Editing Distance geometry (section)

== History ==
The first result in distance geometry is [[Heron's formula]], from 1st century AD, which gives the area of a triangle from the distances between its 3 vertices. [[Brahmagupta's formula]], from 7th century AD, generalizes it to [[cyclic quadrilateral]]s. [[Niccolò Fontana Tartaglia|Tartaglia]], from 16th century AD, generalized it to give the [[Niccolò Fontana Tartaglia#Volume of a tetrahedron|volume of tetrahedron]] from the distances between its 4 vertices.

The modern theory of distance geometry began with [[Arthur Cayley]] and [[Karl Menger]].<ref>{{Cite journal|last1=Liberti|first1=Leo|last2=Lavor|first2=Carlile|date=2016|title=Six mathematical gems from the history of distance geometry|journal=International Transactions in Operational Research|language=en|volume=23|issue=5|pages=897–920|doi=10.1111/itor.12170|issn=1475-3995|arxiv=1502.02816|s2cid=17299562 }}</ref> Cayley published the Cayley determinant in 1841,<ref>{{Cite journal|last=Cayley|first=Arthur|date=1841|title=On a theorem in the geometry of position|journal=Cambridge Mathematical Journal|volume=2|pages=267–271}}</ref> which is a special case of the general Cayley–Menger determinant. Menger proved in 1928 a characterization theorem of all semimetric spaces that are isometrically embeddable in the ''n''-dimensional [[Euclidean space]] <math>\mathbb{R}^n</math>.<ref>{{Cite journal|last=Menger|first=Karl|date=1928-12-01|title=Untersuchungen über allgemeine Metrik|journal=Mathematische Annalen|language=de|volume=100|issue=1|pages=75–163|doi=10.1007/BF01448840|s2cid=179178149 |issn=1432-1807}}</ref><ref name=":0">{{Cite journal|last1=Blumenthal|first1=L. M.|last2=Gillam|first2=B. E.|date=1943|title=Distribution of Points in ''n''-Space|url=https://www.tandfonline.com/doi/pdf/10.1080/00029890.1943.11991349|journal=The American Mathematical Monthly|language=en|volume=50|issue=3|pages=181|doi=10.2307/2302400|jstor=2302400}}</ref> In 1931, Menger used distance relations to give an axiomatic treatment of Euclidean geometry.<ref>{{Cite journal|last=Menger|first=Karl|date=1931|title=New Foundation of Euclidean Geometry|journal=American Journal of Mathematics|volume=53|issue=4|pages=721–745|doi=10.2307/2371222|issn=0002-9327|jstor=2371222}}</ref>

[[Leonard Blumenthal]]'s book<ref name="blumenthal" /> gives a general overview for distance geometry at the graduate level, a large part of which is treated in English for the first time when it was published.