{{Short description|Hyperbolic 3-manifold proposed as a model for the shape of the universe}} 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 [[shape of the universe]] in 2004.<ref name="Aurich0403597" /> The term "horn" is due to [[pseudosphere]] models of hyperbolic space.
==Geometry and topology== 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>
==Cosmology== 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>
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" />
==See also== * [[Gabriel's horn]]
==References== {{reflist|31em|refs= <ref name="Register2004">{{cite news | url = https://www.theregister.co.uk/2004/05/27/universe_picard_topology/ | title = Boffins trumpet horn shaped universe | last = Sherriff | first = Lucy | work = [[The Register]] | date = 2004-05-27 | accessdate = 2006-12-28 }}</ref> <ref name="NSci2004">{{cite news | url=https://www.newscientist.com/article/dn4879-big-bang-glow-hints-at-funnelshaped-universe.html | title = Big Bang glow hints at funnel-shaped Universe | last = Battersby | first = Stephen | work = [[New Scientist]] | date = 2004-04-15 | accessdate = 2007-12-01 }}</ref> }}
{{Manifolds}}
[[Category:3-manifolds]] [[Category:Hyperbolic manifolds]] [[Category:Physical cosmology]]