72 (number)

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

In mathematics

72 is a pronic number, as it is the product of 8 and 9. It is the smallest Achilles number, as it is a powerful number that is not itself a power. 72 is the sum of four consecutive primes (13 + 17 + 19 + 23) and the sum of six consecutive primes (5 + 7 + 11 + 13 + 17 + 19). 72 is the smallest natural number that can be expressed as the difference of the squares of primes in just two distinct ways: 72 = 11² − 7² = 19² − 17².

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. 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.

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.

There are 72 of distinct {7/2} magic heptagrams, all of which have 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).

There are 72 degrees in the central angle of a regular pentagon, which is constructible with a compass and straight-edge.

Hipparchus (greek mathematician‐astronomer c.190 – c.120 BC) is purported to have discovered the phenomenon of the precession of the equinoxes by comparing the position of the vernal equinox against the fixed stars, noting that it shifts westward by about one degree every 72 years.

Abstract algebra

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 6₂₁ made of only regular facets and the 5₂₁ 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 ${\mathrm{E}}_{10}$.

There are 72 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 ${\mathrm{E}}_6$, which as a honeycomb under 2₂₂ forms the ${\mathrm{E}}_6$ 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 ${\mathrm{E}}_7$, 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 ${ℂ}^3$ 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 4₂₁ polytope whose vertices in turn represent the root vectors of the simple Lie group ${\mathrm{E}}_8$.

In religion

In the Talmud (Sanhedrid 87a), "the Muflah - the distinguished/wonderous one" is noted as being a 72nd member of the Sanhedrin council (traditionally limited to 71 members to prevent indecision). This figure is considered to be the most distinguished member of the court, an ordained, expert judge that presides over the council during highly significant and controversial matters. The Mufla according to in Mishnaic traditional teachings may be in reference to Rabbi Elazar ben Azariah.

In other fields

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.

There are 72 micro seasons in the traditional Japanese calendar.

External links

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