Editing Ultrametric space (section)

== Applications ==
* A [[contraction mapping]] may then be thought of as a way of approximating the final result of a computation (which can be guaranteed to exist by the [[Banach fixed-point theorem]]). Similar ideas can be found in [[domain theory]]. [[p-adic analysis|''p''-adic analysis]] makes heavy use of the ultrametric nature of the [[p-adic number|''p''-adic metric]].
* In [[condensed matter physics]], the [[self-averaging]] overlap between spins in the [[Spin glass#Sherrington–Kirkpatrick model|SK Model]] of [[spin glasses]] exhibits an ultrametric structure, with the solution given by the full replica symmetry breaking procedure first outlined by [[Giorgio Parisi]] and coworkers.<ref>Mezard, M; Parisi, G; and Virasoro, M: ''SPIN GLASS THEORY AND BEYOND'', World Scientific, 1986. {{ISBN|978-9971-5-0116-7}}</ref> Ultrametricity also appears in the theory of aperiodic solids.<ref name=physics_apps>{{cite journal |last1=Rammal |first1=R. |last2=Toulouse |first2=G. |last3=Virasoro |first3=M. |title=Ultrametricity for physicists |journal=Reviews of Modern Physics |year=1986 |volume=58 |issue=3 |pages=765–788 |doi=10.1103/RevModPhys.58.765 |url=http://rmp.aps.org/abstract/RMP/v58/i3/p765_1 |access-date=20 June 2011|bibcode=1986RvMP...58..765R }}</ref>
* In [[Taxonomy (biology)|taxonomy]] and [[phylogenetic tree]] construction, ultrametric distances are also utilized by the [[UPGMA]] and [[WPGMA]] methods.<ref>Legendre, P. and Legendre, L. 1998. Numerical Ecology. Second English Edition. Developments in Environmental Modelling 20. Elsevier, Amsterdam.</ref> These algorithms require a constant-rate assumption and produce trees in which the distances from the root to every branch tip are equal. When [[DNA]], [[RNA]] and [[protein]] data are analyzed, the ultrametricity assumption is called the [[molecular clock]].
* Models of [[intermittency]] in three dimensional [[turbulence]] of fluids make use of so-called cascades, and in discrete models of dyadic cascades, which have an ultrametric structure.<ref> {{cite journal |last1=Benzi|first1=R. |last2=Biferale |first2=L. |last3=Trovatore |first3=E. |title=Ultrametric Structure of Multiscale Energy Correlations in Turbulent Models |journal=Physical Review Letters |year=1997 |volume=79 |issue=9 |pages=1670–1674 |doi=10.1103/PhysRevLett.79.1670 |arxiv=chao-dyn/9705018 |bibcode=1997PhRvL..79.1670B |s2cid=53120932 }}</ref>
* In [[geography]] and [[landscape ecology]], ultrametric distances have been applied to measure landscape complexity and to assess the extent to which one landscape function is more important than another.<ref>{{Cite journal|last=Papadimitriou|first=Fivos|year=2013|title=Mathematical modelling of land use and landscape complexity with ultrametric topology|journal=Journal of Land Use Science|language=en|volume=8|issue=2|pages=234–254|doi=10.1080/1747423x.2011.637136|s2cid=121927387|issn=1747-423X|doi-access=free}}</ref>