6

This article is about the number. For the years, see6 BCandAD 6.For other uses, see6 (disambiguation)andNumber Six (disambiguation).

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Cardinal	six
Ordinal	6th (sixth)
Numeral system	senary
Factorization	2 × 3
Divisors	1, 2, 3, 6
Greek numeral	Ϛ´
Roman numeral	VI, vi, ↅ
Greekprefix	hexa-/hex-
Latinprefix	sexa-/sex-
Binary	1102
Ternary	203
Senary	106
Octal	68
Duodecimal	612
Hexadecimal	616
Greek	στ (or ΣΤ or ς)
Arabic,Kurdish,Sindhi,Urdu	٦
Persian	۶
Amharic	፮
Bengali	৬
Chinese numeral	Sáu, lục
Devanāgarī	६
Gujarati	૬
Hebrew	ו
Khmer	៦
Thai	๖
Telugu	౬
Tamil	௬
Saraiki	٦
Malayalam	൬
Armenian	Զ
Babylonian numeral	𒐚
Egyptian hieroglyph	𓏿
Morse code	_....

6(six) is thenatural numberfollowing5and preceding7.It is acomposite numberand the smallest perfect number.

In mathematics

Sixis the smallest positive integer which is neither asquare numbernor aprime number.It is the second smallestcomposite numberafterfour,equal to thesumand theproductof its three properdivisors(1,2and3).As such, six is the only number that is both the sum and product of three consecutive positive numbers. 6 is the smallestperfect number,which are numbers that are equal to theiraliquot sum,or sum of their proper divisors.It is the largest of the fourall-Harshad numbers(1, 2,4,and 6),where it represents the sum between the first prime andcomposite,2 and 4.

6 is apronic numberand the only semiprime to be.It is the first discretebiprime(2 × 3)which makes it the first member of the (2 ×q) discrete biprime family, whereqis a higher prime. All primes above 3 are of the form 6n± 1 forn≥ 1.

As a perfect number:

• 6 is related to theMersenne prime3, since2¹(2²– 1) = 6.(The next perfect number is28.)
• 6 is the only evenperfect numberthat is not the sum of successive odd cubes.
• 6 is the root of the 6-aliquot tree, and is itself thealiquot sumof only one other number; thesquare number25(that is also the number of regions formed by thediagonalsandedgesof aregularsix-sidedhexagon).

Six is the firstunitary perfect number,since it is the sum of its positive properunitary divisors,without including itself. Only five such numbers are known to exist;sixty(10 × 6) andninety(15 × 6) are the next two.

It is the firstprimitivepseudoperfect number,and all integers$n$that are multiples of 6 are pseudoperfect (all multiples of a perfect number are pseudoperfect); six is also the smallestGranville number,or${\mathcal {S}}$-perfect number.

Unrelated to 6's being a perfect number, aGolomb rulerof length 6 is a "perfect ruler".Six is acongruent number.

6 is the secondprimary pseudoperfect number,andharmonic divisor number.It is also the secondsuperior highly composite number,and the last to also be aprimorial.

There are 6 non-equivalent ways in which100can be expressed as the sum of two prime numbers: (3 + 97), (11 + 89), (17 + 83), (29 + 71), (41 + 59) and (47 + 53).

There is not a prime$p$such that the multiplicative order of 2 modulo$p$is 6, that is,$ord_{p}(2)=6$ ByZsigmondy's theorem,if$n$is a natural number that is not 1 or 6, then there is a prime$p$such that$ord_{p}(2)=n$.SeeA112927for such$p$.

The ring of integer of the sixth cyclotomic fieldQ(ζ₆),which is calledEisenstein integer,has 6 units: ±1, ±ω, ±ω²,where$\ Omega ={\frac {1}{2}}(-1+i{\sqrt {3}})=e^{2\pi i/3}$.

Thesix exponentials theoremguarantees (given the right conditions on the exponents) the transcendence of at least one of a set of exponentials.

