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12 (number)

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← 11 12 13 →
Numeral systemduodecimal
Factorization22 × 3
Divisors1, 2, 3, 4, 6, 12
Greek numeralΙΒ´
Roman numeralXII
Greek prefixdodeca-
Latin prefixduodeca-
Hebrew numeralי"ב
Babylonian numeral𒌋𒐖

12 (twelve) is the natural number following 11 and preceding 13. Twelve is a superior highly composite number, divisible by the numbers 2, 3, 4, and 6.

It is the number of years required for an orbital period of Jupiter. It is central to many systems of timekeeping, including the Western calendar and units of time of day, and frequently appears in the world's major religions.


Twelve is the largest number with a single-syllable name in English. Early Germanic numbers have been theorized to have been non-decimal: evidence includes the unusual phrasing of eleven and twelve, the former use of "hundred" to refer to groups of 120, and the presence of glosses such as "tentywise" or "ten-count" in medieval texts showing that writers could not presume their readers would normally understand them that way.[1][2][3] Such uses gradually disappeared with the introduction of Arabic numerals during the 12th-century Renaissance.

Derived from Old English, twelf and tuelf are first attested in the 10th-century Lindisfarne Gospels' Book of John.[note 1][5] It has cognates in every Germanic language (e.g. German zwölf), whose Proto-Germanic ancestor has been reconstructed as *twaliƀi..., from *twa ("two") and suffix *-lif- or *-liƀ- of uncertain meaning.[5] It is sometimes compared with the Lithuanian dvýlika, although -lika is used as the suffix for all numbers from 11 to 19 (analogous to "-teen").[5] Every other Indo-European language instead uses a form of "two"+"ten", such as the Latin duōdecim.[5] The usual ordinal form is "twelfth" but "dozenth" or "duodecimal" (from the Latin word) is also used in some contexts, particularly base-12 numeration. Similarly, a group of twelve things is usually a "dozen" but may also be referred to as a "dodecad" or "duodecad". The adjective referring to a group of twelve is "duodecuple".

As with eleven,[6] the earliest forms of twelve are often considered to be connected with Proto-Germanic *liƀan or *liƀan ("to leave"), with the implicit meaning that "two is left" after having already counted to ten.[5] The Lithuanian suffix is also considered to share a similar development.[5] The suffix *-lif- has also been connected with reconstructions of the Proto-Germanic for ten.[6][7]

As mentioned above, 12 has its own name in Germanic languages such as English (dozen), Dutch (dozijn), German (Dutzend), and Swedish (dussin), all derived from Old French dozaine. It is a compound number in many other languages, e.g. Italian dodici (but in Spanish and Portuguese, 16, and in French, 17 is the first compound number),[dubiousdiscuss] Japanese 十二 jūni.[clarification needed]

Written representation[edit]

In prose writing, twelve, being the last single-syllable numeral, is sometimes taken as the last number to be written as a word, and 13 the first to be written using digits. This is not a binding rule, and in English language tradition, it is sometimes recommended to spell out numbers up to and including either nine, ten or twelve, or even ninety-nine or one hundred. Another system spells out all numbers written in one or two words (sixteen, twenty-seven, fifteen thousand, but 372 or 15,001).[8] In German orthography, there used to be the widely followed (but unofficial) rule of spelling out numbers up to twelve (zwölf). The Duden[year needed] (the German standard dictionary) mentions this rule as outdated.

Mathematical properties[edit]

12 is the sixth composite number and the superfactorial of 3.[9][10] It is the fourth pronic number (equal to 3 × 4),[11] whose digits in decimal are also successive. It is the smallest abundant number, since it is the smallest integer for which the sum of its proper divisors (1 + 2 + 3 + 4 + 6 = 16) is greater than itself,[12] and the second semiperfect number, since there is a subset of the proper divisors of 12 that add up to itself.[13] It is equal to the sum between the second pair of twin primes (5 + 7),[14] while it is also the smallest number with exactly six divisors (1, 2, 3, 4, 6 and 12) which makes it the fifth highly composite number,[15] and since 6 is also one of them, twelve is also the fifth refactorable number.[16] 12, as a number with a perfect number of divisors (six), has a sum of divisors that yields the second perfect number, σ(12) = 28,[17] and as such it is the smallest of two known sublime numbers, which are numbers that have a perfect number of divisors whose sum is also perfect.[18] 12 is the fifth Pell number (preceded by 0, 1, 2, and 5)[19] as well as the third pentagonal number,[20] and a Harshad number in all bases except octal.

