View source for 27 (number)

{{Use mdy dates|date=October 2022}}{{Infobox number
| number = 27
| divisor = 1, 3, 9, 27
|ordinal=27th}}
'''27''' ('''twenty-seven''') is the natural number following [[26 (number)|26]] and preceding [[28 (number)|28]].

== Mathematics ==
Including the null-motif, there are 27 distinct [[Network motif|hypergraph motifs]].<ref>{{Cite book |editor1-last=Balazinska  |editor1-first=Magdalena |editor2-last=Zhou |editor2-first=Xiaofang  |last1=Lee |first1=Geon  |last2=Ko |first2=Jihoon |last3=Shin |first3=Kijung |chapter=Hypergraph Motifs: Concepts, Algorithms, and Discoveries |title=46th International Conference on Very Large Data Bases |url=http://www.vldb.org/pvldb/volumes/13/ |series=Proceedings of the VLDB Endowment |volume=13 |year=2020 |pages= 2256–2269 |publisher=[[Association for Computing Machinery#Portal and Digital Library|ACM Digital Library]] |isbn=9781713816126 |oclc=1246551346 |arxiv=2003.01853 |doi=10.14778/3407790.3407823 |s2cid=221779386 }}</ref>

[[File:Clebsch diagonal cubic surface.png|260px|thumb|The [[Clebsch surface]], with 27 straight lines]]

There are exactly [[27 lines on a cubic surface|''twenty-seven straight lines'']] on a smooth [[cubic surface]],<ref>{{Cite web |last=Baez |first=John Carlos |author-link=John C. Baez |title=27 Lines on a Cubic Surface |website=AMS Blogs |url=https://blogs.ams.org/visualinsight/2016/02/15/27-lines-on-a-cubic-surface/ |publisher=[[American Mathematical Society]] |date=February 15, 2016 |access-date=2023-10-31 }}</ref> which give a basis of the [[fundamental representation]] of [[E6 (mathematics)|Lie algebra]] <math>\mathrm {E_{6}}</math>.<ref>{{Cite journal |last=Aschbacher |first=Michael |author-link=Michael Aschbacher |title=The 27-dimensional module for E<sub>6</sub>. I |journal=[[Inventiones Mathematicae]] |publisher=[[Springer Science+Business Media|Springer]] |location=Heidelberg, DE |year=1987 |volume=89 |pages=166–172 |doi=10.1007/BF01404676 |bibcode=1987InMat..89..159A |mr=0892190 |zbl=0629.20018  |s2cid=121262085 }}</ref><ref>{{Cite OEIS |A121737 |Dimensions of the irreducible representations of the simple Lie algebra of type E6 over the complex numbers, listed in increasing order. |access-date=2023-10-31 }}</ref>

The unique simple formally real [[Jordan algebra]], the exceptional Jordan algebra of self-adjoint [[Square matrix|3 by 3 matrices]] of [[quaternion]]s, is 27-dimensional;<ref>{{Cite journal |last=Kac |first=Victor Grigorievich |author-link=Victor Kac |title=Classification of Simple Z-Graded Lie Superalgebras and Simple Jordan Superalgebras |journal=[[Communications in Algebra]] |volume=5 |issue=13 |publisher=[[Taylor & Francis]] |year=1977 |page=1380 |doi=10.1080/00927877708822224 |mr=0498755 |s2cid=122274196 |zbl=0367.17007 }}</ref> its [[automorphism group]] is the 52-dimensional [[exceptional Lie algebra]] <math>\mathrm {F_{4}}.</math><ref>{{Cite journal |last1=Baez |first1=John Carlos |author-link=John Baez |title=The Octonions |url=https://www.ams.org/journals/bull/2002-39-02/S0273-0979-01-00934-X/ |journal=[[Bulletin of the American Mathematical Society]] |volume=39 |issue=2 |year=2002 |pages=189–191 |publisher=[[American Mathematical Society]] |location=Providence, RI |doi=10.1090/S0273-0979-01-00934-X |doi-access=free |mr=1886087 |zbl=1026.17001 |s2cid=586512 }}</ref>

