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.
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. 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. It is also a
perfect indexed
Harshad number in
decimal (twenty-eighth), as it is divisible by the sum of its digits (
9). 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 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
\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 421 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
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}. 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 literature==