# 22 (number)

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

 ← 21 22 23 →
Cardinaltwenty-two
Ordinal22nd
(twenty-second)
Factorization2 × 11
Divisors1, 2, 11, 22
Greek numeralΚΒ´
Roman numeralXXII
Binary101102
Ternary2113
Senary346
Octal268
Duodecimal1A12

## In mathematics The first 22 numbers can be arranged on a graph such that select sums between two numbers in the set yield all primes from 3 to 43. The graph has near-perfect vertical and horizontal reflective symmetry.

22 is a palindromic number. 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 sixth distinct semiprime, and the fouth of the form $2\times q$  where $q$  is a higher prime. It is the second member of the second cluster of discrete biprimes (21, 22), where the next such cluster is (38, 39). It contains an aliquot sum of 14 (itself semiprime), within an aliquot sequence of four composite numbers (22, 14, 10, 8, 7, 1, 0) that are rooted in the prime 7-aliquot tree.

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.

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

${\frac {22}{7}}$  is a commonly used approximation of the irrational number π, the ratio of the circumference of a circle to its diameter, where in particular 22 and 7 are consecutive hexagonal pyramidal numbers. Also,

${\sqrt[{4}]{\frac {2143}{22}}}=3.141\;592\;65{\color {red}2\;582\;\ldots }$  from an approximate construction of the squaring of the circle by Srinivasa Ramanujan, correct to eight decimal places.

Natural logarithms of integers in binary are known to have Bailey–Borwein–Plouffe type formulae for $\pi$  for all integers $n=\{1,2,3,4...,22\}$ .

22 is the number of partitions of 8, as well as the sum of the totient function over the first eight integers, with φ(n) 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, ..."

All regular polygons with $n$  < $22$  edges can be constructed with an angle trisector, with the exception of the 11-sided hendecagon.

There is an elementary set of twenty-two single-orbit convex tilings that tessellate two-dimensional space with face-transitive, edge-transitive, and/or vertex-transitive properties: eleven of these are regular and semiregular Archimedean tilings, while the other eleven are their dual Laves tilings. Twenty-two edge-to-edge star polygon tilings exist in the second dimension that incorporate regular convex polygons: eighteen involve specific angles, while four involve angles that are adjustable. Finally, there are also twenty-two regular complex apeirohedra of the form p{a}q{b}r: eight are self-dual, while fourteen exist as dual polytope pairs; twenty-one belong in $\mathbb {C} ^{2}$  while one belongs in $\mathbb {C} ^{3}$ .

There are twenty-two 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:

The family of k21 polytopes can be extended backward to include the rectified 5-cell and the three-dimensional triangular prism, which is the simplest semiregular polytope.
On the other hand, k22 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, whose 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 ${\bar {T}}_{7}$ , with 222 made of 122 facets, and so forth. The Coxeter symbol for these figures is of the form kij, 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.

There are twenty-two Coxeter groups in the sixth dimension that generate uniform polytopes: four of these generate uniform non-prismatic figures, while the remaining eighteen generate uniform prisms, duoprisms and triaprisms.

The number 22 appears prominently within sporadic groups. Mathieu group M22 is one of 26 such sporadic finite simple groups, 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, |HS| = 100|M22|. Conway group Co1 and Fischer group Fi24 both have 22 different conjugacy classes.

The extended binary Golay code $\mathbb {B} _{24}$ , which is related to Steiner system W24, is constructed as a vector space of F2 from the words:

$c_{j}=e({\overline {\rm {x}}})\cdot {\overline {\rm {x}}}^{j}(j=0,...,22),{\text{ }}$  and ${\text{ }}$ $c_{23}=\sum _{i=0}^{22}{\overline {\rm {x}}}^{i}+{\overline {\rm {x}}}^{\infty }$
with $c\in F_{2}$ , and $e({\overline {\rm {x}}})$  the quadratic residue code of the binary Golay code $\mathbb {B} _{23}$  (with ${\overline {\rm {x}}}^{\infty }$  its parity check). M23 is the automorphism group of $\mathbb {B} _{23}$ .

The extended ternary Golay code [12, 6, 6], whose root is the ternary Golay code [11, 6, 5] over F3, has a complete weight enumerator value equal to:

$x^{12}+y^{12}+z^{12}+22\left(x^{6}y^{6}+y^{6}z^{6}+z^{6}x^{6}\right)+220\left(x^{6}y^{3}z^{3}+x^{3}y^{6}z^{3}+x^{3}y^{3}z^{6}\right).$

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 $(0.{\overline {142857}}).$  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 aircraft

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

## In art, entertainment, and media

### In other fields

• Catch-22 (1961), Joseph Heller's novel, and its 1970 film adaptation gave rise to the expression of logic "catch-22".
• Revista 22 is a magazine published in Romania.
• There are 22 stars in the Paramount Pictures logo.
• "Twenty Two" (February 10, 1961) is Season 2–episode 17 (February 10, 1961) of the 1959–1964 TV series The Twilight Zone, in which a hospitalized dancer has nightmares about a sinister nurse inviting her to Room 22, the hospital morgue.
• Traditional Tarot decks have 22 cards with allegorical subjects. These serve as trump cards in the game. The Fool is usually a kind of wild-card among the trumps and unnumbered, so the highest trump is numbered 21. Occult Tarot decks usually have 22 similar cards which are called Major Arcana by fortune-tellers. Occultists have related this number to the 22 letters of the Hebrew alphabet and the 22 paths in the Kabbalistic Tree of Life.
• "22" is the number assigned to the unborn soul who serves as a prominent character in the Pixar film Soul.

## In sports

• In both American football and association football, a total of 22 players (counting both teams) start the game, and this is also the maximum number of players that can be legally involved in play at any given time.
• In men's Australian rules football, each team is allowed a squad of 22 players (18 on the field and 4 interchanges).
• The length of a cricket pitch is 22 yards.
• In rugby union, the "22" is a line in each half of the field which is 22 meters from the respective try line. It has significance in a number of laws particularly relating to kicking the ball away.
• A snooker game (called a "frame") starts with 22 coloured balls at specified locations on the table (15 red balls and 7 others).

## In other uses

Twenty-two may also refer to: