22 (number)

← 21 22 23 →
← 20 21 22 23 24 25 26 27 28 29 → List of numbers Integers ← 0 10 20 30 40 50 60 70 80 90 →
Cardinal	twenty-two
Ordinal	22nd (twenty-second)
Factorization	2 × 11
Divisors	1, 2, 11, 22
Greek numeral	ΚΒ´
Roman numeral	XXII
Binary	101102
Ternary	2113
Senary	346
Octal	268
Duodecimal	1A12
Hexadecimal	1616

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

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.

Properties

22 is a palindromic number. ^{[2] [3]} 22 is the sixth distinct semiprime, ^[4] and the fourth of the form ${\displaystyle 2\times q}$ where ${\displaystyle q}$ 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:

• the maximum number of regions into which five intersecting circles divide the plane, ^[5]
• the quantity of pieces in a disc that can be created with six straight cuts, which makes 22 the seventh central polygonal number, ^{[6] [7]}
• the fourth pentagonal number, the third hexagonal pyramidal number, and the third centered heptagonal number, ^{[8] [9] [10]}
• 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. ^{[11] [12]}

22 is also a Perrin number, from a sum of 10 and 12, ^[13] and the second Smith number, the second Erdős–Woods number, and the fourth large Schröder number. ^{[14] [15] [16]}

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, ..." ^[17]

Permutable and unique primes

The are 22 permutable primes in decimal: ^[18]

$${\displaystyle {\begin{aligned}&(2,3,5,7),\\&(R_{2},13,17,31,37,71,73,79,97),\\&(113,131,199,311,337,373,733,919,991)\\\end{aligned}}}$$

that precede the infinite (conjectured) sequence of prime repunits ${\displaystyle R_{19},R_{23},R_{317},R_{1031},R_{49081},\ldots }$, where ${\displaystyle R_{2}}$ represents ${\displaystyle 11.}$

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 ${\displaystyle (0.{\overline {142857}}).}$ ^[19]

Being 84 digits long with a period length of 294 digits, it is the number:

${\displaystyle 142,857,157,142,857,142,856,999,999,985,714,285,714,285,857,142,857,142,855,714,285,571,428,571,428,572,857,143.}$

The sum of all two-digit permutable primes in decimal — that are pairs, without including ${\displaystyle R_{2}}$ — 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).

Geometry

Polytopes

All regular polygons with ${\displaystyle n}$ < ${\displaystyle 22}$ edges can be constructed with an angle trisector, with the exception of the 11-sided hendecagon. ^[20]

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. ^[21] 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 ${\displaystyle \mathbb {C} ^{2}}$ while one belongs in ${\displaystyle \mathbb {C} ^{3}}$. ^[22]

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 ${\displaystyle {\bar {T}}_{7}}$, 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.

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). ^[23] 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.

Binary and ternary Golay codes

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

${\displaystyle c_{j}=e({\overline {\rm {x}}})\cdot {\overline {\rm {x}}}^{j}(j=0,...,22),{\text{ }}}$ and ${\displaystyle {\text{ }}}$${\displaystyle c_{23}=\sum _{i=0}^{22}{\overline {\rm {x}}}^{i}+{\overline {\rm {x}}}^{\infty }}$

with ${\displaystyle c\in F_{2}}$, and ${\displaystyle e({\overline {\rm {x}}})}$ the quadratic residue code of the binary Golay code ${\displaystyle \mathbb {B} _{23}}$ (with ${\displaystyle {\overline {\rm {x}}}^{\infty }}$ its parity check). M₂₃ is the automorphism group of ${\displaystyle \mathbb {B} _{23}}$.

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: ^[25]

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

Calculations for π

${\displaystyle {\frac {22}{7}}=3.14{\color {red}28}\ldots }$ 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,

${\displaystyle {\sqrt[{4}]{\frac {2143}{22}}}=3.141\;592\;65{\color {red}2\;582\;\ldots }}$ from an approximate construction of the squaring of the circle by Srinivasa Ramanujan, correct to eight decimal places. ^[26]

Natural logarithms of integers in binary are known to have Bailey–Borwein–Plouffe type formulae for ${\displaystyle \pi }$ for all integers ${\displaystyle n=\{1,2,3,4...,22\}}$. ^{[27] [28]}

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 art, entertainment, and media