There are twenty-seven [[sporadic group]]s, if the [[Tits group|''non-strict'' group of Lie type]] <math>\mathrm {T}</math> (with an [[Faithful representation|irreducible representation]] that is twice that of <math>\mathrm {F_{4}}</math> in 104 dimensions)<ref>{{Cite journal |last = Lubeck |first = Frank |title = Smallest degrees of representations of exceptional groups of Lie type |url = https://www.tandfonline.com/doi/abs/10.1081/AGB-100002175 |journal = [[Communications in Algebra]] |volume = 29 |issue = 5 |page = 2151 |publisher = [[Taylor & Francis]] |location = Philadelphia, PA |year = 2001 |doi = 10.1081/AGB-100002175 |mr = 1837968 |s2cid = 122060727 |zbl = 1004.20003 }}</ref> is included.<ref>{{Cite journal |last1=Hartley |first1=Michael I. |last2=Hulpke |first2=Alexander |title=Polytopes Derived from Sporadic Simple Groups |url=https://cdm.ucalgary.ca/article/view/61945/46662  |journal=Contributions to Discrete Mathematics |volume=5 |number=2 |publisher=[[University of Calgary]] Department of Mathematics and Statistics |issn=1715-0868 |location=Alberta, CA |year=2010 |page=27 |doi=10.11575/cdm.v5i2.61945 |doi-access=free |mr=2791293 |zbl=1320.51021 |s2cid=40845205 }}</ref>

In [[Robin's theorem]] for the [[Riemann hypothesis]], twenty-seven integers fail to hold <math>\sigma(n) < e^\gamma n \log \log n </math> for values <math>n \leq 5040,</math> where <math>\gamma</math> is the [[Euler–Mascheroni constant]]; this hypothesis is true [[if and only if]] this inequality holds for every larger <math>n.</math><ref>{{Cite journal |last=Axler |first=Christian |title=On Robin's inequality |journal=[[The Ramanujan Journal]] |publisher=[[Springer Science+Business Media|Springer]] |location=Heidelberg, GE |year=2023 |volume=61 |issue=3 |pages=909–919 |doi=10.1007/s11139-022-00683-0 |doi-access=free |arxiv=2110.13478 |bibcode=2021arXiv211013478A |zbl=1532.11010 |s2cid=239885788 }}</ref><ref>{{Cite journal |last1=Robin |first1=Guy |title=Grandes valeurs de la fonction somme des diviseurs et hypothèse de Riemann |language=French |url=http://zakuski.utsa.edu/~jagy/Robin_1984.pdf |journal=[[Journal de Mathématiques Pures et Appliquées]] |series= Neuvième Série |volume=63 |issue=2 |year=1984 |pages=187–213|issn=0021-7824 |mr=774171 |zbl=0516.10036 }}</ref><ref>{{Cite OEIS |A067698 |Positive integers such that sigma(n) is greater than or equal to exp(gamma) * n * log(log(n)). |access-date=2023-10-31 }}</ref>

The [[Clebsch surface]] has 27 [[Exceptional line|exceptional lines]] can be defined over the real numbers.

It is possible to arrange 27 vertices and connect them with edges to create the [[Holt graph]].

==In other fields==
*The [[27 club]] refers to the age when many popular music figures died. It connects the age 27 with connotations of an early end.{{Citation needed|date=January 2025}}

== See also ==
* [[72 (number)]] &ndash; 27 reversed

== Notes ==
{{Notelist}}

== References ==
{{Reflist}}

=== Further reading ===
Wells, D. ''[[The Penguin Dictionary of Curious and Interesting Numbers]]'' London: Penguin Group. (1987), p.&nbsp;106.

== External links ==
* [http://primes.utm.edu/curios/page.php/27.html Prime Curios! 27] from the [[Prime Pages]]
* [http://27.chrismore.com Mystery of the number 27] - Large collection of 27 related trivia and facts.
* [http://www.27project.org The 27 Project] - collection of sightings of 27 in movies, TV, culture and art

{{Integers|zero}}

{{DEFAULTSORT:27 (Number)}}
[[Category:Integers]]