedare far from being back-to-back. Unfortunately, the spherical spaces were, in fact, more promising. Theyincrease in the number of degrees of freedom that went on to prove that the fit between the power spec-results from such a scenario means that a full-circle trum predicted by the Poincaré dodecahedron model Mapping spacesearch in the WMAP data is beyond current comput- and that observed by WMAP was even better than we Launched in 2001,ing capabilities. had previously thought. But the German team also the Wilkinson extended its calculations to well-proportioned tetra- Microwave AnisotropyCosmic horn hedral and octahedral spherical spaces in which Ω > 1 Probe hasIn June 2004, however, Boud Roukema and colleagues (see figure 3). changed our viewat the Torun Centre for Astronomy in Poland inde- These spaces are somewhat easier to understand of the cosmos.pendently searched for circles in the WMAP data. By than a dodecahedral space, but they require higher val-only looking for back-to-back circles within a limited ues of the density: Ω > 1.015 for octahedral spaces andrange of angular sizes and neglecting all other possible Ω > 1.025 for tetrahedral spaces, compared withmatches, the computer time was reduced drastically. Ω > 1.009 for dodecahedral spaces. However, theseRemarkably, the Polish team found six pairs of values are still compatible with the WMAP data.matched circles distributed in a dodecahedral pattern Furthermore, Steiner and co-workers found that theand twisted by 36°, each with an angular size of about signal for pairs of matched circles could have be missed11°. This implied that Ω = 1.010 ± 0.001, which is per- by current analyses of the cosmic microwave back-fectly consistent with our dodecahedral model, ground due to various measurement effects that dam-although the result was much less publicized than the age or even destroy the temperature matching.earlier negative results. Another active area of cosmic topology is “cosmic In fact, the statistical significance of the match still crystallography”, which was initially devised by the pre-needs to be improved, which means that the validity of sent author and co-workers in 1996 and is now beingthe Poincaré dodecahedron model is still open to pursued by, among others, Germán Gomero of thedebate. In the last few months, however, there has been Universidade Estadual Paulista in Brazil and Marcelomuch theoretical progress on well-proportioned spaces Reboucas of the Brazilian Center for Research inin general. Early this year, for example, Frank Steiner Physics. In cosmic crystallography researchers look forand co-workers at the University of Ulm in Germany repeating patterns in the 3D distribution of high-red-proposed a multiply connected hyperbolic topology shift sources, such as galaxy clusters and quasars, muchcalled the Picard hyperbolic space. Like the Poincaré like the repeating patterns of atoms observed in crys-dodecahedron, this horn-shaped space belongs to the tals. By building so-called pair-separation histograms,family of well-proportioned spaces and it also correctly cosmologists are in most cases able to detect a multi-fits the low vibrational modes of the WMAP data. connected topology of space in the form of spikes thatHowever, since the topology requires the density para- clearly stand out above the distribution expected formeter to have a value of Ω = 0.95, and thus a negatively the simply connected case.Physics World September 2005 7
- 7. Feature: Cosmic topology physicsweb.org A Pandora’s box for physics Ω with a precision of 1%. A value lower than 1.01 will Finite well-proportioned spaces, especially the Poin- rule out the Poincaré dodecahedron model, since the caré dodecahedron, open something of a Pandora’s size of the corresponding dodecahedron would become box for the physics of the early universe. The standard greater than the observable universe and would not model of cosmology relies in the main on the hypothe- leave any observable imprint on the microwave back- sis that the early universe underwent a phase of expo- ground. A value greater than 1.01, on the other hand, nential expansion called inflation, which produced would strengthen the models’ cosmological pertinence.The second density fluctuations on all scales. In the simplest in- Whether or not some multiply connected model ofround of WMAP flationary models, space is supposed to have become space such as the Poincaré dodecahedron is refuted bydata may hint immensely larger than the observable universe. There- future astronomical data, cosmic topology will con- fore, a positive curvature (i.e. Ω > 1), even if weak, tinue to remain at the heart of our understanding aboutat additional implies a finite space and sets strong constraints on the ultimate structure of our universe.trouble to inflationary models.come It is possible to build “low scale” inflationary uni- More about: cosmic topology verses in which the inflation phase ends more quickly W Aurich et al. 2005 CMB anisotropy of the Poincaré than it does in general inflationary modes, leading to a dodecahedron arXiv.org/abs/astro-ph/0412569 detectable space curvature. In other words, even if space N Cornish et al. 2004 Constraining the topology of the universe is not flat, a multiconnected topology does not contra- Phys. Rev. Lett. 92 201302 dict the general idea of inflation. However, no convinc- J P Luminet et al. 2002 Is space finite? The Once and Future ing physical scenario for this has yet been proposed. Cosmos, Scientific American (special edition) pp58–65 Perhaps the most fundamental challenge is to link J P Luminet et al. 2003 Dodecahedral space topology as an the present-day topology of space to a quantum origin, explanation for weak wide-angle temperature correlations in the since general relativity does not allow for topological cosmic microwave background Nature 425 593 changes during the course of cosmic evolution. A quan- A Riazuelo et al. 2004 Cosmic microwave background in tum theory of gravity could allow us to address this multiconnected flat spaces Phys. Rev. D 69 103518 problem, but there is currently no indication about how B Roukema et al. 2004 A hint of Poincaré dodecahedral such a unified theory might actually describe the emer- topology in the WMAP first year sky map Astron. Astrophy. gence of multiply connected spaces. 423 821 Data from the European Planck Surveyor, which is J Weeks 2001 The Shape of Space (New York, Dekker) scheduled for launch in 2007, will be able to determine WMAP results: map.gsfc.nasa.gov8 Physics World September 2005

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