90 (number)
90 (ninety) is the natural number following 89 and preceding 91.
In the English language, the numbers 90 and 19 are often confused, as they sound very similar.
When carefully enunciated, they differ in which syllable is stressed: 19 /naɪnˈtiːn/ vs 90 /ˈnaɪnti/.
However, in dates such as 1999, and when contrasting numbers in the teens and when counting, such as 17, 18, 19, the stress shifts to the first syllable: 19 /ˈnaɪntiːn/.
Ninety is a pronic number as it is the product of 9 and 10,[1] and along with 12 and 56, one of only a few pronic numbers whose digits in decimal are also successive.
90 is divisible by the sum of its base-ten digits, which makes it the thirty-second Harshad number.
[2] The twelfth triangular number 78[11] is the only number to have an aliquot sum equal to 90, aside from the square of the twenty-fourth prime, 892 (which is centered octagonal).
[14] It is also twice 45, which is the ninth triangular number, and the second-smallest sum of twelve non-zero integers, from two through thirteen
90 can be expressed as the sum of distinct non-zero squares in six ways, more than any smaller number (see image):[15]
The square of eleven 112 = 121 is the ninetieth indexed composite number,[16] where the sum of integers
is 65, which in-turn represents the composite index of 90.
[16] In the fractional part of the decimal expansion of the reciprocal of 11 in base-10, "90" repeats periodically (when leading zeroes are moved to the end).
[17] The eighteenth Stirling number of the second kind
of 3, as the number of ways of dividing a set of six objects into three non-empty subsets.
[18] 90 is also the sixteenth Perrin number from a sum of 39 and 51, whose difference is 12.
Since prime sextuplets are formed from prime members of lower order prime k-tuples, 90 is also a record maximal gap between various smaller pairs of prime k-tuples (which include quintuplets, quadruplets, and triplets).
[a] 90 is the third unitary perfect number (after 6 and 60), since it is the sum of its unitary divisors excluding itself,[22] and because it is equal to the sum of a subset of its divisors, it is also the twenty-first semiperfect number.
[24] In normal space, the interior angles of a rectangle measure 90 degrees each, while in a right triangle, the angle opposing the hypotenuse measures 90 degrees, with the other two angles adding up to 90 for a total of 180 degrees.
ratio, and another 30 slim rhombi with diagonals in
It is the dual polyhedron to the rectified truncated icosahedron, a near-miss Johnson solid.
On the other hand, the final stellation of the icosahedron has 90 edges.
It also has 92 vertices like the rhombic enneacontahedron, when interpreted as a simple polyhedron.
The self-dual Witting polytope contains ninety van Oss polytopes such that sections by the common plane of two non-orthogonal hyperplanes of symmetry passing through the center yield complex 3{4}3 Möbius–Kantor polygons.
[25] The root vectors of simple Lie group E8 are represented by the vertex arrangement of the
By Coxeter, the incidence matrix configuration of the Witting polytope can be represented as: This Witting configuration when reflected under the finite space
splits into 85 = 45 + 40 points and planes, alongside 27 + 90 + 240 = 357 lines.
[25] Whereas the rhombic enneacontahedron is the zonohedrification of the regular dodecahedron,[26] a honeycomb of Witting polytopes holds vertices isomorphic to the E8 lattice, whose symmetries can be traced back to the regular icosahedron via the icosian ring.
[27] The maximal number of pieces that can be obtained by cutting an annulus with twelve cuts is 90 (and equivalently, the number of 12-dimensional polyominoes that are prime).