Twelve is the number of divisors of 60 and 90, the second and third unitary perfect numbers (6 is the first). It is also the number of distinct prime factors that belong to the fifth unitary perfect number, the largest known,


The second perfect number, 28, is the arithmetic mean of the twelve divisors of the fourth harmonic divisor number, 140 (like 6, and 28), which generates an integer harmonic mean of 5.[23][24][25]

If an odd perfect number is not divisible by 3, it will have at least twelve distinct prime factors.[26]

There are 12 Latin squares of size 3 × 3, where symbols appear exactly once in each row and exactly once in each column.[27]

There are twelve Jacobian elliptic functions and twelve cubic distance-transitive graphs.

A twelve-sided polygon is a dodecagon. In its regular form, it is the largest polygon that can uniformly tile the plane alongside other regular polygons, as with the truncated hexagonal tiling or the truncated trihexagonal tiling. There are 12 regular and semiregular tilings when enantiomorphic forms of the snub hexagonal tiling are counted separately.[28]

A regular dodecahedron has twelve pentagonal faces. Regular cubes and octahedrons both have 12 edges, while regular icosahedrons have 12 vertices. The rhombic dodecahedron has twelve rhombic faces and is able to tessellate three-dimensional space; it is the only Catalan solid to generate a honeycomb with copies of itself. Its dual polyhedron, the cuboctahedron, has 12 vertices with radial equilateral symmetry, and is one of two quasiregular polyhedra.

The densest three-dimensional lattice sphere packing has each sphere touching twelve other spheres, and this is almost certainly true for any arrangement of spheres (the Kepler conjecture). Twelve is also the kissing number in three dimensions.

There are twelve complex apeirotopes in dimensions five and higher, which include van Oss polytopes in the form of complex -orthoplexes.[29] There are also twelve paracompact hyperbolic Coxeter groups of uniform polytopes in five-dimensional space.

Bring's curve is a Riemann surface of genus four, with a domain that is a regular hyperbolic 20-sided icosagon.[30] By the Gauss-Bonnet theorem, the area of this fundamental polygon is equal to .

The Leech lattice, which holds the solution to the kissing number in twenty-four dimensions,[31] has a density equal to:


Its quaternionic representation contains vectors modulo that are congruent to either one of coordinate-frames, or zero;[33][34] with 1,365 the twelfth Jacobsthal number, and 144 equal to 122.

Fischer group is a sporadic group with a total of twelve maximal subgroups, the smallest of which is Mathieu group .[35][36] holds standard generators equal to (2A, 13, 11),[37] with a further condition where .[38] Furthermore, its faithful complex representation is 78-dimensional,[39] where 78 is the twelfth triangular number.[40] Otherwise, the largest alternating group represented inside any sporadic groups is , as a maximal subgroup inside the third-largest third generation sporadic group, Harada-Norton group .[41][42] While or are not maximal subgroups of the largest sporadic group, the friendly giant , one of its maximal subgroup is .[43] More deeply, the double cover is a maximal subgroup of ,[44][45] which is the third-largest maximal subgroup inside ;[46][47] with the double cover as the largest maximal subgroup inside .[43] The smallest second generation sporadic group, Janko group , holds standard generators (2A, 3B, 7) that yield .[38]

Twelve is the smallest weight for which a cusp form exists. This cusp form is the discriminant whose Fourier coefficients are given by the Ramanujan -function and which is (up to a constant multiplier) the 24th power of the Dedekind eta function:

This fact is related to a constellation of interesting appearances of the number twelve in mathematics ranging from the fact that the abelianization of special linear group has twelve elements, to the value of the Riemann zeta function at being , which stems from the Ramanujan summation

Although the series is divergent, methods such as Ramanujan summation can assign finite values to divergent series.

  • 12 is an Anti-Meertens Number. If we power the digits from the end to the prime numbers starting from 2 and then multiply, then the result will be the number Itself.

