Cos-top

cos-top@cosmo.torun.pl

19 discussions

Helio Fagundes
by Boud Roukema 01 Aug '19

01 Aug '19

hi cos-top For those who haven't heard the news yet, Helio Fagundes passed away on 30 July. For those not fully familiar with his pioneering work in pushing for cosmic topology as an observational field within cosmology/extragalactic astronomy, here is the ADS list of his work: https://ui.adsabs.harvard.edu/search/q=%20author%3A%22fagundes%2C%20h%22&so… His first paper with a closed hyperbolic spatial section and an urge to make observational tests with quasars seems to be his 1982 paper: https://ui.adsabs.harvard.edu/abs/1982LMaPh...6..417F Boud

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hyperbolic 3D spaces with cubical FDs?
by Boud Roukema 04 Mar '19

04 Mar '19

hi cos-top Do there exist any hyperbolic multiply connected constant-curvature compact spaces for which the fundamental domain is a cube (with flat faces, of course) for some points in the space? Or is there a proof that this is impossible? We know that there are (at least) two such spherical spaces, that Peter Kramer calls C_2 and C_3: Kramer09: https://ui.adsabs.harvard.edu/#abs/2009PhyS...80b5902K/ Are there any hyperbolic ones? Cheers Boud [Reminder - this is an opt-in list - you can get info at http://cosmo.torun.pl/mailman/listinfo/cos-top for unsubscribing or subscribing; information should be included in this email for doing these or other operations via email.]

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Please post: Royal Astronomical Society Specialist Discussion meeting on topology Dec 13
by Glenn Starkman 05 Nov '13

05 Nov '13

Date: 13 Dec 2013 Time: 10:30am Location: Burlington House, London < http://www.ras.org.uk/about-the-ras/burlington-house> We are pleased to announce a Royal Astronomical Society Specialist Discussion meeting on topology. General relativity connects the local geometry of spacetime to the energy contents, but the topology of space is unconstrained. Moreover, it is expected that any theory linking quantum mechanics and relativity is likely to exhibit topology change at scales that may have become cosmologically relevant. Cosmological observations have shown that the Universe is very nearly flat, and that the topology is trivial nearly out to the last scattering surface, but we have no direct measurement of its topology on scales larger than that. This discussion meeting will give an overview of current research in mathematical topology, its application to cosmology and particle physics, and a discussion of recent observations from Planck and other missions. Confirmed keynote speakers include Jeff Weeks (independent) and Philip Candelas (Oxford).

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eight cosmic topology papers for first semester 2013
by Boud Roukema 23 Jun '13

23 Jun '13

hi cos-top, Cosmic topology seems to be turning into a hot topic: we have at least eight papers - a mix of observational and theoretical papers - listed by the ADS for 2013. Feel free to reply to add any missing papers to this list! http://cdsads.u-strasbg.fr/abs/2013arXiv1306.2737F A search for nontoroidal topological lensing in the Sloan Digital Sky Survey quasar catalog Authors: Fujii, Hirokazu; Yoshii, Yuzuru http://cdsads.u-strasbg.fr/abs/2013arXiv1305.5576B Solving the Einstein constraints in periodic spaces with a multigrid approach Authors: Bentivegna, Eloisa http://cdsads.u-strasbg.fr/abs/2013arXiv1304.1811A The Topology and Size of the Universe from CMB Temperature and Polarization Data Authors: Aslanyan, Grigor; Manohar, Aneesh V.; Yadav, Amit P. S. http://cdsads.u-strasbg.fr/abs/2013arXiv1303.5086P Planck 2013 results. XXVI. Background geometry and topology of the Universe Authors: Planck Collaboration http://cdsads.u-strasbg.fr/abs/2013arXiv1303.4226A A search for cosmic topology in the final WMAP data Authors: Aurich, Ralf; Lustig, Sven http://cdsads.u-strasbg.fr/abs/2013arXiv1302.6537B Wave computation on the Poincar\'e dodecahedral space Authors: Bachelot-Motet, Agnes http://cdsads.u-strasbg.fr/abs/2013arXiv1302.4425R Deep redshift topological lensing: strategies for the T^3 candidate Authors: Roukema, Boudewijn F.; France, Martin J.; Kazimierczak, Tomasz A.; Buchert, Thomas http://cdsads.u-strasbg.fr/abs/2013PhRvD..87d3521R Topological implications of inhomogeneity Authors: Roukema, Boudewijn F.; Blanloeil, Vincent; Ostrowski, Jan J. Cheers Boud