In music

• "Twenty Two" is a song by:
  • Karma to Burn (2007) ^[29]
  • The Vicar (2013) ^[30]
  • Jordan Sweeney (2008) ^[31]
  • The Good Life (2000) ^[32]
  • Sweet Nectar (1996) ^[33]
  • American Generals (2004) ^[34]
  • Dan Anderson (2007) ^[35]
  • Bad Cash Quartet (2006) ^[36]
  • Millencolin (1999) ^[37]
  • Enter the Worship Circle (2005) ^[38]
  • Blank Dogs (2008) ^[39]
  • Al Candello (2002) ^[40]
  • Amen Dunes (2018) ^[41]
• In Jay-Z's song "22 Two's", he rhymes the words: too, to, and two, 22 times in the first verse. ^[42]
• "22 Acacia Avenue" is a song by Iron Maiden on the album The Number of the Beast. ^[43]
• Catch 22 is an album by death metal band Hypocrisy. ^[44]
• "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. ^[45]
• The Norwegian electronica project Ugress uses 22 as a recurring theme. All four albums feature a track with 22 in the title. ^{[46] [47] [48]}
• "22" is a song by Taylor Swift on her fourth album Red. ^[30]
• "The Number 22" is a song by Ashbury Heights on the album The Looking Glass Society. ^[49]
• 22, A Million is an album by Bon Iver. The first track of the album is called "22 (OVER SOON)". ^[50]
• Cubic 22 was a Belgian techno duo. ^[51]
• "22" is a song by the English alternative rock band Deaf Havana on their album Old Souls. ^[52]
• "22" is a song by the Irish singer Sarah McTernan. ^[53] 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". ^[54]
• Revista 22 is a magazine published in Romania.
• There are 22 stars in the Paramount Pictures logo. ^[55]
• "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. ^[56]
• 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. ^[57]

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. ^[58]

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, 'the two little ducks' after its shape. ^{[59] [60]}

See also

• Catch 22 (disambiguation)
• List of highways numbered 22
• Synchronicity ^{[ relevant? ]}

