# 72 (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. It is the smallest Achilles number, as it's a powerful number that is not itself a power.
72 is an abundant number. With exactly twelve positive divisors, including 12 (one of only two sublime numbers), 72 is also the twelfth member in the sequence of refactorable numbers. As no smaller number has more than 12 divisors, 72 is a largely composite number. 72 has an Euler totient of 24. It is a highly totient number, as there are 17 solutions to the equation φ(x) = 72, more than any integer under 72. 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. 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. It also is a perfect indexed Harshad number in decimal (twenty-eighth), as it is divisible by the sum of its digits (9).
* 72 is the second multiple of 12, after 48, that is not a sum of twin primes. It is, however, the sum of four consecutive primes (13 + 17 + 19 + 23), as well as the sum of six consecutive primes (5 + 7 + 11 + 13 + 17 + 19).
* 72 is the first number that can be expressed as the difference of the squares of primes in just two distinct ways: 11$^2$ − 7$^2$ = 19$^2$ − 17$^2$.
* 72 is the sum of the first two sphenic numbers (30, 42), which have a difference of 12, that is also their abundance.
* 72 is the magic constant of the first non-normal, full prime reciprocal magic square in decimal, based on ⁠1/17⁠ in a 16 × 16 grid.
* 72 is the sum between 60 and 12, the former being the second unitary perfect number before 6 (and the latter the smallest of only two sublime numbers). More specifically, twelve is also the number of divisors of 60, as the smallest number with this many divisors.
* 72 is the number of distinct {7/2} magic heptagrams, all with a magic constant of 30.
* 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). As such, this row is the third and largest to be in equivalence with a sum of consecutive k row sums, after (1, 2, 3; 6) and (6, 10, 20; 36).
* 72 is the number of degrees in the central angle of a regular pentagon, which is constructible with a compass and straight-edge.
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 $\mathrm {E} _{n}$ Lie algebras:
* 72 is the number of vertices of the six-dimensional 1$_{22}$ 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 $\mathrm {E} _{6}$, which as a honeycomb under 2$_{22}$ forms the $\mathrm {E} _{6}$ lattice. 1$_{22}$ is part of a family of k$_{22}$ 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, 3$_{21}$ ∈ k$_{21}$ 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 $\mathrm {E} _{7}$, which are contained in the vertices of 2$_{31}$ ∈ k$_{31}$, also with 576 or 24$^2$ 6-simplexes like 3$_{21}$. The triangular prism is the root polytope in the k$_{21}$ family of polytopes, which is the simplest semiregular polytope, with k$_{31}$ rooted in the analogous four-dimensional tetrahedral prism that has four triangular prisms alongside two tetrahedra as cells.
* The complex Hessian polyhedron in $\mathbb {C} ^{3}$ 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 4$_{21}$ polytope whose vertices in turn represent the root vectors of the simple Lie group $\mathrm {E} _{8}$.
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 ${\tilde {T}}_{9}$: it contains the final semiregular hyperbolic honeycomb 6$_{21}$ made of only regular facets and the 5$_{21}$ Euclidean honeycomb as its vertex figure, which is the geometric representation of the $\mathrm {E} _{8}$ lattice. Furthermore, ${\tilde {T}}_{9}$ shares the same fundamental symmetries with the Coxeter-Dynkin over-extended form $\mathrm {E} _{8}$$^{++}$ equivalent to the tenth-dimensional symmetries of Lie algebra $\mathrm {E} _{10}$.
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 $\mathbb {F_{1}}$), and 73 the largest indexed member of a definite quadratic integer matrix representative of all prime numbers that is also the number of distinct orders (without multiplicity) inside all 194 conjugacy classes of $\mathbb {F_{1}}$. Sporadic groups are a family of twenty-six finite simple groups, where $\mathrm {E} _{6}$, $\mathrm {E} _{7}$, and $\mathrm {E} _{8}$ 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.

## In religion

* In Islam 72 is the number of beautiful wives that are promised to martyrs in paradise, according to Hadith (sayings of Muhammad).

## In other fields

Seventy-two is also:
* In typography, a point is 1/72 inch.
* 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