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half-turn space analysis; new spherical 3-manifolds ?
by Boud Roukema 11 May '12

11 May '12

hi cosmic topologists, Today we have *two* cosmic topology articles on arXiv on the same day! (1) http://arxiv.org/abs/1009.5880 Cosmic microwave anisotropies in an inhomogeneous compact flat universe Authors: R. Aurich, S. Lustig This shows that the half-turn space E_2 also seems to provide a nice fit, at least with the S_60deg statistic, to the WMAP 7yr data, and it even seems to be a bit better than T^3. (2) http://arxiv.org/abs/1009.5825 Multipole analysis in cosmic topology Authors: Peter Kramer This seems to claim that the author has found 3 new spherical 3-manifolds, "N8, N9, N10". It's not clear to me if he claims that they can be given constant curvature, but maybe it's obvious to someone who knows the mathematics a bit better. i had thought that the constant curvature spherical 3-manifolds were already completely classified. Are N8, N9, and N10 new constant-curvature spherical 3-manifolds, in addition to those in Gausmann et al. 2001 http://arxiv.org/abs/gr-qc/0106033 ? cheers boud

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recent cosmic topology work
by Boud Roukema 07 Apr '09

07 Apr '09

hi everyone on cos-top, The Ulm group continue to work on T^3 models - here's a nice new paper: http://arXiv.org/abs/0903.3133 Other news: there's an observational signature of cosmic topology which occurs for flat compact models but not for curved compact models (assuming standard physical components of the Universe in both cases). The good news for cosmic topology in general: this observational signature has been detected to extremely high significance! :) The bad news for the Poincare space: this observational signature has been detected to extremely high significance! :( Details: http://arxiv.org/abs/0904.0975 A Poincare space is certainly not excluded by the argument, but it would require an exotic explanation for the observational signature rather than a simple explanation. Occam's razor suggests that we may have to accept T^3 (or another flat compact space). Anyway, see the paper to find out what this observational signature is. cheers boud

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the Poincare dodecahedral space is better balanced than other spaces
by Boud Roukema 20 Feb '09

20 Feb '09

Dear cos-top friends, It seems that not only is there at least some interaction between cosmic topology and gravity - the residual gravity acceleration effect as derived heuristically for the 1-torus and 3-torus[1], but the effect distinguishes 3 well-proportioned spaces (T^3, S^3/T^*, S^3/O^*) as special in that the effect cancels to third order in the Taylor expansion, and even more interesting, the Poincare dodecahedral space cancels even better than these other three well-proportioned spaces. The effect in S^3/I^* cancels down to fifth order in the Taylor expansion![2] In this sense, it's better balanced than the other spaces. Maybe we could say that the former three spaces are well-proportioned and "well-balanced", while the Poincare space is well-proportioned and "super-well-balanced"? This dynamical/geometrical/topological result is independent of the empirical arguments in favour of the Poincare space. Enjoy! :) boud PS: There'll be a cosmic topology session at the 12-th Marcel Grossmann Meeting in Paris, 12-18 July 2009. http://www.icra.it/MG/mg12/en/ Contact Marek Demianski and register if you're interested. [1] RBBSJ, 2007, http://arxiv.org/abs/astro-ph/0602159 Astron.Astrophys.463:861 [2] RR 2009, http://arxiv.org/abs/0902.3402, subm A&A

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draft of a response to Steven Weinberg's "Cosmology" page 9
by Boud Roukema 08 Jun '08

