72 (number)

Template:Infobox number 72 (seventy-two) is the natural number following 71 and preceding 73. It is half a gross or six dozen (i.e., 60 in duodecimal).

In mathematics

Seventy-two is a pronic number, as it is the product of 8 and 9.<ref>Template:Cite OEIS</ref> It is the smallest Achilles number, as it is a powerful number that is not itself a power.<ref>Template:Cite OEIS</ref>

72 is an abundant number.<ref>Template:Cite OEIS</ref> With exactly twelve positive divisors, including 12 (one of only two sublime numbers),<ref>Template:Cite OEIS</ref> 72 is also the twelfth member in the sequence of refactorable numbers.<ref>Template:Cite OEIS

The sequence of refactorable numbers goes: 1, 2, 8, 9, 12, 18, 24, 36, 40, 56, 60, 72, 80, 84, 88, 96, ...</ref> As no smaller number has more than 12 divisors, 72 is a largely composite number.<ref name="OEIS-A067128">Template:Cite OEIS</ref> 72 has an Euler totient of 24.<ref>Template:Cite OEIS</ref> It is a highly totient number, as there are 17 solutions to the equation φ(x) = 72, more than any integer under 72.<ref name="highly totient">Template:Cite OEIS</ref> It is equal to 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 divisors. 144, or twice 72, is also highly totient, as is 576, the 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 integers is 72.<ref>Template:Cite OEIS</ref> It is also a perfect indexed Harshad number in decimal (twenty-eighth), as it is divisible by the sum of its digits (9).<ref>Template:Cite OEIS</ref>

72 plays a role in the Rule of 72 in economics when approximating annual compounding of interest rates 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 vertices of the six-dimensional 1₂₂ polytope, which also contains as facets 720 edges, 702 polychoral 4-faces, of which 270 are four-dimensional 16-cells, and two sets of 27 demipenteract 5-faces. These 72 vertices are the root vectors of the simple Lie group <math>\mathrm E_{6}</math>, which as a honeycomb under 2₂₂ forms the <math>\mathrm E_{6}</math> lattice. 1₂₂ is part of a family of k₂₂ polytopes whose first member is the fourth-dimensional 3-3 duoprism, of symmetry order 72 and made of six triangular prisms. On the other hand, 3₂₁ ∈ k₂₁ is the only semiregular polytope in the seventh dimension, also featuring a total of 702 6-faces of which 576 are 6-simplexes and 126 are 6-orthoplexes that contain 60 edges and 12 vertices, or collectively 72 one-dimensional and two-dimensional elements; with 126 the number of root vectors in <math>\mathrm E_{7}</math>, which are contained in the vertices of 2₃₁ ∈ k₃₁, also with 576 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 that has four triangular prisms alongside two tetrahedra as cells.
• The complex Hessian polyhedron in <math>\mathbb{C}^3</math> contains 72 regular complex triangular edges, as well as 27 polygonal Möbius–Kantor faces and 27 vertices. It is notable for being the vertex figure of the complex Witting polytope, which shares 240 vertices with the eight-dimensional semiregular 4₂₁ polytope whose vertices in turn represent the root vectors of the simple Lie group <math>\mathrm E_{8}</math>.

There are 72 compact and paracompact Coxeter groups of ranks four through ten: 14 of these are compact finite representations in only three-dimensional and 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 ninth dimension, of which the most important is <math>\tilde {T}_{9}</math>: it contains the final semiregular hyperbolic honeycomb 6₂₁ made of only regular facets and the 5₂₁ Euclidean honeycomb as its vertex figure, which is the geometric representation of the <math>\mathrm E_{8}</math> lattice. Furthermore, <math>\tilde {T}_{9}</math> shares the same fundamental symmetries with the Coxeter-Dynkin over-extended form <math>\mathrm E_{8}</math>⁺⁺ equivalent to the tenth-dimensional symmetries of Lie algebra <math>\mathrm E_{10}</math>.

72 lies between the 8th pair of twin primes (71, 73), where 71 is the largest supersingular prime that is a factor of the largest sporadic group (the friendly giant <math>\mathbb {F_{1}}</math>), and 73 the largest indexed member of a definite quadratic integer matrix representative of all prime numbers<ref>Template:Cite OEIS

{2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 67, 73}</ref>Template:Efn that is also the number of distinct orders (without multiplicity) inside all 194 conjugacy classes of <math>\mathbb {F_{1}}</math>.<ref>Template:Cite arXiv</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 groups that are part of sixteen finite Lie groups that are also simple, or non-trivial groups whose only normal subgroups are the trivial group and the groups themselves.Template:Efn

In religion

• In Islam 72 is the number of beautiful wives that are promised to martyrs in paradise, according to Hadith (sayings of Muhammad).<ref>{{#invoke:citation/CS1|citation

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In other fields

Seventy-two is also:

• In typography, a point is 1/72 inch.<ref>{{#invoke:citation/CS1|citation

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• The Rule of 72 in finance.
• 72 equal temperament is a tuning used in Byzantine music and by some modern composers.
• The number of micro seasons in the traditional Japanese calendar<ref>{{#invoke:citation/CS1|citation

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External links

• Go Figure: What can 72 tell us about life, BBC News, 20 July 2011