2^2 * 3^1 = 12

List of basic calculations[edit]

Multiplication 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 50 100 1000
12 × x 12 24 36 48 60 72 84 96 108 120 132 144 156 168 180 192 204 216 228 240 252 264 276 288 300 600 1200 12000
Division 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
12 ÷ x 12 6 4 3 2.4 2 1.714285 1.5 1.3 1.2 1.09 1 0.923076 0.857142 0.8 0.75
x ÷ 12 0.083 0.16 0.25 0.3 0.416 0.5 0.583 0.6 0.75 0.83 0.916 1 1.083 1.16 1.25 1.3
Exponentiation 1 2 3 4 5 6 7 8 9 10 11 12
12x 12 144 1728 20736 248832 2985984 35831808 429981696 5159780352 61917364224 743008370688 8916100448256
x12 1 4096 531441 16777216 244140625 2176782336 13841287201 68719476736 282429536481 1000000000000 3138428376721 8916100448256

In other bases[edit]

The duodecimal system (1210 [twelve] = 1012), which is the use of 12 as a division factor for many ancient and medieval weights and measures, including hours, probably originates from Mesopotamia.

In base thirteen and higher bases (such as hexadecimal), twelve is represented as C.

In nature[edit]

Notably, twelve is the number of full lunations in a solar year, hence the number of months in a solar calendar, as well as the number of signs in the Western, Islamic and the Chinese zodiac. Twelve is also the number of years for an orbital period of Jupiter.


The number twelve carries religious, mythological and magical symbolism, generally representing perfection, entirety, or cosmic order in traditions since antiquity.[48]

Ancient Greek religion[edit]

Judaism and Christianity[edit]

  • The significance is especially pronounced in the Hebrew Bible.

Ishmael – the first-born son of Abraham – has 12 sons/princes (Genesis 25:16), and Jacob also has 12 sons, who are the progenitors of the Twelve Tribes of Israel.[50] This is reflected in Christian tradition, notably in the twelve Apostles. When Judas Iscariot is disgraced, a meeting is held (Acts) to add Saint Matthias to complete the number twelve once more. The Book of Revelation contains much numerical symbolism, and many of the numbers mentioned have 12 as a divisor. 12:1 mentions a woman—interpreted as the people of Israel, the Church and the Virgin Mary—wearing a crown of twelve stars (representing each of the twelve tribes of Israel). Furthermore, there are 12,000 people sealed from each of the twelve tribes of Israel (the Tribe of Dan is omitted while Manasseh is mentioned), making a total of 144,000 (which is the square of 12 multiplied by a thousand).

12 was the only number considered to be religiously divine in the 1600s causing many Catholics to wear 12 buttons to church every Sunday. Some extremely devout Catholics would always wear this number of buttons to any occasion on any type of clothing.[citation needed]


Twelve is referred to in a few different verses of the Quran. Two are in reference to the Twelve Tribes of Israel.

And ˹remember˺ when Moses prayed for water for his people, We said, "Strike the rock with your staff." Then twelve springs gushed out, ˹and˺ each tribe knew its drinking place. ˹We then said,˺ "Eat and drink of Allah’s provisions, and do not go about spreading corruption in the land."

— Surah Al-Baqarah (The Heifer):60[51]

The second reference is:

We divided them into twelve tribes—each as a community. And We revealed to Moses, when his people asked for water, "Strike the rock with your staff." Then twelve springs gushed out. Each tribe knew its drinking place. We shaded them with clouds and sent down to them manna and quails,1 ˹saying˺, "Eat from the good things We have provided for you." They ˹certainly˺ did not wrong Us, but wronged themselves.

— Surah Al-A'raf (The Heights):160[52]

Note 1: Manna (heavenly bread) and quails (chicken-like birds) sustained the children of Israel in the wilderness after they left Egypt.

The third reference is to the number of months and the sacred ones amongst them:

Indeed, the number of months with Allāh is twelve [lunar] months in the register of Allāh [from] the day He created the heavens and the earth; of these, four are sacred.2

— Surah At-Tawbah (The Repentance):36[53]

Note 2: The four sacred months of the Islamic calendar are Dhu al-Qa'dah, Dhu al-Hijjah, Muharram, and Rajab (months 11, 12, 1 and 7).


  • There are twelve Jyotirlinga (Self-formed Lingas) of Lord Shiva in Hindu temples across India according to the Shaiva tradition.
  • The Sun god Surya has 12 names.
  • The god Hanuman has 12 names.
  • There are 12 Petals in Anahata or "heart chakra".
  • There are frequently said to be 12 Âdityas.


