72 (number)
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72 (seventy-two) is the
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called '' cardinal ...
following 71 and preceding 73. It is half a gross or 6
dozen A dozen (commonly abbreviated doz or dz) is a grouping of twelve. The dozen may be one of the earliest primitive integer groupings, perhaps because there are approximately a dozen cycles of the Moon, or months, in a cycle of the Sun, or year ...
(i.e., 60 in
duodecimal The duodecimal system (also known as base 12, dozenal, or, rarely, uncial) is a positional notation numeral system using twelve as its base. The number twelve (that is, the number written as "12" in the decimal numerical system) is instead wr ...
).


In mathematics

Seventy-two is a
pronic number A pronic number is a number that is the product of two consecutive integers, that is, a number of the form n(n+1).. The study of these numbers dates back to Aristotle. They are also called oblong numbers, heteromecic numbers,. or rectangular number ...
, as it is the product of 8 and 9. 72 is an
abundant number In number theory, an abundant number or excessive number is a number for which the sum of its proper divisors is greater than the number. The integer 12 is the first abundant number. Its proper divisors are 1, 2, 3, 4 and 6 for a total of 16. Th ...
, with a total of 12 factors, and a
Euler totient In number theory, Euler's totient function counts the positive integers up to a given integer that are relatively prime to . It is written using the Greek letter phi as \varphi(n) or \phi(n), and may also be called Euler's phi function. In o ...
of 24. 72 is also a
highly totient number A highly totient number k is an integer that has more solutions to the equation \phi(x) = k, where \phi is Euler's totient function, than any integer below it. The first few highly totient numbers are 1, 2, 4, 8, 12, 24, 48, 72, 144, 240, 4 ...
, as there are 17 solutions to the equation φ(''x'') = 72, more than any integer below 72. It is equal to the sum the sum of its preceding smaller highly totient numbers 24 and 48, and contains the first six highly totient numbers 1, 2, 4, 8, 12 and 24 as a subset of its
proper divisor In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...
s. 144, or twice 72, is also highly totient, as is 576, the
square In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90- degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length a ...
of 24. While 17 different integers have a totient value of 72, the sum of
Euler's totient function In number theory, Euler's totient function counts the positive integers up to a given integer that are relatively prime to . It is written using the Greek letter phi as \varphi(n) or \phi(n), and may also be called Euler's phi function. In ...
φ(''x'') over the first 15 integers is 72. 72 is also a
Harshad number In mathematics, a harshad number (or Niven number) in a given number base is an integer that is divisible by the sum of its digits when written in that base. Harshad numbers in base are also known as -harshad (or -Niven) numbers. Harshad number ...
in
decimal The decimal numeral system (also called the base-ten positional numeral system and denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hindu–Arabic numeral ...
, as it is divisible by the sum of its digits. *72 is the smallest
Achilles number An Achilles number is a number that is powerful but not a perfect power. A positive integer is a powerful number if, for every prime factor of , is also a divisor. In other words, every prime factor appears at least squared in the factori ...
, as it's a powerful number that is not itself a power. *72 is the sum of four consecutive
primes A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
(13 + 17 + 19 + 23), as well as the sum of six consecutive primes (5 + 7 + 11 + 13 + 17 + 19). *72 is the smallest number whose fifth power is the sum of five smaller fifth powers: 195 + 435 + 465 + 475 + 675 = 725. *72 is the sum of the eighth row of Lozanić's triangle. *72 is the number of distinct magic heptagrams, all with a
magic constant The magic constant or magic sum of a magic square is the sum of numbers in any row, column, or diagonal of the magic square. For example, the magic square shown below has a magic constant of 15. For a normal magic square of order ''n'' – that is ...
of 30. *72 is the number of degrees in the
central angle A central angle is an angle whose apex (vertex) is the center O of a circle and whose legs (sides) are radii intersecting the circle in two distinct points A and B. Central angles are subtended by an arc between those two points, and the arc le ...
of a
regular pentagon In geometry, a pentagon (from the Greek πέντε ''pente'' meaning ''five'' and γωνία ''gonia'' meaning ''angle'') is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°. A pentagon may be simpl ...
, which is constructible with a compass and straight-edge. 72 plays a role in the Rule of 72 in
economics Economics () is the social science that studies the production, distribution, and consumption of goods and services. Economics focuses on the behaviour and interactions of economic agents and how economies work. Microeconomics anal ...
when approximating annual
compounding In the field of pharmacy, compounding (performed in compounding pharmacies) is preparation of a custom formulation of a medication to fit a unique need of a patient that cannot be met with commercially available products. This may be done for me ...
