XXII Army Corps (Spain)

22 (twenty-two) is the natural number following 21 and preceding 23.

In mathematics

22 is a palindromic number and the eighth semiprime; its proper divisors are 1, 2, and 11. It is the second Smith number, the second Erdős–Woods number, and the fourth large Schröder number. It is also a Perrin number, from a sum of 10 and 12.

22 is the fourth pentagonal number, the third hexagonal pyramidal number, and the third centered heptagonal number.

The maximum number of regions into which five intersecting circles divide the plane is 22. 22 is also the quantity of pieces in a disc that can be created with six straight cuts, which makes 22 the seventh central polygonal number.

$\frac$ is a commonly used approximation of the irrational number , the ratio of the circumference of a circle to its diameter; where both 22 and 7 are consecutive hexagonal pyramidal numbers. 22 also features in another approximation for pi, here by Srinivasa Ramanujan from an approximate construction of the squaring of the circle, and correct to eight decimal digits:
\sqrt = 3.141\;592\;65

Natural logarithms of integers in binary are known to have Bailey–Borwein–Plouffe type formulae for $\pi$ for all integers $n = \$.

22 is the number of partitions of 8, as well as the sum of the totient function for the first eight integers, with for 22 returning 10.

22 can read as "two twos", which is the only fixed point of John Conway's look-and-say function. In other words, "22" generates the infinite repeating sequence "22, 22, 22, ..."

There is an elementary set of 22 single-orbit convex tilings that tessellate two dimensional space with face-transitive, edge-transitive, and/or vertex-transitive properties: 11 of these are regular and semiregular Archimedean tilings, while the other 11 are their dual Laves tilings. 22 edge-to-edge star polygon tilings exist in the second dimension that incorporate regular convex polygons: 18 involve specific angles, while 4 involve angles that are adjustable. Finally, there are also 22 regular complex apeirohedra of the form _pqr: 8 are self-dual, while 14 exist as dual polytope pairs; 21 belong in $\mathbb^2$ while one belongs in $\mathbb^3$.

There are 22 different subgroups that describe full icosahedral symmetry. Three groups are generated by particular inversions, five groups by reflections, and nine groups by rotations, alongside three mixed groups, the pyritohedral group, and the full icosahedral group.

There are 22 finite semiregular polytopes through the eighth dimension, aside from the infinite families of prisms and antiprisms in the third dimension and inclusive of 2 enantiomorphic forms. Defined as vertex-transitive polytopes with regular facets, there are:
*15 Archimedean semiregular solids in 3-space, which include two chiral forms, one from the snub cube and one from the snub dodecahedron. In other words, from symmetries of 13 distinct semiregular polyhedra, two of which have mirror images.
*3 semiregular polychora in 4-space: the rectified 5-cell, the rectified 600-cell and the snub 24-cell.
*4 semiregular polytopes from 5-space through 8-space that are part of the family of ''k''₂₁ polytopes: the demipenteract, 2₂₁, 3₂₁, and 4₂₁; with the final figure representing the root vectors of simple Lie group E₈. The family of ''k''₂₁ polytopes can be extended backward to include the rectified 5-cell and the three-dimensional triangular prism, which is the simplest semiregular polytope.

: ''k''₂₂ polytopes are a family of five different polytopes up through the eighth dimension, that include three finite polytopes and two honeycombs. Its root figure is the first proper duoprism, the 3-3 duoprism (-1₂₂), which is made of six triangular prisms. The second figure is the birectified 5-simplex (0₂₂), and the last finite figure is the 6th-dimensional 1₂₂ polytope. 1₂₂ is highly symmetric, with 720 edges, two sets of 27 demipenteract 5-faces, and 702 polychoral faces of which 270 are 16-cells; its 72 vertices represent the root vectors of the simple Lie group E₆. 3₂₂ is a paracompact infinite honeycomb that contains 2₂₂ Euclidean honeycomb facets under Coxeter group symmetry $_7$, with 2₂₂ made of 1₂₂ facets, and so forth. The Coxeter symbol for these figures is of the form ''k''_''ij'', where each letter represents a length of order-3 branches on a Coxeter–Dynkin diagram with a single ring on the end node of a ''k''-length sequence of branches.

The number 22 appears prominently within sporadic groups. Mathieu group M₂₂ is one of 26 such sporadic finite simple group, defined as the 3-transitive permutation representation on 22 points. It is the monomial of the McLaughlin sporadic group, McL, and the unique index 2 subgroup of the automorphism group of Steiner system S(3,6,22). Mathieu group M₂₃ contains M₂₂ as a point stabilizer, and has a minimal irreducible complex representation in 22 dimensions, like McL. M₂₃ has two rank 3 actions on 253 points, with 253 equal to the sum of the first 22 non-zero positive integers, or the 22nd triangular number. Both M₂₂ and M₂₃ are maximal subgroups within Mathieu group M₂₄, which works inside the lexicographic generation of Steiner system S(5,8,24) W₂₄, where single elements within 759 octads of 24-element sets occur 253 times throughout its entire set. On the other hand, the Higman–Sims sporadic group HS also has a minimal faithful complex representation in 22 dimensions, and is equal to 100 times the group order of M₂₂, . Conway group Co₁ and Fischer group Fi₂₄ both have 22 different conjugacy classes.

The extended binary Golay code $\mathbb B_$, which is related to Steiner system W₂₄, is constructed as a vector space of ''F₂'' from the words:

:$c_j = e(\overline)\cdot\overline^j (j=0,...,22),\text$ and $\text$$c_ = \sum_^\overline^+\overline^\infty$

:with $c\in F_2$, and $e(\overline)$ the quadratic residue code of the binary Golay code $\mathbb B_$ (with $\overline^\infty$ its parity check). M₂₃ is the automorphism group of $\mathbb B_$.

The extended ternary Golay code 2, 6, 6 whose root is the ternary Golay code 1, 6, 5over ''F₃'', has a complete weight enumerator value equal to:

:$x^+y^+z^+22\left(x^6y^6+y^6z^6+z^6x^6\right)+220\left(x^6y^3z^3+x^3y^6z^3+x^3y^3z^6\right).$

The 22nd unique prime in base ten is notable for having starkly different digits compared to its preceding (and latter) unique primes, as well as for the similarity of its digits to those of the reciprocal of 7 $(0.\overline).$ Being 84 digits long with a period length of 294 digits, it is the number:
:$142,857,157,142,857,142,856,999,999,985,714,285,714,285,857,142,857,142,855,714,285,571,428,571,428,572,857,143$

In science

*22 is the atomic number of titanium.
*22 is the number of bones in the human skull: 14 belong to the facial skeleton and 8 to the neurocranium.

In aircraft

*22 is the designation of the USAF stealth fighter, the F-22 Raptor.

In art, entertainment, and media

In music

* "Twenty Two" is a song by:
** Karma to Burn (2007)
** The Vicar (2013)
** Jordan Sweeney (2008)
** The Good Life (2000)
** Sweet Nectar (1996)
** American Generals (2004)
** Dan Anderson (2007)
** Bad Cash Quartet (2006)
** Millencolin (1999)
** Enter the Worship Circle (2005)
** Blank Dogs

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