There are six basictrigonometric functions:sin, cos,sec, csc, tan, and cot.

The smallest non-abelian groupis thesymmetric group$\mathrm {S_{3}}$which has3!= 6 elements.

Six is atriangular numberand so is its square (36). It is the firstoctahedral number,preceding19.

A regularcube,with sixfaces

A six-sidedpolygonis ahexagon,one of the threeregular polygonscapable oftiling the plane.Figurate numbersrepresenting hexagons (including six) are calledhexagonal numbers.Because 6 is the product of a power of 2 (namely 2¹) with nothing but distinctFermat primes(specifically 3), a regular hexagon is aconstructible polygonwith acompassandstraightedgealone. Ahexagramis a six-pointed geometricstar figure(with theSchläfli symbol{6/2}, 2{3}, or {{3}}).

Six similar coins can be arranged around a central coin of the same radius so that each coin makes contact with the central one (and touches both its neighbors without a gap), but seven cannot be so arranged. This makes 6 the answer to the two-dimensionalkissing number problem.The densestsphere packingof the plane is obtained by extending this pattern to thehexagonallatticein which eachcircletouches just six others.

There is only one non-trivialmagic hexagon:it is of order-3 and made of nineteen cells, with amagic constantof 38. All rows and columns in a 6 × 6magic squarecollectively generate amagic sumof666(which isdoubly triangular). On the other hand,Graeco-Latin squareswith order 6 do not exist; if$n$is anatural numberthat is not2or 6, then there is a Graeco-Latin square of order$n$.

Thecubeis one of fivePlatonic solids,with a total of sixsquaresasfaces.It is the only regular polyhedron that can generate auniform honeycombon its own, which is alsoself-dual.Thecuboctahedron,which is anArchimedeansolid that is one of twoquasiregular polyhedra,has eight triangles and six squares as faces. Inside, itsvertex arrangementcan be interpreted as three hexagons that intersect to form an equatorial hexagonalhemi-face,by-which the cuboctahedron is dissected intotriangular cupolas.This solid is also the only polyhedron withradial equilateral symmetry,where its edges and long radii are of equal length; its one of only fourpolytopeswith this property — the others are the hexagon, thetesseract(as the four-dimensionalanalogueof the cube), and the24-cell.Only six polygons arefacesof non-prismaticuniform polyhedrasuch as the Platonic solids or the Archimedean solids: thetriangle,thesquare,thepentagon,the hexagon, theoctagon,and thedecagon.Ifself-dualimages of the tetrahedron are considered distinct, then there are a total of six regular polyhedra that are formed by three differentWeyl groupsin the third dimension (based ontetrahedral,octahedralandicosahedralsymmetries).

How closely the shape of an object resembles that of a perfectsphereis called itssphericity,calculated by:

$$\Psi ={\frac {\pi ^{\frac {1}{3}}\left(6V_{p}\right)^{\frac {2}{3}}}{A_{p}}}={\frac {A_{s}}{A_{p}}},$$where$A_{s}$is thesurface areaof the sphere,$V_{p}$the volume of the object, and$A_{p}$the surface area of the object.

Infour dimensions,there are a total of sixconvex regular polytopes:the5-cell,8-cell,16-cell,24-cell,120-cell,and600-cell.

$\mathrm {S_{6}}$,with 720 = 6!elements,is the onlyfinitesymmetric group which has anouter automorphism.This automorphism allows us to construct a number of exceptionalmathematical objectssuch as the S(5,6,12)Steiner system,theprojective planeof order 4, the four-dimensional5-cell,and theHoffman-Singleton graph.A closely related result is the following theorem: 6 is the only natural number$n$for which there is a construction of$n$isomorphicobjects on an$n$-set$A$,invariant under all permutations of$A$,but not naturally inone-to-one correspondencewith the elements of$A$.This can also be expressedcategory theoretically:consider thecategorywhose objects are the$n$element sets and whose arrows are thebijectionsbetween the sets. This category has a non-trivialfunctorto itself only for$n=6$.