Ancient Hittite relief carving from Yazılıkaya, a sanctuary at Hattusa, depicting twelve gods of the underworld[54]


  • The number of twelve jurors in jury trials is depicted by Aeschylus in the Eumenides. In the play, the innovation is brought about by the goddess Athena, who summons twelve citizens to sit as jury.
  • In English Common Law, the tradition of twelve jurors harks back to the 10th-century law code introduced by Aethelred the Unready.


  • The lunar year is 12 lunar months. Adding 11 or 12 days completes the solar year.[56]
  • Most calendar systems – solar or lunar – have twelve months in a year.
  • The Chinese use a 12-year cycle for time-reckoning called Earthly Branches.
  • There are twelve hours in a half day, numbered one to twelve for both the ante meridiem (a.m.) and the post meridiem (p.m.). 12:00 p.m. is midday or noon, and 12:00 a.m. is midnight.
  • The basic units of time (60 seconds, 60 minutes, 24 hours) are evenly divisible by twelve into smaller units.

In numeral systems[edit]

۱۲ Arabic ១២ Khmer ԺԲ Armenian
১২ Bangla ΔΙΙ Attic Greek 𝋬 Maya
יב Hebrew
१२ Indian and Nepali (Devanāgarī) 十二 Chinese and Japanese
௧௨ Tamil XII Roman and Etruscan
๑๒ Thai IIX Chuvash
౧౨ Telugu and Kannada ١٢ Urdu
ιβʹ Ionian Greek ൧൨ Malayalam

In science[edit]

Image of the globular cluster Messier 12 by Hubble Space Telescope

In sports[edit]

  • In both soccer and American football, the number 12 can be a symbolic reference to the fans because of the support they give to the 11 players on the field. Texas A&M University reserves the number 12 jersey for a walk-on player who represents the original "12th Man", a fan who was asked to play when the team's reserves were low in a college American football game in 1922. Similarly, Bayern Munich, Hammarby, Feyenoord, Atlético Mineiro, Flamengo, Seattle Seahawks, Portsmouth and Cork City do not allow field players to wear the number 12 on their jersey because it is reserved for their supporters. It also serves as the jersey number for some the National Football League's best and most well-known quarterback, Tom Brady.
  • In Canadian football, 12 is the maximum number of players that can be on the field of play for each team at any time.
  • In cricket, another sport with eleven players per team, teams may select a "12th man", who may replace an injured player for the purpose of fielding (but not batting or bowling).
  • In women's lacrosse, each team has 12 players on the field at any given time, except in penalty situations.
  • In rugby league, one of the starting second-row forwards wears the number 12 jersey in most competitions. An exception is in the Super League, which uses static squad numbering.
  • In rugby union, one of the starting centres, most often but not always the inside centre, wears the 12 shirt.
  • In an NBA game, a quarter lasts 12 minutes.
  • In pool:
    • The pool ball 12 is the 12th in pool and its color is purple.

In technology[edit]

In the arts[edit]


Films with the number twelve or its variations in their titles include:





Music theory[edit]

Pop music[edit]

Art theory[edit]

  • There are twelve basic hues in the color wheel: three primary colors (red, yellow, blue), three secondary colors (orange, green, purple) and six tertiary colors (names for these vary, but are intermediates between the primaries and secondaries).


  • In the game of craps, a dice roll of two sixes (value 12) on the come-out roll constitutes a "craps" and the shooter (dice thrower) loses immediately.
  • Twelve is a character in the Street Fighter video game series.
  • Games such as Backgammon have a long history of 12 points on each side of the gaming board, as evidenced in the XII scripta board in the museum at Ephesus.[57]

In other fields[edit]

