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{{Infobox number
| number = 72
|divisor=1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72
}}
'''72''' ('''seventy-two''') is the [[Natural number|natural number]] following [[71 (number)|71]] and preceding [[73 (number)|73]]. It is half a [[Gross (unit)|gross]] or 6 [[Dozen|dozen]] (i.e., 60 in [[Duodecimal|duodecimal]]).

==In mathematics==
Seventy-two is a [[Pronic number|pronic number]], as it is the product of [[8|8]] and [[9|9]].<ref>{{Cite OEIS |A002378 |Oblong (or promic, pronic, or heteromecic) numbers |access-date=2023-06-15 }}</ref> It is the smallest [[Achilles number]], as it's a [[Powerful number|powerful number]] that is not itself a [[Exponentiation|power]].<ref>{{Cite OEIS |A052486 |Achilles numbers - powerful but imperfect. |access-date=2022-10-22 }}</ref>

72 is an [[Abundant number|abundant number]].<ref>{{Cite OEIS |A005101 |Abundant numbers (sum of divisors of m exceeds 2m). |access-date=2022-10-22 }}</ref> With exactly twelve positive divisors, including [[12 (number)|12]] (one of only two [[Sublime number|sublime number]]s),<ref>{{Cite OEIS |A081357 |Sublime numbers, numbers for which the number of divisors and the sum of the divisors are both perfect. |access-date=2023-06-15 }}</ref> 72 is also the twelfth member in the sequence of [[Refactorable number|refactorable number]]s.<ref>{{Cite OEIS |A033950 |Refactorable numbers: number of divisors of k divides k. Also known as tau numbers. |access-date=2023-06-15 }}
:The sequence of refactorable numbers goes: 1, 2, 8, 9, 12, 18, 24, 36, 40, 56, 60, 72, 80, 84, 88, 96, ...</ref> 72 has a Euler totient of [[24 (number)|24]],<ref>{{Cite OEIS |A000010 |Euler totient function. |access-date=2022-10-22 }}</ref> which makes it a [[Highly totient number|highly totient number]], as there are [[17 (number)|17]] solutions to the equation φ(''x'') = 72, more than any integer below 72.<ref name="highly totient">{{Cite OEIS |A097942 |Highly totient numbers. |access-date=2022-10-22 }}</ref> It is equal to the sum of its preceding smaller highly totient numbers 24 and [[48 (number)|48]], and contains the first [[6|six]] highly totient numbers [[1|1]], [[2|2]], [[4|4]], [[8|8]], 12 and 24 as a subset of its proper divisors. [[144 (number)|144]], or twice 72, is also highly totient, as is 576, the [[Square (algebra)|square]] of 24.<ref name="highly totient" /> While 17 different integers have a totient value of 72, the sum of [[Euler's totient function]] φ(''x'') over the first [[15 (number)|15]] integers is 72.<ref>{{Cite OEIS |A002088 |Sum of totient function. |access-date=2022-10-22 }}</ref> It also is a ''[[Perfect number|perfect]]'' indexed [[Harshad number]] in [[Decimal|decimal]] (twenty-eighth), as it is divisible by the sum of its digits ([[9|9]]).<ref>{{Cite OEIS |A005349 |Niven (or Harshad, or harshad) numbers: numbers that are divisible by the sum of their digits. |access-date=2022-10-22 }}</ref>

{{Bullet list

| 72 is the second multiple of 12, after 48, that is not a sum of [[Twin prime|twin prime]]s.<br />
It is, however, the sum of four consecutive [[Prime number|primes]] {{math|1=(13 + 17 + 19 + 23) }},<ref>{{Cite OEIS |A034963 |Sums of four consecutive primes. |access-date=2023-12-02 }}</ref> as well as the sum of six consecutive primes {{math|1=(5 + 7 + 11 + 13 + 17 + 19) }}.<ref>{{Cite OEIS |A127333 |Numbers that are the sum of 6 consecutive primes. |access-date=2023-12-02 }}</ref>
| 72 is the first number that can be expressed as the difference of the squares of primes in just two distinct ways: {{math|1=11<sup>2</sup> − 7<sup>2</sup> = 19<sup>2</sup> − 17<sup>2</sup> }}.<ref>{{Cite OEIS |A090788 |Numbers that can be expressed as the difference of the squares of primes in just two distinct ways. |access-date=2024-01-03 }}</ref>
| 72 is the smallest number whose fifth power is the sum of five smaller fifth powers: 19<sup>5</sup> + 43<sup>5</sup> + 46<sup>5</sup> + 47<sup>5</sup> + 67<sup>5</sup> = 72<sup>5</sup>.<ref>David Wells: The Penguin Dictionary of Curious and Interesting Numbers</ref>
| 72 is the [[Magic constant|magic constant]] of the first non-normal, full [[Prime reciprocal magic square|prime reciprocal magic square]] in [[Decimal|decimal]], based on {{math|1={{sfrac|1|17}} }} in a 16 × 16 grid.<ref>{{Cite journal |last=Subramani |first=K. |title=On two interesting properties of primes, p, with reciprocals in base 10 having maximum period p - 1. |url=https://jmscm.smartsociety.org/volume1_issue2/Paper4.pdf |journal=J. Of Math. Sci. & Comp. Math. |eissn=2644-3368 |volume=1 |issue=2 |year=2020 |pages=198–200 |publisher=S.M.A.R.T. |location=Auburn, WA |doi=10.15864/jmscm.1204 |s2cid=235037714 }}</ref><ref>{{Cite OEIS |A007450 |Decimal expansion of 1/17. |access-date=2023-11-24 }}</ref><br />
72 is the number of distinct {{math|{7/2} }} [[Magic star|magic heptagrams]], all with a magic constant of 30.<ref>{{Cite OEIS |A200720 |Number of distinct normal magic stars of type {n/2}. |access-date=2022-12-09 }}</ref>
| 72 is the sum of the eighth row of [[Lozanić's triangle]], and equal to the sum of the previous four rows (36, 20, 10, 6).<ref>{{Cite OEIS |A005418 |...row sums of Losanitsch's triangle. |access-date=2022-10-22 }}</ref><br />
As such, this row is the third and largest to be in equivalence with a sum of consecutive {{math|1=''k'' }} row sums, after (1, 2, 3; 6) and (6, 10, 20; 36).
| 72 is the number of degrees in the [[Central angle|central angle]] of a regular pentagon, which is [[Constructible polygon|constructible]] with a compass and straight-edge.

}}

72 plays a role in the [[Rule of 72]] in [[Finance:Economics|economics]] when approximating annual [[Compound interest|compounding]] of [[Finance:Interest rate|interest rate]]s of a round 6% to 10%, due in part to its high number of divisors.

'''Inside <math>\mathrm E_{n}</math> Lie algebras''':