In theclassification of finite simple groups,twenty of twenty-sixsporadic groupsin thehappy familyare part of three families of groups which divide the order of thefriendly giant,the largest sporadic group: fivefirst generationMathieu groups,sevensecond generationsubquotientsof theLeech lattice,and eightthird generationsubgroupsof the friendly giant. The remainingsixsporadic groups do not divide the order of the friendly giant, which are termed thepariahs(Ly,O'N,Ru,J₄,J₃,andJ₁).

List of basic calculations

Multiplication	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	25	50	100	1000
6 ×x	6	12	18	24	30	36	42	48	54	60	66	72	78	84	90	96	102	108	114	120	150	300	600	6000

Division	1	2	3	4	5	6	7	8	9	10		11	12	13	14	15
6 ÷x	6	3	2	1.5	1.2	1	0.857142	0.75	0.6	0.6		0.54	0.5	0.461538	0.428571	0.4
x÷ 6	0.16	0.3	0.5	0.6	0.83	1	1.16	1.3	1.5	1.6		1.83	2	2.16	2.3	2.5

Exponentiation	1	2	3	4	5	6	7	8	9	10		11	12	13
6x	6	36	216	1296	7776	46656	279936	1679616	10077696	60466176		362797056	2176782336	13060694016
x6	1	64	729	4096	15625	46656	117649	262144	531441	1000000		1771561	2985984	4826809

Greek and Latin word parts

Hexa

Hexais classicalGreekfor "six".Thus:

• "Hexadecimal"combineshexa-with the Latinatedecimalto name anumber baseof 16
• Ahexagonis aregular polygonwith six sides
  • L'Hexagoneis a French nickname for the continental part ofMetropolitan Francefor its resemblance to aregular hexagon
• Ahexahedronis apolyhedronwith six faces, with acubebeing a special case
• Hexameteris a poetic form consisting of six feet per line
• A "hex nut" is anutwith six sides, and a hexbolthas a six-sided head
• The prefix "hexa-"also occurs in thesystematic nameof manychemical compounds,such ashexanewhich has 6 carbon atoms (C₆H₁₄).

The prefixsex-

Sex-is aLatinprefixmeaning "six".Thus:

• Senaryis the ordinal adjective meaning "sixth"
• People withsexdactylyhave six fingers on each hand
• The measuring instrument called asextantgot its name because its shape forms one-sixth of a wholecircle
• A group of six musicians is called asextet
• Six babies delivered in one birth aresextuplets
• Sexy primepairs – Prime pairs differing by six aresexy,because sex is the Latin word for six.

TheSI prefixfor 1000⁶isexa-(E), and for its reciprocalatto-(a).

Evolution of the Hindu-Arabic digit

The first appearance of 6 is in theEdicts of Ashokac. 250 BCE.These areBrahmi numerals,ancestors of Hindu-Arabic numerals.

The first known digit "6" in the number "256" in Ashoka'sMinor Rock EdictNo.1 inSasaram,c. 250 BCE

The evolution of our modern digit 6 appears rather simple when compared with the other digits. The modern 6 can be traced back to theBrahmi numeralsofIndia,which are first known from theEdicts of Ashokac. 250 BCE.It was written in one stroke like a cursive lowercase e rotated 90 degrees clockwise. Gradually, the upper part of the stroke (above the central squiggle) became more curved, while the lower part of the stroke (below the central squiggle) became straighter. The Arabs dropped the part of the stroke below the squiggle. From there, the European evolution to our modern 6 was very straightforward, aside from a flirtation with a glyph that looked more like an uppercase G.

On theseven-segment displaysof calculators and watches, 6 is usually written with six segments. Some historical calculator models use just five segments for the 6, by omitting the top horizontal bar. This glyph variant has not caught on; for calculators that can display results in hexadecimal, a 6 that looks like a "b" is not practical.