08 Jun '08

Dear Marcelo, everyone, The topic of Steven Weinberg's rather not-fully-accurate page 9 of his new book is IMHO rather well-suited to discussing on our public mailing list rather than privately - this has nothing to do with revealing unpublished results that we're still working on, it's about correcting some inaccurate statements regarding cosmic topology. i also think it would be worth it that the effort we spend in explaining things to Steven Weinberg can potentially help educate others directly, i.e. those who may read the mailing list archives or get pointed there by Big Brother Google Inc. An alternative (and reasonable) place to discuss this IMHO would be http://cosmocoffee.info, which is a phpBB based forum for cosmology research in general, while this mailing list is a mailman mailing list specifically for cosmic topology. Anyway, to people who do not have a copy of the book "Cosmology" published by Steven Weinberg a few days/weeks ago, just email one of us in private and we can send you a scan of page 9 ("for educational and research purposes only", to avoid any problems with copyright etc.). My general opinion: Yes, i think it would be a good idea to send a message to Steven, and a consensus message signed by several of us would be a good idea. However, it would require a bit of work, since ideally it should be: * clear * complete * well-referenced * but also... reasonably short This is not so easy, especially keeping it short while satisfying the first three criteria... i've posted a suggested draft on our wiki: http://cosmo.torun.pl/Cosmo/SWeinbergCosmologyPage9Response and i'm also posting the same initial draft text below here. Personally, i don't see that there's a huge hurry - i don't imagine OUP issuing a second edition on a time scale of days or weeks - so i haven't suggested any time scale. My suggestion is we give people a week or so to update/correct/ complain. If we wait too long, it will soon by the northern summer break and not much point expecting anyone to respond quickly. If you want to correct/edit my suggested text and/or sign it, please speak up! i've tried to be complete in the reference list while limiting to the precise problems in Steven's text. cheers boud suggested text, openly editable at http://cosmo.torun.pl/Cosmo/SWeinbergCosmologyPage9Response ---------------------------------------------------------------------- Dear Steven, We are glad that your cosmology text published by Oxford University Press a few months ago (March 2008) is, to the best of our knowledge, the first graduate level type cosmology text in which it is clearly stated that there are many more than three possible shapes for the spatial section of our Universe, assuming that the Friedmann-Lemaitre-Robertson-Walker metric is correct apart from density perturbations. Nevertheless, some of the statements on page 9 of your book are probably not as accurate as they could be. Here are some suggestions which should hopefully clarify these issues. 1. "There is no sign of [the same patterns of the distribution of matter and radiation in opposite directions] in the observed distribution of galaxies or cosmic microwave background fluctuations, so any periodicity lengths such as |L_i| must be larger than about 10^10 light years." * It is correct that based on cosmic microwave background constraints (WMAP), it is clear that the in-diameter of the Universe (see e.g. Fig. 10 of Luminet & Roukema 1999 for various definitions of the comoving size of the Universe) is greater than 2 h^-1 Gpc. * However, on a scale of about 12-15 h^-1 Gpc, a large number of papers have been recently published which suggest that, in particular, a Poincare dodecahedral space (for K=+1) or an equi-length 3-torus model (for K=0) better fit the WMAP data than an infinite K=0 model. There is not yet any clear consensus on what the data tell us - some papers argue that the infinite K=0 model cannot be significantly rejected based on the WMAP data. Please see the reference list below for the main papers in this series. 