# Picard horn

> Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Picard_horn
> Markdown URL: https://mediated.wiki/source/Picard_horn.md
> Source: https://en.wikipedia.org/wiki/Picard_horn
> Source revision: 1351988943
> License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/)

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](/source/Hyperbolic_geometry) [3-manifolds](/source/Manifold), first described by [Émile Picard](/source/%C3%89mile_Picard)[1] in 1884.[2] The manifold is the quotient of the [upper half-plane model of hyperbolic 3-space](/source/Poincar%C3%A9_half-plane_model) by the [projective special linear group](/source/Projective_linear_group), PSL 2 ⁡ ( Z [ i ] ) {\displaystyle \operatorname {PSL} _{2}(\mathbf {Z} [i])} . It was proposed as a model for the [shape of the universe](/source/Shape_of_the_universe) in 2004.[3] The term "horn" is due to [pseudosphere](/source/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](/source/Laplacian), tabulated on page 72, where Υ 1 {\displaystyle \Upsilon _{1}} is a fundamental domain of the Picard space.[4]

## 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*.[3]

The model was chosen in an attempt to describe the [microwave background radiation](/source/Microwave_background_radiation) apparent in the universe, and has finite [volume](/source/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](https://en.wiktionary.org/wiki/finite) into the bell of the horn. The curve along any side of horn is considered to be a [negative curve](/source/Hyperbola). The other end extends to infinity.[5][6]

## See also

- [Gabriel's horn](/source/Gabriel's_horn)

## References

1. **[^](#cite_ref-EmilePicard_1-0)** ["Émile Picard - Académie des sciences"](https://web.archive.org/web/20120330102950/http://www.academie-sciences.fr/activite/archive/dossiers/Picard/Picard_oeuvre.htm). Archived from [the original](http://www.academie-sciences.fr/activite/archive/dossiers/Picard/Picard_oeuvre.htm) on 2012-03-30. Retrieved 2011-09-26.

1. **[^](#cite_ref-picard1884_2-0)** [Émile Picard](/source/%C3%89mile_Picard) (1884-03-07). ["Sur un groupe de transformations des points de l'espace situés du même côté d'un plan"](http://www.numdam.org/item?id=BSMF_1884__12__43_0). *Bulletin de la Société Mathématique de France* (in French). **12**: 43–47. Retrieved 2011-08-24.

1. ^ [***a***](#cite_ref-Aurich0403597_3-0) [***b***](#cite_ref-Aurich0403597_3-1) Aurich, Ralf; Lustig, S.; Steiner, F.; Then, H. (2004). "Hyperbolic Universes with a Horned Topology and the CMB Anisotropy". *[Classical and Quantum Gravity](/source/Classical_and_Quantum_Gravity)*. **21** (21): 4901–4926. [arXiv](/source/ArXiv_(identifier)):[astro-ph/0403597](https://arxiv.org/abs/astro-ph/0403597). [Bibcode](/source/Bibcode_(identifier)):[2004CQGra..21.4901A](https://ui.adsabs.harvard.edu/abs/2004CQGra..21.4901A). [doi](/source/Doi_(identifier)):[10.1088/0264-9381/21/21/010](https://doi.org/10.1088%2F0264-9381%2F21%2F21%2F010). [S2CID](/source/S2CID_(identifier)) [17619026](https://api.semanticscholar.org/CorpusID:17619026).

1. **[^](#cite_ref-GrunHunte_4-0)** Fritz Grunewald and Wolfgang Huntebrinker, *[A numerical study of eigenvalues of the hyperbolic Laplacian for polyhedra with one cusp](http://projecteuclid.org/euclid.em/1047591148)*, Experiment. Math. Volume 5, Issue 1 (1996), 57-80

1. **[^](#cite_ref-Register2004_5-0)** Sherriff, Lucy (2004-05-27). ["Boffins trumpet horn shaped universe"](https://www.theregister.co.uk/2004/05/27/universe_picard_topology/). *[The Register](/source/The_Register)*. Retrieved 2006-12-28.

1. **[^](#cite_ref-NSci2004_6-0)** Battersby, Stephen (2004-04-15). ["Big Bang glow hints at funnel-shaped Universe"](https://www.newscientist.com/article/dn4879-big-bang-glow-hints-at-funnelshaped-universe.html). *[New Scientist](/source/New_Scientist)*. Retrieved 2007-12-01.

v t e Manifolds (Glossary, List, Category) Basic concepts Topological manifold Atlas Differentiable/Smooth manifold Differential structure Smooth atlas Submanifold Riemannian manifold Smooth map Submersion Pushforward Tangent space Differential form Vector field Main theorems (list) Atiyah–Singer index Darboux's De Rham's Frobenius Generalized Stokes Hopf–Rinow Noether's Sard's Whitney embedding Maps Curve Diffeomorphism Local Geodesic Exponential map in Lie theory Foliation Immersion Integral curve Lie derivative Section Submersion Types of manifolds Calabi–Yau Closed Collapsing Complete (Almost) Complex (Almost) Contact Einstein Fibered Finsler (Almost, Ricci-) Flat G-structure Hadamard Hermitian Hyperbolic (Hyper) Kähler Kenmotsu Lie group Lie algebra Manifold with boundary Nilmanifold Oriented Parallelizable Poisson Prime Quaternionic Hypercomplex (Pseudo-, Sub-) Riemannian Rizza Sasakian Stein (Almost) Symplectic Tame Tensors Vectors Distribution Lie bracket Pushforward Tangent space bundle Torsion Vector field Vector flow Covectors Closed/Exact Covariant derivative Cotangent space bundle De Rham cohomology Differential form Complex Vector-valued One-form Exterior derivative Interior product Pullback Ricci curvature flow Riemann curvature tensor Tensor field density Volume form Wedge product Bundles Adjoint Affine Associated Cotangent Dual Fiber (Co-) Fibration Jet Lie algebra (Stable) Normal Principal Spinor Subbundle Tangent Tensor Vector Connections Affine Cartan Ehresmann Form Generalized Koszul Levi-Civita Principal Vector Parallel transport Related Classification of manifolds Gauge theory History Morse theory Moving frame Singularity theory Generalizations Banach Diffeology Diffiety Fréchet Hilbert K-theory Non-Hausdorff Orbifold Secondary calculus over commutative algebras Sheaf Stratifold Supermanifold Stratified space

---
Adapted from the Wikipedia article [Picard horn](https://en.wikipedia.org/wiki/Picard_horn) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Picard_horn?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.