12 stars are featured on the Flag of Europe.
  • There are 12 troy ounces in a troy pound (used for precious metals).
  • Twelve of something is called a dozen.
  • In the former British currency system, there were twelve pence in a shilling.[58]
  • In Greek mythology, the number of Labours of Heracles was increased from ten to make twelve.
  • In English, twelve is the number of greatest magnitude that has just one syllable.
    The numerical range on the analog clock ends at 12.
  • 12 is the last number featured on the analogue clock, and also the starting point of the transition from A.M. to P.M. hours or vice-versa.
  • There are twelve months within a year, with the last one being December.
  • The level of grades in which one must attend school typically ends at 12 (although some jurisdictions may include a thirteenth grade depending on the country).[citation needed]
  • Twelve hours form half a day, and twelve hours away from another lead to the same time but with a different period (ex. Twelve hours away from 6:00 AM leads to 6:00 PM).
  • There are normally twelve pairs of ribs in the human body.
  • The Twelve Tables or Leges Duodecim Tabularum, more informally simply Duodecim Tabulae, was the ancient legislation underlying Roman law.
  • In the United States, twelve people are appointed to sit on a jury for felony trials in all but four states, and in federal and Washington, D.C. courts. The number of jurors gave the title to the play (and subsequent films) Twelve Angry Men.
  • Twelve men have walked on Earth's moon.
  • The United States is divided into twelve Federal Reserve Districts (Boston, New York, Philadelphia, Cleveland, Richmond, Atlanta, Chicago, St. Louis, Minneapolis, Kansas City, Dallas, and San Francisco); American paper currency has serial numbers beginning with one of twelve different letters, A through L, representing the Federal Reserve Bank from which the currency originated.
  • According to UFO conspiracy theory, Majestic 12 is a secret committee, allegedly set up by U.S. President Harry S. Truman to investigate the Roswell UFO incident and cover up future extraterrestrial contact.
  • 12 is the number of the French department Aveyron.
  • King Arthur's Round Table had 12 knights plus King Arthur himself.
  • 12 inches in a foot.
  • Alcoholics Anonymous has 12 steps, 12 traditions and 12 concepts for world service.
  • Wilhelm Heinrich Schüßler developed a list of 12 biochemical cell salts, also known as tissue salts.
  • In the People's Republic of China, 12 Core Socialist Values were promoted as part of a campaign beginning in 2012.

See also[edit]


  1. ^ Specially, a passage referring to Judas Iscariot as "one of the twelve" (an of ðæm tuelfum).[4]