* 72 is the number of [[Vertex (geometry)|vertices]] of the [[Uniform 6-polytope|six-dimensional]] [[1 22 polytope|'''1<sub>22</sub>''']] [[Polytope|polytope]], which also contains as [[Facet (geometry)|facets]] [[720 (number)|720]] [[Edge (geometry)|edges]], 702 polychoral 4-faces, of which [[270 (number)|270]] are four-dimensional [[16-cell|16-cell]]s, and two sets of [[27 (number)|27]] demipenteract ''5''-faces. These 72 vertices are the ''root vectors'' of the [[Simple group|simple]] [[Lie group]] [[E6 (mathematics)|<math>\mathrm E {6}</math>]], which as a [[Honeycomb (geometry)|honeycomb]] under [[2 22 honeycomb|'''2<sub>22</sub>''']] forms the <math>\mathrm E {6}</math> lattice. '''1<sub>22</sub>''' is part of a family of [[Duoprism#k 22 polytopes|'''k<sub>22</sub>''']] polytopes whose first member is the fourth-dimensional [[3-3 duoprism|3-3 duoprism]], of symmetry order 72 and made of six [[Triangular prism|triangular prism]]s. On the other hand, [[3 21 polytope|'''3<sub>21</sub>''']] ∈ '''k<sub>21</sub>''' is the only [[Semiregular polytope|semiregular polytope]] in the seventh dimension, also featuring a total of 702 ''6''-faces of which 576 are [[6-simplex|6-simplex]]es and 126 are [[6-orthoplex|6-orthoplex]]es that contain [[60 (number)|60]] edges and [[12 (number)|12]] vertices, or collectively '''72''' one-dimensional and two-dimensional [[Simplex#Elements|elements]]; with [[126 (number)|126]] the number of ''root vectors'' in <math>\mathrm E_{7}</math>, which are contained in the vertices of [[2 31 polytope|2<sub>31</sub>]] ∈ [[6-demicube#Related polytopes|'''k<sub>31</sub>''']], also with 576 or 24<sup>2</sup> 6-simplexes like '''3<sub>21</sub>'''. The triangular prism is the root polytope in the '''k<sub>21</sub>''' family of polytopes, which is the simplest semiregular polytope, with '''k<sub>31</sub>''' rooted in the analogous four-dimensional [[Tetrahedral prism|tetrahedral prism]] that has four triangular prisms alongside two tetrahedra as cells.
*The complex [[Hessian polyhedron]] in [[Complex coordinate space|<math>\mathbb{C}^3</math>]] contains 72 regular complex triangular edges, as well as 27 [[Polygon|polygon]]al [[Möbius–Kantor polygon|Möbius–Kantor]] faces and 27 vertices. It is notable for being the [[Vertex figure|vertex figure]] of the [[Witting polytope|complex Witting polytope]], which shares [[240 (number)|240]] vertices with the eight-dimensional [[Semiregular polytope|semiregular]] '''4<sub>21</sub>''' polytope whose vertices in turn represent the ''root vectors'' of the [[Simple group|simple]] [[Lie group]] <math>\mathrm E {8}</math>.

There are 72 compact and paracompact [[Coxeter group]]s of ranks four through ten: 14 of these are compact finite representations in only [[Three-dimensional space|three-dimensional]] and [[Four-dimensional space|four-dimensional]] spaces, with the remaining 58 paracompact or noncompact ''infinite'' representations in dimensions three through nine. These terminate with three paracompact groups in the [[E9 honeycomb|ninth dimension]], of which the most important is <math>\tilde {T}_{9}</math>: it contains the final [[Semiregular polytope|semiregular]] hyperbolic honeycomb [[E9 honeycomb|'''6<sub>21</sub>''']] made of only [[Facet (geometry)|regular facets]] and the [[5 21 honeycomb|'''5<sub>21</sub>''']] [[Euclidean space|Euclidean]] honeycomb as its [[Vertex figure|vertex figure]], which is the geometric representation of the [[E8 lattice|<math>\mathrm E {8}</math> lattice]]. Furthermore, <math>\tilde {T}_{9}</math> shares the same fundamental symmetries with the Coxeter-Dynkin [[En (Lie algebra)|over-extended]] form <math>\mathrm E_{8}</math><sup>'''++'''</sup>  equivalent to the [[Uniform 10-polytope#Regular and uniform hyperbolic honeycombs|tenth-dimensional]] symmetries of Lie algebra <math>\mathrm E_{10}</math>.

72 lies between the '''8'''th pair of [[Twin prime|twin prime]]s ([[71 (number)|71]], [[73 (number)|73]]), where 71 is the largest [[Supersingular prime (moonshine theory)|supersingular prime]] that is a factor of the largest [[Sporadic group|sporadic group]] (the [[Monster group|friendly giant]] <math>\mathbb {F_{1}}</math>), and 73 the largest [[Sequence|indexed member]] of a [[Definite quadratic form|definite quadratic]] [[Integer matrix|integer matrix]] representative of all prime numbers<ref>{{Cite OEIS |A154363 |Numbers from Bhargava's prime-universality criterion theorem }} 
:{2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 67, '''73'''}</ref>{{efn|1=Where 71 is also the largest prime number less than 73 that is not a member of this set. }} that is also the number of distinct orders (without [[Multiplicity (mathematics)|multiplicity]]) inside all 194 [[Conjugacy class|conjugacy class]]es of <math>\mathbb {F_{1}}</math>.<ref>{{Cite arXiv |last1=He |first1=Yang-Hui |last2=McKay |first2=John |title=Sporadic and Exceptional |year=2015 |pages=20 |class=math.AG |eprint=1505.06742  }}</ref> Sporadic groups are a family of twenty-six finite simple groups, where <math>\mathrm E_{6}</math>, <math>\mathrm E_{7}</math>, and <math>\mathrm E_{8}</math> are associated [[Exceptional Lie algebra|exceptional groups]] that are part of sixteen [[Finite group|finite]] Lie groups that are also '''simple''', or non-trivial groups whose only [[Normal subgroup|normal subgroup]]s are the [[Trivial group|trivial group]] and the groups themselves.{{efn|1=The only other finite simple groups are the infinite families of [[Cyclic group|cyclic group]]s and alternating groups. An exception is the [[Tits group]] <math>\mathbb T</math>, which is sometimes considered a 17th ''non-strict'' group of Lie type that can otherwise more loosely classify as a 27th sporadic group. }}