2. "[Most 3-manifolds] ... seem ill-motivated. In imposing conditions of periodicity we give up the rotation (though not translation) symmetry that led to the Robertson-Walker metric in the first place, so there seems little reason to impose these periodicity conditions while limiting the local spacetime geometry to that described by the Robertson-Walker metric." * This is a little unclear. If by "symmetry" you mean the existence of isometries in the covering space, then there are no global translation symmetries at all for K=-1 or K=+1. Translations only exist for K=0. (Clifford translations exist for some K=+1 spaces, but that's a separate issue.) If you want to retain translational isometries in 3-space, then only K=0 is possible, and your words would seem to describe both K=-1 and K=+1 simply connected spaces as "ill-motivated". * More importantly, the Friedmann-Lemaitre-Robertson-Walker (FLRW) metric only requires local homogeneity and isotropy, not global homogeneity and isotropy. The FLRW metric is intrinsically local, it's about the limit of what happens towards a point. To write this in terms of practical observational cosmology statistics, consider a more realistic model of the Universe, i.e. with a perturbed FLRW metric rather than a perfectly homogeneous one. In this model, "local homogeneity and isotropy" means that various n-point auto-correlation functions of structure tracers within a "neighbourhood" of a Gpc or so should give identical results to within observational uncertainties. This is what is observed, but it does not constrain global homogeneity and isotropy. So it is unclear why a multiply connected 3-manifold with an FLRW metric is in any way "limited" or "ill-motivated". * We are only aware of one paper so far showing that global topology could have an effect (albeit small) on the FLRW metric through the addition of an additional acceleration effect (Roukema et al. 2007). Best regards, Boud Roukema, ... References o "size of the Universe": + Luminet, J.-P., Roukema, B. F., 1999, Topology of the Universe: Theory and Observation, NATO ASIC Proc. 541: Theoretical and Observational Cosmology , 117, http://cdsads.u-strasbg.fr/abs/1999toc..conf..117L o WMAP: PDS and 3-torus models (and other spherical models) vs K=0 infinite model: + Aurich, R., Lustig, S., Steiner, F., 2005, CMB anisotropy of spherical spaces, Classical and Quantum Gravity, 22, 3443, http://cdsads.u-strasbg.fr/abs/2005CQGra..22.3443A + Aurich, R., Lustig, S., Steiner, F., 2005, CMB anisotropy of the PoincarĂŠ dodecahedron, Classical and Quantum Gravity, 22, 2061, http://cdsads.u-strasbg.fr/abs/2005CQGra..22.2061A + Aurich, R., Lustig, S., Steiner, F., 2006, The circles-in-the-sky signature for three spherical universes, Monthly Notices of the Royal Astronomical Society, 369, 240, http://cdsads.u-strasbg.fr/abs/2006MNRAS.369..240A + Caillerie, S., LachiĂ¨ze-Rey, M., Luminet, J.-P., Lehoucq, R., Riazuelo, A., Weeks, J., 2007, A new analysis of the PoincarĂŠ dodecahedral space model, Astronomy and Astrophysics, 476, 691, http://cdsads.u-strasbg.fr/abs/2007A%26A...476..691C + Gundermann, J., 2005, Predicting the CMB power spectrum for binary polyhedral spaces, ArXiv Astrophysics e-prints, arXiv:astro-ph/0503014, http://cdsads.u-strasbg.fr/abs/2005astro.ph..3014G + Key, J. S., Cornish, N. J., Spergel, D. N., Starkman, G. D., 2007, Extending the WMAP bound on the size of the Universe, Physical Review D, 75, 084034, http://cdsads.u-strasbg.fr/abs/2007PhRvD..75h4034K + Lew, B., Roukema, B., 2008, A test of the PoincarĂŠ dodecahedral space topology hypothesis with the WMAP CMB data, Astronomy and Astrophysics, 482, 747, http://cdsads.u-strasbg.fr/abs/2008A%26A...482..747L + Luminet, J.-P., Weeks, J. R., Riazuelo, A., Lehoucq, R., Uzan, J.-P., 2003, Dodecahedral space topology as an explanation for weak wide-angle temperature correlations in the cosmic microwave background, Nature, 425, 593, http://cdsads.u-strasbg.fr/abs/2003Natur.425..593L + Niarchou, A., Jaffe, A., 2007, Imprints of Spherical Nontrivial Topologies on the Cosmic Microwave Background, Physical Review Letters, 99, 081302, http://cdsads.u-strasbg.fr/abs/2007PhRvL..99h1302N + Riazuelo, A., Weeks, J., Uzan, J.-P., Lehoucq, R., Luminet, J.-P., 2004, Cosmic microwave background anisotropies in multiconnected flat spaces, Physical Review D, 69, 103518, http://cdsads.u-strasbg.fr/abs/2004PhRvD..69j3518R + Roukema, B. F., Bulinski, Z., Szaniewska, A., Gaudin, N. E., 2008, The optimal phase of the generalised Poincare dodecahedral space hypothesis implied by the spatial cross-correlation function of the WMAP sky maps, Astronomy and Astrophysics, in press, arXiv:0801.0006, http://cdsads.u-strasbg.fr/abs/2008arXiv0801.0006R + Roukema, B. F., Lew, B., Cechowska, M., Marecki, A., Bajtlik, S., 2004, A hint of PoincarĂŠ dodecahedral topology in the WMAP first year sky map, Astronomy and Astrophysics, 423, 821, http://cdsads.u-strasbg.fr/abs/2004A%26A...423..821R o effect of global topology on the metric: + Roukema, B. F., Bajtlik, S., Biesiada, M., Szaniewska, A., Jurkiewicz, H., 2007, A weak acceleration effect due to residual gravity in a multiply connected universe, Astronomy and Astrophysics, 463, 861, http://cdsads.u-strasbg.fr/abs/2007A%26A...463..861R ----------------------------------------------------------------------