of
interest rate An interest rate is the amount of interest due per period, as a proportion of the amount lent, deposited, or borrowed (called the principal sum). The total interest on an amount lent or borrowed depends on the principal sum, the interest rate, t ...
s of a round 6% to 10%, due in part to its high number of divisors. Inside \mathrm E_
Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identi ...
s: * 72 is the number of vertices of the six-dimensional 122
polytope In elementary geometry, a polytope is a geometric object with flat sides ('' faces''). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions as an ...
, which also contains as
facets A facet is a flat surface of a geometric shape, e.g., of a cut gemstone. Facet may also refer to: Arts, entertainment, and media * ''Facets'' (album), an album by Jim Croce * ''Facets'', a 1980 album by jazz pianist Monty Alexander and his tri ...
720 __NOTOC__ Year 720 ( DCCXX) was a leap year starting on Monday (link will display the full calendar) of the Julian calendar. The denomination 720 for this year has been used since the early medieval period, when the Anno Domini calendar era ...
edges,
702 __NOTOC__ Year 702 ( DCCII) was a common year starting on Sunday (link will display the full calendar) of the Julian calendar. The denomination 702 for this year has been used since the early medieval period, when the Anno Domini calendar era b ...
polychoral An antiphon (Greek ἀντίφωνον, ἀντί "opposite" and φωνή "voice") is a short chant in Christian ritual, sung as a refrain. The texts of antiphons are the Psalms. Their form was favored by St Ambrose and they feature prominently ...
4-faces, of which
270 __NOTOC__ Year 270 ( CCLXX) was a common year starting on Saturday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Antiochianus and Orfitus (or, less frequently, year 102 ...
are
four-dimensional A four-dimensional space (4D) is a mathematical extension of the concept of three-dimensional or 3D space. Three-dimensional space is the simplest possible abstraction of the observation that one only needs three numbers, called '' dimensions'' ...
16-cell In geometry, the 16-cell is the regular convex 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol . It is one of the six regular convex 4-polytopes first described by the Swiss mathematician Ludwig Schläfli in the ...
s, and two sets of 27
demipenteract In five-dimensional geometry, a demipenteract or 5-demicube is a semiregular 5-polytope, constructed from a ''5-hypercube'' ( penteract) with alternated vertices removed. It was discovered by Thorold Gosset. Since it was the only semiregular ...
''5''-faces. These 72 vertices are the '' root vectors'' of the
simple Simple or SIMPLE may refer to: *Simplicity, the state or quality of being simple Arts and entertainment * ''Simple'' (album), by Andy Yorke, 2008, and its title track * "Simple" (Florida Georgia Line song), 2018 * "Simple", a song by Johnn ...
Lie group In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the addi ...
\mathrm E_, which as a
honeycomb A honeycomb is a mass of hexagonal prismatic wax cells built by honey bees in their nests to contain their larvae and stores of honey and pollen. Beekeepers may remove the entire honeycomb to harvest honey. Honey bees consume about of honey ...
under 222 forms the \mathrm E_ lattice. 122 is part of a family of k22 polytopes whose first member is the fourth-dimensional
3-3 duoprism In the geometry of 4 dimensions, the 3-3 duoprism or triangular duoprism is a four-dimensional convex polytope. It can be constructed as the Cartesian product of two triangles and is the simplest of an infinite family of four-dimensional polytopes ...
, of symmetry order 72 and made of six
triangular prism In geometry, a triangular prism is a three-sided prism; it is a polyhedron made of a triangular base, a translated copy, and 3 faces joining corresponding sides. A right triangular prism has rectangular sides, otherwise it is ''oblique''. A ...
s. On the other hand, 321k21 is the only
semiregular polytope In geometry, by Thorold Gosset's definition a semiregular polytope is usually taken to be a polytope that is vertex-transitive and has all its facets being regular polytopes. E.L. Elte compiled a longer list in 1912 as ''The Semiregular Polyt ...
in the seventh dimension, also featuring a total of 702 ''6''-faces of which 576 are
6-simplex In geometry, a 6-simplex is a self-dual regular 6-polytope. It has 7 vertices, 21 edges, 35 triangle faces, 35 tetrahedral cells, 21 5-cell 4-faces, and 7 5-simplex 5-faces. Its dihedral angle is cos−1(1/6), or approximately 80.41°. Alte ...
es and 126 are
6-orthoplex In geometry, a 6-orthoplex, or 6-cross polytope, is a regular 6-polytope with 12 vertices, 60 edges, 160 triangle faces, 240 tetrahedron cells, 192 5-cell ''4-faces'', and 64 ''5-faces''. It has two constructed forms, the first being regular wi ...
es that contain 60 edges and 12 vertices, or collectively 72 one-dimensional and two-dimensional elements; with 126 the number of ''root vectors'' in \mathrm E_, which are contained in the vertices of 231k31, also with
576 __NOTOC__ Year 576 ( DLXXVI) was a leap year starting on Wednesday (link will display the full calendar) of the Julian calendar. The denomination 576 for this year has been used since the early medieval period, when the Anno Domini calendar era ...
