22 (number)

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

Properties
22 is a palindromic number. 22 is the sixth distinct semiprime, and the fourth 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.

Twenty-two is also:

22 is also a Perrin number, from a sum of 10 and 12, and the second Smith number, the second Erdős–Woods number, and the fourth large Schröder number.

22 can also 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, ..."

Permutable and unique primes
There are 22 permutable primes in decimal: $$ \begin{align} & (2,3,5,7), \\ & (R_{2},13,17,31,37,71,73,79,97), \\ & (113,131,199,311,337,373,733,919,991) \\ \end{align} $$ that precede the infinite (conjectured) sequence of prime repunits $$R_{19}, R_{23}, R_{317}, R_{1031}, R_{49081}, \ldots$$, where $$R_{2}$$ represents $$11.$$

The twenty-second 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 $$7$$, equal to $$0.\overline{142857}.$$

Being 84 = 7 × 12 digits long with a period length of 294 = 14 × 21 digits, it is the number:
 * $$142,857,157,142,857,142,856,999,999,985,714,285,714,285,857,142,857,142,855,714,285,571,428,571,428,572,857,143.$$

The sum of all two-digit permutable primes in decimal — that are pairs, without including $$R_{2}$$ — is 418, which is the sum of the digits of the twenty-second unique prime in base ten (all repunit primes are unique, where 3 and 37 are permutable as well as unique).

Free polyominoes
22 is the number of distinct tilings of a four by four square with free 4-polyominoes.

This sequence has as its first terms {1, 1, 2, 22, 515, …}, where the sum of the first five terms yields the 100th prime number 541.

Polytopes
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, that is based on the regular icosahedron. 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 k21 polytopes: the demipenteract, 221, 321, and 421; with the final figure representing the root vectors of simple Lie group E8.
 * 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.

Sporadic groups
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$|M_{22}|$. Conway group Co1 and Fischer group Fi24 both have 22 different conjugacy classes.

Binary and ternary Golay codes
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:


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

Calculations for $φ(n)$
$$\frac{22}{7} = 3.14{\color{red}28}\ldots$$ is a commonly used approximation of the irrational number $\pi$, 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.

On the other hand, where twenty-two is the second-smallest numerator in convergents of pi, the difference between the next two larger numerators (355, 333) is also 22. In the true approximation for $$\pi$$, the first six digits of its fractional part in decimal representation sum to 22, strings of digits which also end in 2. Natural logarithms of integers in binary are known to have Bailey–Borwein–Plouffe type formulae for $$\pi$$ for all integers $$n = \{1, 2, 3, 4 ..., 22\}$$.

In science

 * 22 is the atomic number of titanium.
 * 22 is the number of proteinogenic amino acids.
 * 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 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 (2008)
 * Al Candello (2002)
 * Amen Dunes (2018)
 * In Jay-Z's song "22 Two's", he rhymes the words: too, to, and two, 22 times in the first verse.
 * "22 Acacia Avenue" is a song by Iron Maiden on the album The Number of the Beast.
 * Catch 22 is an album by death metal band Hypocrisy.
 * "22" is a song by Lily Allen on the album It's Not Me, It's You.
 * 22 Dreams is a song and album by Paul Weller. The album has 22 songs on it.
 * The Norwegian electronica project Ugress uses 22 as a recurring theme. All four albums feature a track with 22 in the title.
 * "22" is a song by Taylor Swift on her fourth album Red.
 * "The Number 22" is a song by Ashbury Heights on the album The Looking Glass Society.
 * 22, A Million is an album by Bon Iver. The first track of the album is called "22 (OVER SOON)".
 * Cubic 22 was a Belgian techno duo.
 * "22" is a song by the English alternative rock band Deaf Havana on their album Old Souls.
 * "22" is a song by the Irish singer Sarah McTernan. She represented Ireland with this song at Eurovision 2019.

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 computing and technology

 * 22 is the standard port number for the Secure Shell protocol.

In culture and religion

 * There are 22 letters in the Hebrew alphabet.
 * In the Kabbalah, there are 22 paths between the Sephirot.
 * There are 22 chapters in the book: The Revelation of Jesus Christ, Book of Revelation.
 * 22 is a master number in numerology.

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 weights and measures

 * The number of yards in a chain.

In other uses
Twenty-two may also refer to:
 * 22 is the number of the French department Côtes-d'Armor
 * "22" is a common name for the .22 calibre .22 Long Rifle cartridge.
 * In French jargon, "22" is used as a phrase to warn of the coming of the police (typically "22, v'là les flics !" (In English: "5-0! Cops!")
 * In photography, f/22 is the largest f-stop (and thus smallest aperture) available on most lenses made for single-lens reflex cameras
 * In Spanish lottery and bingo, 22 is nicknamed los dos patitos after its shape.