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Benasque meeting, anyone?
by Boud Roukema 04 Apr '08

04 Apr '08

hi everyone on the cos-top mailing list, There are at least two cosmology workshop-type meetings this northern summer at which cosmic topology is likely to be discussed: * Pune, Maharashtra (India, monsoon time but the monsoon in Pune is mild (not strong) ;) "Cosmology with CMB and LSS" 21 July - 31 Aug (full = 6 weeks = 3 * 2-week sessions, but can stay for a shorter time) http://icts.tifr.res.in/sites/cmb/index * Benasque, Pyrenees (mountains on spanish/french border) "Modern Cosmology" 27 July - 14 August 2008 http://benasque.ecm.ub.es/2008cosmology/2008cosmology.htm http://cosmocoffee.info/viewtopic.php?t=1093 Tarun Souradeep is one of the main organisers of the Pune meeting and cosmic topology is listed for the third two-week part of the full session; Glenn Starkman and Dominik Schwarz are listed as participants for the Benasque meeting. i'm thinking of going to the Benasque meeting (leaving a day or two before the end). Is anyone thinking of going to one of these meetings? i can't imagine that i'll be over two weeks together with Glenn and Dominik without having any useful discussions on cosmic topology, and it would probably make sense that more of us are together than less. Both the Pune and Benasque meetings sound like they'll be as much workshops where real work will get done rather than just standard conference presentations. The deadlines for both are 15 April. cheers boud

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Happy New +36 degree Year!
by Boud Roukema 08 Jan '08

08 Jan '08

hi everyone, i guess maybe only Marcelo on this list is feeling +36 degrees or so Celsius right now, but PDS +36 degrees seems now to at least qualitatively agree with Ralf, Frank and Sven's work and the Caillerie et al. paper. http://arXiv.org/abs/0801.0006 In other words, we managed to kill off our previous "hint" candidate, but another one better agreeing with other people's work has turned up. Are we converging? Enjoy :). Best wishes for the New Year boud

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1 - sad news (most already know); 2 - equalising of T^3 lengths
by Boud Roukema 29 Jan '07

29 Jan '07

hi cos-top, Probably most of us already know the sad news - Jesper Gundermann is no longer with us - since June 2006. Jeff has submitted an interesting article with Jesper as posthumous co-author: http://arxiv.org/abs/astro-ph/0611640 It's not much of a consolation. Jesper was a nice guy and did good work in cosmic topology. A year ago he (together with Jeff) responded with lightning fast rapidity regarding my attempt to understand the Poincare dodecahedral space, showing (if any more evidence were necessary) that he was someone who thought about the subject deeply: http://cosmo.torun.pl/pipermail/cos-top/2006-February/thread.html My other theme (more pleasant) in this email is that i'm wondering if people have any thoughts on this paper? http://arxiv.org/abs/astro-ph/0602159 In this paper (A&A in press), we show heuristically that contrary to what most of us (well, me anyway) believe(d) and have stated and told to others for many years, topology and dynamics of a nearly FLRW universe are *not* independent of one another. In other words, there is some dependence between them. It's a weak dependence, but there is some link there. And it's related to the difference between an idealised perfectly FLRW (homogeneous) universe and a real one containing perturbations represented statistically in an approximate way by a power spectrum. i was also surprised when we did the calculation and found the result that the effect tends to equalise the three lengths of a T^3 universe, whereas all the papers that used T^3 models assumed this simply because it's simple, without mentioning the fact that it might have a physical justification. For me the obvious question to follow is whether or not the lengths of a PDS universe would also tend to be equalised - by slightly unequal evolution of the scale factor - keeping them to an ideal equilibrium, and avoiding the PDS universe becoming slightly "oblate" or "prolate". Regards boud

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Re: 1 - sad news (most already know); 2 - equalising of T^3 lengths
by weeks＠geometrygames.org 29 Jan '07

29 Jan '07

Your message to Jeff Weeks has *not* been delivered. My old e-mail address (weeks(a)geometrygames.org) was once publicly available on the web, so as a result it gets lots of spam. To shake off the spam, I've stopped using that address. To contact me please go to the page http://www.geometrygames.org/contact.html Thanks, Jeff Weeks