or 242 6-simplexes like 321. The triangular prism is the root polytope in the k21 family of polytopes, which is the simplest semiregular polytope, with k31 rooted in the analogous four-dimensional
tetrahedral prism In geometry, a tetrahedral prism is a convex uniform 4-polytope. This 4-polytope has 6 polyhedral cells: 2 tetrahedra connected by 4 triangular prisms. It has 14 faces: 8 triangular and 6 square. It has 16 edges and 8 vertices. It is one of 18 u ...
that has four triangular prisms alongside two tetrahedra as
cells Cell most often refers to: * Cell (biology), the functional basic unit of life Cell may also refer to: Locations * Monastic cell, a small room, hut, or cave in which a religious recluse lives, alternatively the small precursor of a monastery w ...
. *The
complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...
Hessian polyhedron In geometry, the Hessian polyhedron is a regular complex polyhedron 333, , in \mathbb^3. It has 27 vertices, 72 3 edges, and 27 33 faces. It is self-dual. Coxeter named it after Ludwig Otto Hesse for sharing the '' Hessian configuration'' \ ...
in \mathbb^3 contains 72 regular complex triangular edges, as well as 27
polygon In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed '' polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two ...
al Möbius–Kantor faces and 27 vertices. It is notable for being the
vertex figure In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off. Definitions Take some corner or vertex of a polyhedron. Mark a point somewhere along each connected edge. Draw line ...
of the complex Witting polytope, which shares
240 __NOTOC__ Year 240 ( CCXL) was a leap year starting on Wednesday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Sabinus and Venustus (or, less frequently, year 993 ''Ab u ...
vertices with the eight-dimensional semiregular 421 polytope whose vertices in turn represent the ''root vectors'' of the
simple Simple or SIMPLE may refer to: *Simplicity, the state or quality of being simple Arts and entertainment * ''Simple'' (album), by Andy Yorke, 2008, and its title track * "Simple" (Florida Georgia Line song), 2018 * "Simple", a song by Johnn ...
Lie group In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the addi ...
\mathrm E_. There are 72
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in Britis ...
and
paracompact In mathematics, a paracompact space is a topological space in which every open cover has an open refinement that is locally finite. These spaces were introduced by . Every compact space is paracompact. Every paracompact Hausdorff space is normal ...
Coxeter group In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean refle ...
s of ranks four through ten: 14 of these are compact finite representations in only
three-dimensional Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called '' parameters'') are required to determine the position of an element (i.e., point). This is the inform ...
and
four-dimensional A four-dimensional space (4D) is a mathematical extension of the concept of three-dimensional or 3D space. Three-dimensional space is the simplest possible abstraction of the observation that one only needs three numbers, called '' dimensions'' ...
spaces, with the remaining 58 paracompact or noncompact ''infinite'' representations in dimensions three through nine. These terminate with three paracompact groups in the ninth dimension, of which the most important is \tilde _: it contains the final semiregular hyperbolic honeycomb 621 made of only regular facets and the 521 Euclidean honeycomb as its
vertex figure In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off. Definitions Take some corner or vertex of a polyhedron. Mark a point somewhere along each connected edge. Draw line ...
, which is the geometric representation of the \mathrm E_ lattice. Furthermore, \tilde _ shares the same fundamental symmetries with the Coxeter-Dynkin over-extended form \mathrm E_++ equivalent to the tenth-dimensional symmetries of Lie algebra \mathrm E_. 72 lies between the eighth pair of
twin prime A twin prime is a prime number that is either 2 less or 2 more than another prime number—for example, either member of the twin prime pair (41, 43). In other words, a twin prime is a prime that has a prime gap of two. Sometimes the term ''twin p ...
s ( 71, 73), where 71 is the largest supersingular prime that is a factor of the largest
sporadic group In mathematics, a sporadic group is one of the 26 exceptional groups found in the classification of finite simple groups. A simple group is a group ''G'' that does not have any normal subgroups except for the trivial group and ''G'' itself. The ...
, the friendly giant, with all primes
greater than or equal to In mathematics, an inequality is a relation which makes a non-equal comparison between two numbers or other mathematical expressions. It is used most often to compare two numbers on the number line by their size. There are several different ...
73 non-supersingular. Sporadic groups are a family of twenty-six
finite simple group Finite is the opposite of infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected or marke ...
s, where \mathrm E_, \mathrm E_, and \mathrm E_ 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 subgroups are the trivial group and the groups themselves.


In science

*The atomic number of hafnium *In degrees Fahrenheit considered to be room temperature.


In astronomy

*Messier object Messier 72, M72, a visual magnitude, magnitude 10.0 globular cluster in the constellation Aquarius (constellation), Aquarius. *The New General Catalogue]
object
NGC 72, a magnitude 13.5 barred spiral galaxy in the constellation 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 "Pseudo-Aristeas, Letter of Aristeas". *The conventional number of Seventy Apostles, disciples sent forth by Jesus in ''Luke'' 10 in some manuscripts (seventy in others). *The number of names of God, according to Kabbalah (see names of God in Judaism). *The Shemhamphorasch related to the number of the names of God. *The total number of books in the Bible in the Catholic version if the ''Book of Lamentations'' is considered part of the ''Book of Jeremiah''. *The current distribution of the 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 Battle of Karbala. *The number of houri each Muslim martyr (or every Muslim male, according to some ahadith) shall receive as companions in Paradise. *The degrees of the Jacob's Ladder were to the number of 72, according to the Zohar. *The 72 disciples of Confucius who mastered his teachings (also given as 77). *Mahavira, the twenty-fourth and last tirthankara of Jainism, is said to have attained nirvana after his physical death at the age of 72. *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 (goddess), Nut, the Sky Goddess. He uses these portions to make the five intercalary days on which the remaining Gods and Goddesses are born. *The good god Osiris was enclosed in a coffin by 72 evil disciples and accomplices of Set (mythology), Set. *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 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).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" *The number of demons sealed away by King Solomon with The Lesser Key of Solomon.


In other fields

Seventy-two is also: *In dots per inch (dpi), the default screen resolution for an image or graphic on an 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 aircraft carrier , named after U.S. President Abraham Lincoln. *The designation of the Soviet T-72 tank. *The Rule of 72 in 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 Santoor (Persian instrument), 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 goetia ''The Lesser Key of Solomon''. *A form of radio shorthand that roughly translates as meaning "Best wishes" in the QRP (low power) community *A common limit for characters per line in computing *72 equal temperament is a tuning used in Byzantine music and by some modern composers.


In sports and games

*The usual par for an 18-hole golf course, especially those in tournament play. *The number of victories the Chicago Bulls achieved during the 1995-96 NBA season, which was at the time the National Basketball Association record. *The number of spaces in a game of Parcheesi, from start space to "home."


Footnotes


External links


Go Figure: What can 72 tell us about life
''BBC News'', 20 July 2011 {{Integers, zero Integers