References

1. Barton, James. "The Number 22: Properties and Meanings". Virtue Science. Archived from the original on 2023-07-23. Retrieved 2022-04-17.
2. Sloane, N. J. A. (ed.). "SequenceA002113(Palindromes in base 10)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2022-04-16.
3. Weisstein, Eric W. "Semiprime". mathworld.wolfram.com. Retrieved 2020-08-12.
4. Sloane, N. J. A. (ed.). "SequenceA001358". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
5. Sloane, N. J. A. (ed.). "SequenceA014206". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2022-04-16.
6. Sloane, N. J. A. (ed.). "SequenceA000124(Central polygonal numbers (the Lazy Caterer's sequence))". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2016-05-31.
7. Wells, D. The Penguin Dictionary of Curious and Interesting Numbers London: Penguin Group. (1986): 31
8. Sloane, N. J. A. (ed.). "SequenceA000326(Pentagonal numbers)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2016-05-31.
9. Sloane, N. J. A. (ed.). "SequenceA069099(Centered heptagonal numbers)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2016-05-31.
10. Sloane, N. J. A. (ed.). "SequenceA002412(Hexagonal pyramidal numbers, or greengrocer's numbers)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2022-04-16.
11. Sloane, N. J. A. (ed.). "SequenceA002088(Sum of totient function)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2016-05-31.
12. Sloane, N. J. A. (ed.). "SequenceA000010(Euler totient function phi(n))". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2022-04-16.
13. Sloane, N. J. A. (ed.). "SequenceA001608(Perrin sequence)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2016-05-31.
14. Sloane, N. J. A. (ed.). "SequenceA006753(Smith numbers)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2016-05-31.
15. Sloane, N. J. A. (ed.). "SequenceA059756(Erdős-Woods numbers)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2016-05-31.
16. Sloane, N. J. A. (ed.). "SequenceA006318(Large Schröder numbers)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2022-06-01.
17. Sloane, N. J. A. (ed.). "SequenceA010861(Look-and-say constant sequence 22)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2022-07-21.
18. Sloane, N. J. A. (ed.). "SequenceA003459(Absolute primes (or permutable primes): every permutation of the digits is a prime.)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2024-02-19.
19. Sloane, N. J. A. (ed.). "SequenceA040017(Unique period primes)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2022-05-20.
20. Gleason, Andrew M. (1988). "Angle trisection, the heptagon, and the triskaidecagon". American Mathematical Monthly . Taylor & Francis, Ltd. 95 (3): 191–194. doi:10.2307/2323624. JSTOR   2323624. MR   0935432. S2CID   119831032.
21. Tilings and patterns Branko Gruenbaum, G.C. Shephard, 1987. 2.5 Tilings using star polygons, pp.82-85.
22. Coxeter, H.S.M. (1991), Regular Complex Polytopes, Cambridge University Press, p. 140, ISBN   0-521-39490-2
23. Weisstein, Eric W. "Mathieu Groups". mathworld.wolfram.com. Retrieved 2022-07-02.
24. Bernhardt, Frank; Landrock, Peter; Manz, Olaf (1990). "The Extended Golay Codes Considered as Ideals". Journal of Combinatorial Theory . Series A. 55 (2): 237. doi:10.1016/0097-3165(90)90069-9.
25. Ostergard, Patrick R. K.; Svanstrom, Mattias (2002). "Ternary Constant Weight Codes". The Electronic Journal of Combinatorics. 9: 13 (R41). doi: 10.37236/1657 .
26. Ramanujan, S. (1914). "Modular equations and approximations to π" (PDF). Quarterly Journal of Mathematics . 45: 350–372.
27. Bailey, David H.; Borwein, Peter B.; Plouffe, Simon (1997). "On the Rapid Computation of Various Polylogarithmic Constants". Mathematics of Computation . 66 (218): 905. doi: 10.1090/S0025-5718-97-00856-9 . hdl: 2060/19970009337 . MR   1415794.
28. Chamberland, Marc (2003). "Binary BBP-Formulae for Logarithms and Generalized Gaussian-Mersenne Primes" (PDF). Journal of Integer Sequences. 6 (3.3.7): 5. Bibcode:2003JIntS...6...37C.
29. Twenty Two – Karma to Burn | Song Info | AllMusic , retrieved 2020-08-12
30. Twenty Two – The Vicar | Song Info | AllMusic , retrieved 2020-08-12
31. Twenty Two – Jordan Sweeney | Song Info | AllMusic , retrieved 2020-08-12
32. Twenty Two – The Good Life | Song Info | AllMusic , retrieved 2020-08-12
33. Twenty Two – Sweet Nectar | Song Info | AllMusic , retrieved 2020-08-12
34. Twenty Two – American Generals | Song Info | AllMusic , retrieved 2020-08-12
35. Twenty Two – Dan Anderson | Song Info | AllMusic , retrieved 2020-08-12
36. Twenty Two – Bad Cash Quartet | Song Info | AllMusic , retrieved 2020-08-12
37. Twenty Two – Millencolin | Song Info | AllMusic , retrieved 2020-08-12
38. Twenty Two – Enter the Worship Circle | Song Info | AllMusic , retrieved 2020-08-12
39. Twenty Two – Blank Dogs | Song Info | AllMusic , retrieved 2020-08-12
40. Twenty Two – Al Candello | Song Info | AllMusic , retrieved 2020-08-12
41. Twenty Two – Amen Dunes | Song Info | AllMusic , retrieved 2020-08-12
42. 22 Two's – Jay-Z | Song Info | AllMusic , retrieved 2020-08-12
43. 22 Acacia Avenue – Iron Maiden | Song Info | AllMusic , retrieved 2020-08-12
44. Catch 22 – Hypocrisy | Songs, Reviews, Credits | AllMusic , retrieved 2020-08-12
45. 22 Dreams – Paul Weller | Songs, Reviews, Credits | AllMusic , retrieved 2020-08-12
46. "Ugress | Album Discography". AllMusic. Retrieved 2020-08-12.
47. Cinematronics – Ugress | Songs, Reviews, Credits | AllMusic , retrieved 2020-08-12
48. Unicorn – Ugress | Songs, Reviews, Credits | AllMusic , retrieved 2020-08-12
49. Number 22 – Ashbury Heights | Song Info | AllMusic , retrieved 2020-08-12
50. 22, A Million – Bon Iver | Songs, Reviews, Credits | AllMusic , retrieved 2020-08-12
51. "Cubic 22 | Songs". AllMusic. Retrieved 2020-08-12.
52. 22 – Deaf Havana | Song Info | AllMusic , retrieved 2020-08-12
53. 22 – Sarah McTernan | Song Info | AllMusic , retrieved 2020-08-12
54. "Definition of CATCH-22". www.merriam-webster.com. Retrieved 2020-08-12.
55. "Paramount Pictures' Logo Started as a Desktop Doodle, and Has Endured for 105 Years". 4 March 2019. Retrieved 2020-08-12.
56. González-Wippler, Migene (1991). The Complete Book of Amulets & Talismans. Llewellyn Worldwide. p. 87. ISBN   978-0-87542-287-9.
57. Sharp, Damian (2001). Simple Numerology: A Simple Wisdom book (A Simple Wisdom Book series). Red Wheel. p. 7. ISBN   978-1-57324-560-9.
58. "Definition of CHAIN". www.merriam-webster.com. Retrieved 2020-08-19. a unit of length equal to 66 feet
59. Cuartas, Javier (1990-01-05). "La suerte de los dos patitos" [The luck of the two little ducks]. El País (in Spanish). Oviedo. Retrieved 17 September 2020.
60. Sanz, Elena (26 April 2010). "Los dos patitos, la niña bonita, la mala pata..." Muy Interesante (in Spanish). Retrieved 17 September 2020. Lo más normal es que el nombre tuviera que ver con la forma del número. Por ejemplo, el 11 era las banderillas, y el 22, los dos patitos o las monjas arrodilladas.

External links

• "The Number 22 at The Database of Number Correlations". Virtuescience.