Editing Octonion (section)

==Applications==
The octonions play a significant role in the classification and construction of other mathematical entities. For example, the [[exceptional Lie group]] {{math|[[G2 (mathematics)|''G''<sub>2</sub>]]}} is the automorphism group of the octonions, and the other exceptional Lie groups {{math|[[F4 (mathematics)|''F''<sub>4</sub>]]}}, {{math|[[E6 (mathematics)|''E''<sub>6</sub>]]}}, {{math|[[E7 (mathematics)|''E''<sub>7</sub>]]}} and {{math|[[E8 (mathematics)|''E''<sub>8</sub>]]}} can be understood as the isometries of certain [[projective plane]]s defined using the octonions.<ref>Baez (2002), section 4.</ref> The set of [[self-adjoint]] 3 × 3 octonionic [[matrix (mathematics)|matrices]], equipped with a symmetrized matrix product, defines the [[Albert algebra]]. In [[discrete mathematics]], the octonions provide an elementary derivation of the [[Leech lattice]], and thus they are closely related to the [[sporadic simple groups]].<ref>{{cite journal|last=Wilson |first=Robert A. |author-link=Robert Arnott Wilson |title=Octonions and the Leech lattice |journal=[[Journal of Algebra]] |volume=322 |issue=6 |date=2009-09-15 |pages=2186–2190 |doi=10.1016/j.jalgebra.2009.03.021 |url=http://www.maths.qmul.ac.uk/%7Eraw/pubs_files/octoLeech1rev.pdf}}</ref><ref>{{cite journal|last=Wilson |first=Robert A. |author-link=Robert Arnott Wilson |title=Conway's group and octonions |journal=Journal of Group Theory |date=2010-08-13 |doi=10.1515/jgt.2010.038 |volume=14 |pages=1–8 |s2cid=16590883 |url=http://www.maths.qmul.ac.uk/~raw/pubs_files/octoConway.pdf}}</ref>

Applications of the octonions to physics have largely been conjectural. For example, in the 1970s, attempts were made to understand [[quark]]s by way of an octonionic [[Hilbert space]].<ref>{{cite journal|last1=Günaydin |first1=M. |last2=Gürsey |first2=F. |author-link2=Feza Gürsey |year=1973 |title=Quark structure and octonions |journal=[[Journal of Mathematical Physics]] |volume=14 |issue=11 |pages=1651–1667 |doi=10.1063/1.1666240|bibcode=1973JMP....14.1651G }}<br />{{cite journal|last1=Günaydin |first1=M. |last2=Gürsey |first2=F. |author-link2=Feza Gürsey |year=1974 |title=Quark statistics and octonions |journal=[[Physical Review D]] |volume=9 |issue=12 |pages=3387–3391 |doi=10.1103/PhysRevD.9.3387|bibcode=1974PhRvD...9.3387G }}</ref> It is known that the octonions, and the fact that only four normed division algebras can exist, relates to the [[spacetime]] dimensions in which [[supersymmetry|supersymmetric]] [[quantum field theory|quantum field theories]] can be constructed.<ref>{{cite journal|last1=Kugo |first1=Taichiro |last2=Townsend |first2=Paul |title=Supersymmetry and the division algebras |journal=[[Nuclear Physics B]] |volume=221 |issue=2 |date=1983-07-11 |pages=357–380 |doi=10.1016/0550-3213(83)90584-9|bibcode=1983NuPhB.221..357K |url=https://cds.cern.ch/record/140183 }}</ref><ref>{{cite encyclopedia|last1=Baez |first1=John C. |author-link1=John C. Baez |last2=Huerta |first2=John |title=Division Algebras and Supersymmetry I |arxiv=0909.0551 |encyclopedia=Superstrings, Geometry, Topology, and C*-algebras |publisher=[[American Mathematical Society]] |year=2010 |editor-last1=Doran |editor-first1=R. |editor-last2=Friedman |editor-first2=G. |editor-last3=Rosenberg |editor-first3=J.}}</ref> Also, attempts have been made to obtain the [[Standard Model]] of elementary particle physics from octonionic constructions, for example using the "Dixon algebra" <math>\ \mathbb C \otimes \mathbb H \otimes \mathbb O ~.</math><ref name=wolchover>{{cite magazine |last=Wolchover |first=Natalie |author-link=Natalie Wolchover |date=2018-07-20 |title=The peculiar math that could underlie the laws of nature |website=[[Quanta Magazine]] |url=https://www.quantamagazine.org/the-octonion-math-that-could-underpin-physics-20180720/ |access-date=2018-10-30}}</ref><ref>{{cite journal |last=Furey |first=Cohl |author-link=Cohl Furey |date=2012-07-20 |title=Unified theory of ideals |journal=[[Physical Review D]] |volume=86 |issue=2 |page=025024 |doi=10.1103/PhysRevD.86.025024 |arxiv=1002.1497 |bibcode=2012PhRvD..86b5024F |s2cid=118458623 }}<br />{{cite journal|last=Furey |first=Cohl |author-link=Cohl Furey |date=2018-10-10 |title=Three generations, two unbroken gauge symmetries, and one eight-dimensional algebra |journal=[[Physics Letters B]] |volume=785 |pages=84–89 |doi=10.1016/j.physletb.2018.08.032 |bibcode=2018PhLB..785...84F |arxiv=1910.08395 |s2cid=126205768 }}<br/>{{cite journal|last=Stoica |first=O.C. |title=Leptons, quarks, and gauge from the complex Clifford algebra <math>\mathbb{C}\ell_6</math> |journal=[[Advances in Applied Clifford Algebras]] |year=2018 |doi=10.1007/s00006-018-0869-4 |volume=28 |page=52 |arxiv=1702.04336 |s2cid=125913482 }}<br />{{cite conference |last=Gresnigt |first=Niels G. |title=Quantum groups and braid groups as fundamental symmetries |conference=European Physical Society conference on High Energy Physics, 5–12 July 2017, Venice, Italy |date=2017-11-21 |arxiv=1711.09011}}<br />{{cite book |last=Dixon |first=Geoffrey M. |year=1994 |title=Division Algebras: Octonions, quaternions, complex numbers, and the algebraic design of physics |publisher=[[Springer-Verlag]] |doi=10.1007/978-1-4757-2315-1 |isbn=978-0-7923-2890-2 |oclc=30399883 }}<br/>{{cite web |last=Baez |first=John C. |author-link=John C. Baez |date=2011-01-29 |title=The Three-Fold Way (part 4) |access-date=2018-11-02 |website=[[The n-Category Café]] |url=https://golem.ph.utexas.edu/category/2011/01/the_threefold_way_part_4_1.html}}</ref>