==In science==
*The [[Physics:Atomic number|atomic number]] of [[Chemistry:Hafnium|hafnium]]
*In degrees Fahrenheit considered to be [[Physics:Room temperature|room temperature]].{{citation needed|date=April 2023}}

===In astronomy===
*[[Astronomy:Messier object|Messier object]] [[Astronomy:Messier 72|M72]], a magnitude 10.0 [[Astronomy:Globular cluster|globular cluster]] in the [[Astronomy:Constellation|constellation]] [[Astronomy:Aquarius (constellation)|Aquarius]].
*The [[Astronomy:New General Catalogue|New General Catalogue]] object [[Astronomy:NGC 72|NGC 72]], a magnitude 13.5 [[Astronomy:Barred spiral galaxy|barred spiral galaxy]] in the constellation [[Astronomy:Andromeda (constellation)|Andromeda]].
*The precession of equinoxes traces out a pair of cones joined at their apices in a cycle of approximately 26,000 years, that is 1 degree every 72 years (approximation to the nearmost integer).

==In religion==
*The number of languages spoken at the Tower of Babylon, according to later tradition.
*The conventional number of scholars translating the Septuagint, according to the legendary account in the "Letter of Aristeas".
*The number of companions of [[Biography:Zoroaster|Zoroaster]] who were martyred.<ref>{{cite news |title=صدویک نام خداوند چیست؟ |url=https://amordadnews.com/8276/ |access-date=8 January 2023 |agency=Amordad |date=22 May 2018 |language=fa-IR}}</ref>
*The conventional number of disciples sent forth by Jesus in ''Luke'' 10 in some manuscripts (seventy in others).
*The number of names of God, according to [[Unsolved:Kabbalah|Kabbalah]] (see [[Unsolved:Names of God in Judaism|names of God in Judaism]]).
*The Shemhamphorasch related to the number of the names of God.
*The total number of books in the [[Religion:Bible|Bible]] in the Catholic version if the ''Book of Lamentations'' is considered part of the ''Book of Jeremiah''.
*The current distribution of the [[Religion:Book of Revelation|Book of Revelation]] is 22 chapters, adopted since the 13th century,  but the oldest known division of the text is that of the Greek commentator Andrew of Cesary (6th century) in 72 chapters.
*The number of people martyred along with Imam Hussain at the [[Religion:Battle of Karbala|Battle of Karbala]].
*The degrees of the [[Religion:Jacob's Ladder|Jacob's Ladder]] were to the number of 72, according to the Zohar.
*The 72 disciples of [[Biography:Confucius|Confucius]] who mastered his teachings (also given as 77).
*Mahavira, the twenty-fourth and last [[Religion:Tirthankara|tirthankara]] of [[Philosophy:Jainism|Jainism]], is said to have attained nirvana after his physical death at the age of 72.
*[[Religion:Thoth|Thoth]], in an Egyptian creation myth, wins a 72nd of each day of the year from the Moon in a game of draughts, as a favour to Nut, the Sky Goddess. He uses these portions to make the five intercalary days on which the remaining Gods and Goddesses are born.<ref>{{cite book |last1=Hart |first1=George |title=Egyptian Myths |year=1990 |url=https://archive.org/details/egyptianmyths00hart/page/40/mode/2up?q=plutarch |pages=40–41 |publisher=University of Texas Press |isbn=0-292-72076-9}}</ref><ref>{{cite book |last1=Hart |first1=George |title=A Dictionary of Egyptian Gods and Goddesses |year=1990 |pages=144–145 |publisher=University of Texas Press |isbn=9780292720763 |url=https://archive.org/details/egyptianmyths00hart/page/40/mode/2up?q=plutarch}}</ref><ref>{{cite book |last1=Plutarch |title=Isis and Osiris |publisher=Loeb Classics |pages=LCL 306: 30–31 |url=https://www.loebclassics.com/view/plutarch-moralia_isis_osiris/1936/pb_LCL306.31.xml}}</ref>
*The good god [[Biology:Osiris|Osiris]] was enclosed in a coffin by 72 evil disciples and accomplices of Set.<ref>"Egyptian Myths", George Hart, p41, University of Texas Press, 1990</ref>
*At the age of the puberty, the young Parsee received the investiture of the sacred cord Kucti made of 72 linens in symbol of the community.
*In Cao Đài, the number of planets between hell and heaven.
*There are 72 stupas which comprise [[Astronomy:Borobudur|Borobudur]], the world's largest Buddhist temple.
*72 major temples have been found at Angkor, seat of the ancient Khmer Empire.
*In Islam, 72 is the number of sects or denominations that are doomed to Hell, according to ''Hadith'' (Sayings of prophet Muhammad).<ref name="Islam and Sects">{{cite web |url=http://www.islamandsects.com |title=Sects In Islam - 73 Groups in Islam, Division - Denominations |url-status=dead |access-date=24 March 2013 |archive-url=https://web.archive.org/web/20130506043338/http://www.islamandsects.com/ |archive-date=6 May 2013 }}</ref><ref>Sunan Ibn Maajah, no. 3982 "My Ummah will be divided into seventy-three sects, one of which will be in Paradise and seventy-two will be in the Fire"</ref>
*The number of demons sealed away by King Solomon with [[Unsolved:The Lesser Key of Solomon|The Lesser Key of Solomon]].