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binary polyhedral groups
by Jeff Weeks 10 Feb '06

10 Feb '06

Dear Cos-Top, Jesper pointed out that the final sentence of my earlier message, namely This same relation holds, of course, for the other single-action spherical manifolds. implies something that is not correct. To make that statement more precise (and correct!) please note that in the construction if you travel in the direction of a symmetry axis [of Polyhedron A] of order n, you'll find 2n translates of the fundamental domain [Polyhedron B] spaced pi/n radians apart the polyhedra should be interpreted as follows: Polyhedron A is the polyhedron for which the group is named (that is, a tetrahedron for the binary tetrahedral group, an octahedron for the binary octahedral group, or an icosahedrdon for the binary icosahedral group). Polyhedron B is the fundamental domain itself (respectively an octahedron, a truncated cube or a dodecahedron in the preceding three examples). In the case of the binary icosahedral group, the symmetries of the icosahedron are exactly the symmetries of the dodecahedron, so no confusion is possible. In the case of the binary octahedral group, the symmetries of the octahedron are exactly the symmetries of the truncated cube, so again no confusion is possible. But... in the case of the binary tetrahedral group, the symmetries of the tetrahedron are a proper subset of the symmetries of the fundamental domain (the octahedron). In other words, the fundamental domain (the octahedron) acquires some "accidental" symmetries that are not a priori forced upon it by the symmetries of the basic tetrahedron. In this case, to enumerate the elements of the binary tetrahedral group, it's crucial that we start with the basic tetrahedron, and not with the octahedral fundamental domain. (The reason the binary tetrahedral group's fundamental domain acquires extra symmetry is that the tetrahedron is self-dual. In effect the octahedral fundamental domain is the intersection of the basic tetrahedron and its dual. Thus the doubling of the symmetry.) For an elementary explanation of binary polyhedral groups, see Section 3 (pages 5161-5163) of "Topological lensing in spherical spaces", Classical and Quantum Gravity 18 (2001) 5155-5186 online at www.arxiv.org/gr-qc/0106033 (Someday I plan to write up a different explanation of these groups, but for now I hope the cited reference will serve well enough.) In any case, many thanks to Jesper for pointing out the error in the earlier e-mail. Best wishes to all, Jeff www.geometrygames.org/contact.html

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Jesper tells why
by Jeff Weeks 07 Feb '06

07 Feb '06

P.P.S. I like Jesper's response because it hints at *why* the various levels contain the number of dodecahedra that they do. In effect, Jesper is showing the relation between the binary icosahedral group (which is the holonomy group of the Poincaré dodecahedral space) and the ordinary dodecahedral group (the symmetries of the original dodecahedron at layer 1). The relationship, which Jesper's table clearly shows, is that if you travel in the direction of a symmetry axis (of the original layer 1 dodecahedron) of order n, you'll find 2n translates of the dodecahedron spaced pi/n radians apart. This same relation holds, of course, for the other single-action spherical manifolds.

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120 cell
by Jeff Weeks 07 Feb '06

07 Feb '06

Dear Boud et al., You intuition is correct -- there are some additional in-between layers. The attached images show layers 1 through 5. Layers 6 through 9 are of course symmetrical as you work your way towards the south pole. Note that the cells in layer 5 sit "vertically" with respect to the equatorial hyperplane (i.e. they're orthogonal to the equatorial hyperplane) which is why they appear flat in the attached image (each dark blue hexagon is the 2D shadow of a 3D cell when you project from 4D space to 3D space). Anyhow, here's the accounting: layer 1 1 layer 2 12 layer 3 20 layer 4 12 layer 5 30 layer 6 12 layer 7 20 layer 8 12 layer 9 1 ----------- total 120 Note that all numbers are "dodecahedal numbers", respecting the dodecahedral symmetry of the whole construction. The 120-cell is quite beautiful, isn't it? Best wishes to all of you, Jeff

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"vertical" faces
by Jeff Weeks 07 Feb '06