Octonions have also arisen in the study of [[black hole entropy]], [[quantum information science]],<ref>{{cite journal|last1=Borsten |first1=Leron |last2=Dahanayake |first2=Duminda |last3=Duff |first3=Michael J. |author-link3=Michael Duff (physicist) |last4=Ebrahim |first4=Hajar |last5=Rubens |first5=Williams |title=Black holes, qubits and octonions |journal=[[Physics Reports]] |volume=471 |issue=3–4 |year=2009 |pages=113–219 |arxiv=0809.4685|doi=10.1016/j.physrep.2008.11.002 |bibcode=2009PhR...471..113B |s2cid=118488578 }}</ref><ref>{{cite journal|last1=Stacey |first1=Blake C. |title=Sporadic SICs and the Normed Division Algebras |journal=[[Foundations of Physics]] |year=2017 |volume=47 |issue=8 |pages=1060–1064 |doi=10.1007/s10701-017-0087-2 |arxiv=1605.01426 |bibcode=2017FoPh...47.1060S|s2cid=118438232 }}</ref>  [[string theory]],<ref>{{Cite web|url=https://www.newscientist.com/article/mg20327232-100-beyond-space-and-time-8d-surfers-paradise/|title=Beyond space and time: 8D – Surfer's paradise|website=New Scientist}}</ref> and [[Digital image processing|image processing]].<ref>{{cite journal | url=https://ieeexplore.ieee.org/document/10552342 | doi=10.1109/LSP.2024.3411934 | bibcode=2024ISPL...31.1615J | title=Octonion Phase Retrieval | last1=Jacome | first1=Roman | last2=Mishra | first2=Kumar Vijay | last3=Sadler | first3=Brian M. | last4=Arguello | first4=Henry | journal=IEEE Signal Processing Letters | date=2024 | volume=31 | page=1615 | arxiv=2308.15784 }}</ref>

Octonions have been used in solutions to the [[hand eye calibration problem]] in [[robotics]].<ref>{{cite journal |first1=J. |last1=Wu |first2=Y. |last2=Sun |first3=M. |last3=Wang and |first4=M. |last4=Liu |title=Hand-Eye Calibration: 4-D Procrustes Analysis Approach |journal=IEEE Transactions on Instrumentation and Measurement |volume=69 |issue=6 |pages=2966–81 |date=June 2020 |doi=10.1109/TIM.2019.2930710 |bibcode=2020ITIM...69.2966W |s2cid=201245901 }}</ref>

Deep octonion networks provide a means of efficient and compact expression in machine learning applications.<ref>{{cite journal |first1=J. |last1=Wu |first2=L. |last2=Xu |first3=F. |last3=Wu |first4=Y. |last4=Kong |first5=L. |last5=Senhadji |first6=H. |last6=Shu |title=Deep octonion networks |journal=Neurocomputing |volume=397 |issue= |pages=179–191 |date=2020 |doi=10.1016/j.neucom.2020.02.053 |s2cid=84186686 |id=hal-02865295|doi-access=free }}</ref><ref>{{Cite journal |title=Marine Debris Segmentation Using a Parameter Efficient Octonion-Based Architecture  |date=2023 |doi=10.1109/lgrs.2023.3321177 |last1=Bojesomo |first1=Alabi |last2=Liatsis |first2=Panos |last3=Almarzouqi |first3=Hasan |journal=IEEE Geoscience and Remote Sensing Letters |volume=20 |pages=1–5 |bibcode=2023IGRSL..2021177B |doi-access=free }}</ref>