Just as in most moderntypefaces,in typefaces withtext figuresthe character for the digit 6 usually has anascender,as, for example, in.

This digit resembles an inverted9.To disambiguate the two on objects and documents that can be inverted, the 6 has often been underlined, both in handwriting and on printed labels.

In music

A standardguitarhas sixstrings.

In artists

• Les Six( "The Six" in English) was a group consisting of the French composersGeorges Auric,Louis Durey,Arthur Honegger,Darius Milhaud,Francis PoulencandGermaine Tailleferrein the 1920s
• Bands with the number six in their name includeSix Organs of Admittance,6 O'Clock Saints,Electric Six,Eve 6,Los Xey(seiis Basque for "six" ), Out On Blue Six, Six In Six,Sixpence None the Richer,Slant 6,Vanity 6, andYou Me At Six
• #6 is the pseudonym ofAmericanmusicianShawn Crahan,when performing with the bandSlipknot

In instruments

• A standardguitarhas six strings
• Mostwoodwindinstruments have six basic holes or keys (e.g., bassoon, clarinet, pennywhistle, saxophone); these holes or keys are usually not given numbers or letters in the fingering charts

In music theory

• There are six whole tones in an octave.
• There are six semitones in a tritone.

In works

• "Six geese a-laying" were given as a present on the sixth day in the popularChristmas carol,"The Twelve Days of Christmas".
• Divided in six arias,Hexachordum Apollinisis generally regarded as one of the pinnacles ofJohann Pachelbel's oeuvre.
• The theme of the sixth album by Dream Theater,Six Degrees of Inner Turbulence,was the number six: the album has six songs, and the sixth song—that is, the complete second disc—explores the stories of six individuals suffering from various mental illnesses.
• Aristotlegave six elements of tragedy, the first of which isMythos.

In religion

See also:666

Judaism

• Six points on aStar of David
• Six orders of theMishnah
• Six symbolic foods placed on thePassover Seder Plate
• Godtook six days to create the world in theOld TestamentBook of Genesis;humankind was created on day 6. In theCity of God,Augustine of Hipposuggested (book 11, chapter 30) that God's creation of the world took six days because 6 is aperfect number.
• TheJewishholiday ofShavuotstarts on the sixth day of theHebrewmonth ofSivan
• Seraphshave six wings.

Islam

• There areSix articles of faith
• Six points on aSeal of Solomon
• Fasting six days ofShawwal,together with the month ofRamadan,is equivalent to fasting the whole year
• The heavens and earth were created in six days according to theQuran:

Indeed, We created the heavens and the earth and everything in between in six Days,¹and We were not ˹even˺ touched with fatigue.²

— SurahQaf:38

Note 1: The word day is not always used in the Quran to mean a 24-hour period. According to SurahAl-Hajj(The Pilgrimage):47, a heavenly Day is 1000 years of our time. TheDay of Judgmentwill be 50,000 years of our time - SurahAl-Maarij(The Ascending Stairways):4. Hence, the six Days of creation refer to six eons of time, known only by Allah.

Note 2: SomeIslamic scholarsbelieve this verse comes in response to Exodus 31:17, which says, "The Lord made the heavens and the earth in six days, but on the seventh day He rested and was refreshed."

Others

• InHindu theology,atrasarenuis the combination of six celestialparamānus(atoms).
• In Taoism:
  • Six Lines of aHexagram
  • Six MinistriesofHuang Di

In science

Astronomy

• Messier objectM6,a magnitude 4.5open clusterin the constellationScorpius,also known as theButterfly Cluster
• TheNew General CatalogueobjectNGC 6,aspiral galaxyin theconstellationAndromeda
• The Roman numeral VI:
  • Stands forsubdwarfsin theYerkes spectral classification scheme
  • (Usually) stands for the sixth-discovered satellite of a planet or minor planet (e.g.Jupiter VI)
• 6 Hebe

Biology

The cells of abeehiveare six-sided.