  1. ^ Gordon, E. V. (1957). Introduction to Old Norse. Oxford, England: Clarendon Press. pp. 292–293. Archived from the original on 2016-04-15. Retrieved 2017-09-08.
  2. ^ Stevenson, W. H. (December 1899). "The Long Hundred and its Use in England". Archaeological Review. 4 (5): 313–317.
  3. ^ Goodare, Julian (1993). "The Long Hundred in medieval and early modern Scotland" (PDF). Proceedings of the Society of Antiquaries of Scotland. 123: 395–418. doi:10.9750/PSAS.123.395.418. S2CID 162146336.
  4. ^ John 6:71.
  5. ^ a b c d e f Oxford English Dictionary, 1st ed. "twelve, adj. and n." Oxford University Press (Oxford), 1916.
  6. ^ a b Oxford English Dictionary, 1st ed. "eleven, adj. and n." Oxford University Press (Oxford), 1891.
  7. ^ Dantzig, Tobias (1930), Number: The Language of Science.
  8. ^ "Numbers: Writing Numbers // Purdue Writing Lab". Purdue Writing Lab. Retrieved 25 February 2020.
  9. ^ Sloane, N. J. A. (ed.). "Sequence A002808 (The composite numbers: numbers n of the form x*y for x > 1 and y > 1.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-06-15.
  10. ^ "Sloane's A000178: Superfactorials". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-07-29.
  11. ^ Sloane, N. J. A. (ed.). "Sequence A002378 (Oblong (or promic, pronic, or heteromecic) numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-06-15.
  12. ^ Sloane, N. J. A. (ed.). "Sequence A005101 (Abundant numbers (sum of divisors of m exceeds 2m).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-01-10.
  13. ^ Sloane, N. J. A. (ed.). "Sequence A005835 (Pseudoperfect (or semiperfect) numbers n: some subset of the proper divisors of n sums to n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
  14. ^ Sloane, N. J. A. (ed.). "Sequence A001097 (Twin primes.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-06-15.
  15. ^ Sloane, N. J. A. (ed.). "Sequence A002182 (Highly composite numbers, definition (1): numbers n where d(n), the number of divisors of n (A000005), increases to a record.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-06-19.
  16. ^ Sloane, N. J. A. (ed.). "Sequence A033950 (Refactorable numbers: number of divisors of k divides k. Also known as tau numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-06-15.
  17. ^ Sloane, N. J. A. (ed.). "Sequence A000396 (Perfect numbers k: k is equal to the sum of the proper divisors of k.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-01-10.
  18. ^ "Sloane's A081357 : Sublime numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
  19. ^ Sloane, N. J. A. (ed.). "Sequence A000129 (Pell numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-01-10.
  20. ^ Sloane, N. J. A. (ed.). "Sequence A000326 (Pentagonal numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-01-10.
  21. ^ Wall, Charles R. (1988). "New unitary perfect numbers have at least nine odd components" (PDF). Fibonacci Quarterly. 26 (4): 312. ISSN 0015-0517. MR 0967649. Zbl 0657.10003.
  22. ^ Sloane, N. J. A. (ed.). "Sequence A002827 (Unitary perfect numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-01-10.
  23. ^ Sloane, N. J. A. (ed.). "Sequence A000203 (Sigma n, the sum of the divisors of n)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-01-10.
  24. ^ Sloane, N. J. A. (ed.). "Sequence A001599 (Harmonic or Ore numbers: numbers n such that the harmonic mean of the divisors of n is an integer.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-01-10.
  25. ^ Sloane, N. J. A. (ed.). "Sequence A001600 (Harmonic means of divisors of harmonic numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-01-11.
  26. ^ Pace P., Nielsen (2007). "Odd perfect numbers have at least nine distinct prime factors". Mathematics of Computation. 76 (260). Providence, R.I.: American Mathematical Society: 2109–2126. arXiv:math/0602485. Bibcode:2007MaCom..76.2109N. doi:10.1090/S0025-5718-07-01990-4. MR 2336286. S2CID 2767519. Zbl 1142.11086.
  27. ^ Sloane, N. J. A. (ed.). "Sequence A002860 (Number of Latin squares of order n)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-01-19.
  28. ^ Grünbaum, Branko; Shephard, G. C. (1987). "Section 2.1: Regular and uniform tilings". Tilings and Patterns. New York: W. H. Freeman and Company. p. 59. doi:10.2307/2323457. ISBN 0-7167-1193-1. JSTOR 2323457. OCLC 13092426. S2CID 119730123.
  29. ^ H. S. M. Coxeter (1991). Regular Complex Polytopes (2 ed.). Cambridge University Press. pp. 144–146. doi:10.2307/3617711. ISBN 978-0-521-39490-1. JSTOR 3617711. S2CID 116900933. Zbl 0732.51002.
  30. ^ Weber, Matthias (2005). "Kepler's small stellated dodecahedron as a Riemann surface" (PDF). Pacific Journal of Mathematics. 220 (1): 172. doi:10.2140/pjm.2005.220.167. MR 2195068. S2CID 54518859. Zbl 1100.30036.
  31. ^ Sloane, N. J. A. (ed.). "Sequence A002336 (Maximal kissing number of n-dimensional laminated lattice.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-04-06.
    Equal to 196,560 24-spheres in twenty-four dimensions.
  32. ^ Cohn, Henry; Kumar, Abhinav (2009). "Optimality and uniqueness of the Leech lattice among lattices". Annals of Mathematics. 170 (3). Princeton, NJ: Princeton University & the Institute for Advanced Study: 1003–1050. doi:10.4007/annals.2009.170.1003. JSTOR 25662173. MR 2600869. S2CID 10696627. Zbl 1213.11144.
  33. ^ Wilson, Robert A. (1982). "The Quaternionic Lattice for 2G2(4) and its Maximal Subgroups". Journal of Algebra. 77 (2). Elsevier: 451–453. doi:10.1016/0021-8693(82)90266-6. MR 0673128. S2CID 120032380. Zbl 0501.20013.
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    "The reader should note that each of Wilson's frames [Wilson 82] contains three of ours, with 3 · 48 = 144 vectors, and has slightly larger stabilizer."
  35. ^ Wilson, Robert A. (1984). "On maximal subgroups of the Fischer group Fi22". Mathematical Proceedings of the Cambridge Philosophical Society. 95 (2): 197–222. doi:10.1017/S0305004100061491. ISSN 0305-0041. MR 0735364.
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