07 Feb '06

Boud, >i guess another slightly confusing thing is in the picture of layer 4, >where the hexagons look "flat" whereas if i understand correctly, these >should be concave in order that layer 5 cells can be stuck on here. No, the hexagons on the surface of layer 4 really are flat (not concave) ! Indeed they must be flat, because the 30 dodecahedra in layer 5 are completely flat (zero volume!) in this projection. The situation is analogous to placing a cube on a tabletop and then projecting it orthogonally onto that tabletop. The cube's four side faces project to line segments. Those projections have zero area, even though the side faces themselves have positive area. The analogous thing happens with the orthogonal projection of the dodecahedron from 4D to 3D. The 30 dodecahedra near the equator have positive volume, but because they are perpendicular to the 3D space we're projecting onto, their shadows (projections) have zero volume. I hope that's clear! Best wishes, Jeff P.S. I'm not sure whether my message which bounced yesterday has made it to the list yet or not, so for sake of completeness I'll reattach those images to the present message for the sake of the cc readers. Please forgive the duplication.

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SV: [Cos-top] MG11 - Topology of the Universe (fwd) (and PDS - how many tiles?)
by Gundermann, Jesper 07 Feb '06

07 Feb '06

Dear Boud >A tiling of S^3 by the PDS has 120 copies of the PDS. >If we think of this tiling starting from "here" at the "North Pole" >(sorry Marcelo, this is North bias ;) (image 1) and we "stick" a copy >(image 2) of the PDS next to itself along a North-Pole-South-Pole axis >(e.g. X), and then moving along the X axis, we stick another copy (image 3) next >to image 2, and then another and so on, then it seems to me that image 6 >covers the "South Pole" and is the "antipode" copy of image 1. >In other words, the "distance" between the poles is 5 copies of the FD. >This allows an easy order of magnitude check on the total number of copies, >since the "radius" of one hemi-hypersphere is 2.5 "copies", so that the >total volume is roughly (using flat space arithmetic): >2* (4\pi/3) (2.5)^3 \sim 2* 2^2 * (5/2)^3 = 5^3 = 125 >which is a good approximation to 120. >(AFAIR, Jeff Weeks showed me how to do this - this is not original.) >Now for my question: >We have 6 "layers": >North pole - 1 FD >layer 2 - 12 FD's >layer 3 - n FD's L>ayer 4 - n FD's >layer 5 - 12 FD's >South pole - 1 >Clearly layer 3 or 4 has more than 12 copies, there are some spaces that >need to be filled. >But the value required for the total to be 120, i.e. n= 47, is an odd >number. Intuition says that everything should be even and symmetric. >Can someone give a nice intuitive explanation for why n is odd and why >this 12-symmetry is broken? Maybe a nice diagram? >In fact, i have a partial answer: it seems to me there are some in-between >layers, which we might call 2.5 and 4.5 and i guess also 3.5. But it's >difficult for me to imagine them, and it's still difficult for me to imagine >how the 12-symmetry is broken. >It would be nice if someone had a nice diagram or intuitive >explanation of how the 12-symmetry is broken. You were right about the intermediate layers, in fact, there are 9 layers Note that the central dodecahedron has 12 faces, 30 edges and 20 vertices. The distance of the centers of The 120 dodecahedrons are Distance from origin: direction number 0 1 pi/5 midpoint faces 12 Pi/3 vertices 20 2*pi/5 midpoint faces 12 Pi/2 midpoint edges 30 3*pi/5 midpoint faces 12 2*pi/3 vertices 20 4*pi/5 midpoint faces 12 Pi 1 Sum 120 Best regards Jesper Gundermann

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MG11 - Topology of the Universe (fwd) (and PDS - how many tiles?)
by Boud Roukema 06 Feb '06

