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A '''Picard horn''', also called the '''Picard topology''' or '''Picard model''', is one of the oldest known [[hyperbolic geometry|hyperbolic]] [[manifold|3-manifold]]s, first described by [[Émile Picard]]<ref name="EmilePicard">{{cite web |url=http://www.academie-sciences.fr/activite/archive/dossiers/Picard/Picard_oeuvre.htm |title=Émile Picard - Académie des sciences |accessdate=2011-09-26 |url-status=dead |archiveurl=https://web.archive.org/web/20120330102950/http://www.academie-sciences.fr/activite/archive/dossiers/Picard/Picard_oeuvre.htm |archivedate=2012-03-30 }}</ref> in 1884.<ref name="picard1884">{{cite journal|author= Émile Picard|author-link= Émile Picard | language = French |title= Sur un groupe de transformations des points de l'espace situés du même côté d'un plan |journal= Bulletin de la Société Mathématique de France |volume=12 |pages=43–47 |date=1884-03-07 |url= http://www.numdam.org/item?id=BSMF_1884__12__43_0 |accessdate =2011-08-24}}</ref> The manifold is the quotient of the [[Poincaré half-plane model|upper half-plane model of hyperbolic 3-space]] by the [[projective linear group|projective special linear group]], <math>\operatorname{PSL}_2(\mathbf{Z}[i])</math>. It was proposed as a model for the |
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A '''Picard horn''', also called the '''Picard topology''' or '''Picard model''', is a theoretical model for the |
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[[shape of the |
[[shape of the universe]] in 2004.<ref name="Aurich0403597" /> The term "horn" is due to [[pseudosphere]] models of hyperbolic space. |
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==Geometry and topology== |
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| ⚫ | The term was coined by Ralf Aurich, Sven Lustig, Frank Steiner, and Holger Then in their paper ''Hyperbolic Universes with a Horned Topology and the CMB Anisotropy''.<ref name="Aurich0403597">{{cite journal|last= Aurich |first= Ralf | |
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| ⚫ | A modern description, in terms of fundamental domain and identifications, can be found in section 3.2, page 63 of Grunewald and Huntebrinker, along with the first 80 eigenvalues of the [[Laplacian]], tabulated on page 72, where <math>\Upsilon_1</math> is a fundamental domain of the Picard space.<ref name="GrunHunte">Fritz Grunewald and Wolfgang Huntebrinker, ''[http://projecteuclid.org/euclid.em/1047591148 A numerical study of eigenvalues of the hyperbolic Laplacian for polyhedra with one cusp]'', Experiment. Math. Volume 5, Issue 1 (1996), 57-80</ref> |
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==Cosmology== |
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The space in question is the quotient of the [[Poincaré half-plane model|upper half-plane model of hyperbolic 3-space]] by the group <math>\operatorname{PSL}_2(\mathbf{Z}[i])</math>, which was first described by Émile Picard<ref name="EmilePicard">http://www.academie-sciences.fr/activite/archive/dossiers/Picard/Picard_oeuvre.htm</ref> in 1884.<ref name="picard1884">{{cite journal|author= [[Emile Picard]] | language = [[French language|French]] |title= Sur un groupe de transformations des points de l'espace situés du même côté d'un plan |journal= Bulletin de la Société Mathématique de France |volume=12 |pages=43-37 |date=1884-03-07 |url= http://www.numdam.org/item?id=BSMF_1884__12__43_0 |accessdate =2011-08-24}}</ref> |
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| ⚫ | The term was coined in 2004 by Ralf Aurich, Sven Lustig, Frank Steiner, and Holger Then in their paper ''Hyperbolic Universes with a Horned Topology and the CMB Anisotropy''.<ref name="Aurich0403597">{{cite journal|last= Aurich |first= Ralf |author2=Lustig, S. |author3=Steiner, F. |author4=Then, H. |title= Hyperbolic Universes with a Horned Topology and the CMB Anisotropy |journal= [[Classical and Quantum Gravity]] |volume=21 |issue= 21 |pages=4901–4926|date=2004 |doi= 10.1088/0264-9381/21/21/010 |arxiv = astro-ph/0403597 |bibcode = 2004CQGra..21.4901A |s2cid= 17619026 }}</ref> |
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| ⚫ | The model was chosen in an attempt to describe the [[microwave background radiation]] apparent in the universe, and has finite [[volume]] and useful spectral characteristics (the first several eigenvalues of the Laplacian are computed and in good accord with observation). In this model one end of the figure curves [[wiktionary:finite|finitely]] into the bell of the horn. The curve along any side of horn is considered to be a [[hyperbola|negative curve]]. The other end extends to infinity.<ref name="Register2004" /><ref name="NSci2004" /> |
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| ⚫ | A modern description, in terms of fundamental domain and identifications, can be found in section 3.2, page 63 of Fritz Grunewald and Wolfgang Huntebrinker, ''[http://projecteuclid.org/euclid.em/1047591148 A numerical study of eigenvalues of the hyperbolic Laplacian for polyhedra with one cusp]'', Experiment. Math. Volume 5, Issue 1 (1996), 57-80 |