==In other fields==

'''Seventy-two''' is also:
*In [[Dots per inch|dots per inch]] (dpi), the default screen resolution for an image or graphic on an [[Software:Apple Macintosh|Apple Macintosh]] screen.
*In typography, a [[Point (typography)|point]] is 1/72 inch.
*The number of the French department Sarthe.
*The registry of the U.S. Navy's nuclear [[Engineering:Aircraft carrier|aircraft carrier]] {{USS|Abraham Lincoln|CVN-72}}, named after U.S. President Abraham Lincoln.
*The designation of the Soviet [[Engineering:T-72|T-72]] tank.
*The [[Rule of 72]] in [[Finance:Finance|finance]].
*Book: ''72 Hour Hold'' by Bebe Moore Campbell
*CD: ''Seventy Two & Sunny'' by Uncle Kracker
*Jill Clayburgh and LeVar Burton starred in ''Firestorm: 72 Hours in Oakland'' (1993)
*Alternative music band The Delta 72
*The Persian classical santur, a hammered dulcimer, has 72 strings in 24 triple-stringed courses.
*The Turin Brakes song, also known as Emergency 72
*The number of members in National Senate of Argentina.
*A Civil Air Patrol unit in Laramie, WY, RMR-WY-072.
*There are 72 demons and other spirits in the [[Religion:Goetia|goetia]] ''[[Unsolved:The Lesser Key of Solomon|The Lesser Key of Solomon]]''.
*A common limit for [[Characters per line|characters per line]] in computing
*72 equal temperament is a tuning used in Byzantine music and by some modern composers.
*The number of microseasons in the traditional Japanese calendar<ref>https://www.nippon.com/en/features/h00124/</ref>

==In sports and games==
*The usual par for an 18-hole golf course, especially those in tournament play.
*The number of spaces in a game of Parcheesi, from start space to "home".

==Notes==
{{Notelist}}

==References==

{{Reflist}}

==External links==
* [https://www.bbc.co.uk/news/magazine-14217443 Go Figure: What can 72 tell us about life], ''BBC News'', 20 July 2011


[[Category:Integers]]

{{Sourceattribution|72 (number)}}