06 Feb '06

hi everyone, The following is self-explanatory - you can reply to Marek directly - here is his email in antispam format (this list is publicly archived so better not write people's email addresses in ways that robots can read them): Marek.Demianski at fuw edu pl > ---------- Forwarded message ---------- > Date: Mon, 6 Feb 2006 15:02:38 +0100 (CET) > From: Marek Demianski > Subject: MG11 - Topology of the Universe > > > Dear Colleague, > > I have been asked to organize a parallel session on "Topology of the > Universe" at the XI Marcel Grossmann Meeting in Berlin, July 23 > - 29, 2006. If you are planing to attend the meeting and would like to > present a contribution at this session please let me know. Please spread this > information among your friends and collaborators. > > More information about the Meeting (and on line registration) you can > find at www.icra.it/MG/mg11. Please register by May 31 > to save 50EURO on the conference fee. > > Sincerely yours, > Marek Demianski BTW: i have a question on PDS (Poincare Dodecahedral Space) intuition which someone might be able to answer. A tiling of S^3 by the PDS has 120 copies of the PDS. If we think of this tiling starting from "here" at the "North Pole" (sorry Marcelo, this is North bias ;) (image 1) and we "stick" a copy (image 2) of the PDS next to itself along a North-Pole-South-Pole axis (e.g. X), and then moving along the X axis, we stick another copy (image 3) next to image 2, and then another and so on, then it seems to me that image 6 covers the "South Pole" and is the "antipode" copy of image 1. In other words, the "distance" between the poles is 5 copies of the FD. This allows an easy order of magnitude check on the total number of copies, since the "radius" of one hemi-hypersphere is 2.5 "copies", so that the total volume is roughly (using flat space arithmetic): 2* (4\pi/3) (2.5)^3 \sim 2* 2^2 * (5/2)^3 = 5^3 = 125 which is a good approximation to 120. (AFAIR, Jeff Weeks showed me how to do this - this is not original.) Now for my question: We have 6 "layers": North pole - 1 FD layer 2 - 12 FD's layer 3 - n FD's layer 4 - n FD's layer 5 - 12 FD's South pole - 1 Clearly layer 3 or 4 has more than 12 copies, there are some spaces that need to be filled. But the value required for the total to be 120, i.e. n= 47, is an odd number. Intuition says that everything should be even and symmetric. Can someone give a nice intuitive explanation for why n is odd and why this 12-symmetry is broken? Maybe a nice diagram? In fact, i have a partial answer: it seems to me there are some in-between layers, which we might call 2.5 and 4.5 and i guess also 3.5. But it's difficult for me to imagine them, and it's still difficult for me to imagine how the 12-symmetry is broken. It would be nice if someone had a nice diagram or intuitive explanation of how the 12-symmetry is broken. Going to a 122 tiling would save the tiling symmetry, but intuition says that 120 is better than 122. On the other hand, 2 * (1 + 12 + 48) = 122 seems simpler in terms of tiling symmetry. In the Nature paper, 120 is quoted: http://arxiv.org/abs/astro-ph/0310253 (After all, we *have* made a few minor mistakes in the past in our subject - e.g. when we said that there were uncountably many negatively curved 3-manifolds, and eventually, at MG IX (?), Jeff explained and showed us that the number was countably infinite - the classification is complicated and unfinished, but the number is countable.) cheers boud

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Mota et al. astro-ph/0503683
by Boud Roukema 01 Apr '05

01 Apr '05

hi cosmic topology friends, IMHO this mailing list can be used something like a continuous workshop, with a mix of new results, informal discussion, links to software and data. The question of whether we want to risk talking about "unpublished ideas" is something that i guess will evolve by practice. At a workshop, we almost certainly give each other lots of new ideas - and who gets the credit for a one sentence comment over lunch during many hours of discussion that probably gets redigested into the back of your mind and then you think it's your own idea? In this case, the list is publicly archived - after any of us send an email to cos-top at cosmo.torun.pl, we should be able to check what was sent a few minutes later here: http://cosmo.torun.pl/pipermail/cos-top/ so even though this does not count as formal "journal" publication, at least in our community we know who mentioned what idea, when. In fact, my feeling is serious discussion on this list might lead to many-author papers. After all, SNeIa and CMB people published important papers with 30-40 or however many authors, so surely it's better to publish a very good paper with many authors than to miss results because we're not working together... Anyway, Marcelo and his colleagues just put out a nice article about the observational situation in the limit | Omega_tot - 1 | << 1 (there's an obvious error in the abstract stating that the discussion concerns | Omega_tot - 1 | << 0 ;-) http://arxiv.org/abs/astro-ph/0503683 cheers boud PS: Hi Taro - we had a small workshop in Paris a few weeks ago and decided to start a publicly archived mailing list + a wiki page - feel free to join the list at: http://cosmo.torun.pl/mailman/listinfo/cos-top (see Subscribing to Cos-top) and/or register and edit the twiki page(s): http://cosmo.torun.pl/cgi-bin/twiki/view/Cosmo/TopologyMeudon2005March

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