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==See also== |
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| ⚫ | The model was |
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* [[Gabriel's Horn]] |
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==References== |
==References== |
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{{reflist|31em|refs= |
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* {{ cite news |
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<ref name="Register2004">{{cite news |
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| url = |
| url = https://www.theregister.co.uk/2004/05/27/universe_picard_topology/ |
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| title = Boffins trumpet horn shaped universe |
| title = Boffins trumpet horn shaped universe |
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| last = Sherriff | first = Lucy |
| last = Sherriff | first = Lucy |
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| date = 2004-05-27 |
| date = 2004-05-27 |
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| accessdate = 2006-12-28 |
| accessdate = 2006-12-28 |
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}} |
}}</ref> |
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<ref name="NSci2004">{{cite news |
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| url= |
| url=https://www.newscientist.com/article/dn4879-big-bang-glow-hints-at-funnelshaped-universe.html |
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| title = Big Bang glow hints at funnel-shaped Universe |
| title = Big Bang glow hints at funnel-shaped Universe |
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| last = Battersby | first = Stephen |
| last = Battersby | first = Stephen |
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| date = 2004-04-15 |
| date = 2004-04-15 |
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| accessdate = 2007-12-01 |
| accessdate = 2007-12-01 |
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}}</ref> |
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}} |
}} |
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<references /> |
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{{Manifolds}} |
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| ⚫ | |||
[[Category:3-manifolds]] |
[[Category:3-manifolds]] |
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[[Category:Hyperbolic geometry]] |
[[Category:Hyperbolic geometry]] |
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| ⚫ | |||
Latest revision as of 06:59, 21 August 2022
A Picard horn, also called the Picard topology or Picard model, is one of the oldest known hyperbolic 3-manifolds, first described by Émile Picard[1] in 1884.[2] The manifold is the quotient of the upper half-plane model of hyperbolic 3-space by the projective special linear group, . It was proposed as a model for the shape of the universe in 2004.[3] The term "horn" is due to pseudosphere models of hyperbolic space.
![{\displaystyle \operatorname {PSL} _{2}(\mathbf {Z} [i])}](https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/a5f6f313b6977b5c3b269fd159ea92c7f219133f)
Geometry and topology
[edit]A modern description, in terms of fundamental domain and identifications, can be found in section 3.2, page 63 of Grunewald and Huntebrinker, along with the first 80 eigenvalues of the Laplacian, tabulated on page 72, where is a fundamental domain of the Picard space.[4]
Cosmology
[edit]The term was coined in 2004 by Ralf Aurich, Sven Lustig, Frank Steiner, and Holger Then in their paper Hyperbolic Universes with a Horned Topology and the CMB Anisotropy.[3]
The model was chosen in an attempt to describe the microwave background radiation apparent in the universe, and has finite volume and useful spectral characteristics (the first several eigenvalues of the Laplacian are computed and in good accord with observation). In this model one end of the figure curves finitely into the bell of the horn. The curve along any side of horn is considered to be a negative curve. The other end extends to infinity.[5][6]
See also
[edit]References
[edit]- ^ "Émile Picard - Académie des sciences". Archived from the original on 2012-03-30. Retrieved 2011-09-26.
- ^ Émile Picard (1884-03-07). "Sur un groupe de transformations des points de l'espace situés du même côté d'un plan". Bulletin de la Société Mathématique de France (in French). 12: 43–47. Retrieved 2011-08-24.
- ^ a b Aurich, Ralf; Lustig, S.; Steiner, F.; Then, H. (2004). "Hyperbolic Universes with a Horned Topology and the CMB Anisotropy". Classical and Quantum Gravity. 21 (21): 4901–4926. arXiv:astro-ph/0403597. Bibcode:2004CQGra..21.4901A. doi:10.1088/0264-9381/21/21/010. S2CID 17619026.
- ^ Fritz Grunewald and Wolfgang Huntebrinker, A numerical study of eigenvalues of the hyperbolic Laplacian for polyhedra with one cusp, Experiment. Math. Volume 5, Issue 1 (1996), 57-80
- ^ Sherriff, Lucy (2004-05-27). "Boffins trumpet horn shaped universe". The Register. Retrieved 2006-12-28.
- ^ Battersby, Stephen (2004-04-15). "Big Bang glow hints at funnel-shaped Universe". New Scientist. Retrieved 2007-12-01.