• The cells of abeehivearesix-sided.
• Insectshave sixlegs(seeHexapoda).
• Six kingdoms in thetaxonomic rankbelowdomain (biology);Animalia,Plantae,Fungi,Protista,Archaea/Archaeabacteria,andBacteria/Eubacteria.SeeKingdom (biology).
• The six elements most common in biomolecules are called the CHNOPS elements; the letters stand for the chemical abbreviations ofcarbon,hydrogen,nitrogen,oxygen,phosphorus,andsulfur.SeeCHON.

Chemistry

Amoleculeofbenzenehas aringof sixcarbonand sixhydrogenatoms.

• A benzene molecule has a ring of six carbon atoms.
• 6 is theatomic numberof carbon.
• The sixfoldsymmetryofsnowflakesarises from thehexagonalcrystal structureofordinary ice.
• Ahexameris an oligomer made of six subunits.

Medicine

• There are six tastes in traditional Indian medicine (Ayurveda): sweet, sour, salty, bitter, pungent, and astringent. These tastes are used to suggest a diet based on the symptoms of the body.
• Phase 6 is one of sixpandemic influenza phases.

Physics

In theStandard Modelofparticle physics,there are sixtypesofquarksand six types ofleptons.

• In the Standard Model of particle physics, there are six types ofquarksand six types ofleptons.
• Instatistical mechanics,thesix-vertex modelhas six possible configurations of arrows at each vertex
• There are six colors in the RGB color wheel: (primary) red, blue, green, (secondary) cyan, magenta, and yellow. (SeeTertiary color)
• Inthree-dimensionalEuclidean space,there are six unknownsupport reactionsfor astatically determinatestructure: oneforcein each of the three dimensions, and onemomentthrough each of three possibleorthogonalplanes.

In sports

• TheOriginal Sixteams in the National Hockey League areToronto,Chicago,Montreal,New York,Boston,andDetroit.They are the oldest remaining teams in the league, though not necessarily the first six; they comprised the entire league from1942to1967.
• Number of players:
  • Inassociation football(soccer), the number of substitutes combined by both teams, that are allowed in the game.
  • Inbox lacrosse,the number of players per team, including the goaltender, that are on the floor at any one time, excluding penalty situations.
  • Inice hockey,the number of players per team, including the goaltender, that are on the ice at any one time during regulation play, excluding penalty situations. (Some leagues reduce the number of players on the ice during overtime.)
  • Involleyball:
    • Six players from each team on each side play against each other.
    • Standard rules only allow six total substitutions per team per set. (Substitutions involving the libero, a defensive specialist who can only play in the back row, are not counted against this limit.)
  • Six-man footballis a variant of American or Canadian football, played by smaller schools with insufficient enrollment to field the traditional 11-man (American) or 12-man (Canadian) squad.
• Scoring:
  • In bothAmericanandCanadian football,6 points are awarded for a touchdown.
  • InAustralian rules football,six points are awarded for a goal, scored when a kicked ball passes between the defending team's two inner goalposts without having been touched by another player.
  • Incricket,six runsare scored for the batting team when the ball is hit to the boundary or the ground beyond it without having touched the ground in the field.
• Inbasketball,theballused for women's full-court competitions is designated "size 6".
• Inpoolandsnookerone its table contains six pockets.
• In mostrugby leaguecompetitions (but not theSuper League,which uses static squad numbering), the jersey number 6 is worn by the startingfive-eighth(Southern Hemisphere term) orstand-off(Northern Hemisphere term).
• Inrugby union,the starting blindside flanker wears jersey number 6. (Some teams use "left" and "right" flankers instead of "openside" and "blindside", with 6 being worn by the starting left flanker.)

In technology

6 as a resin identification code, used in recycling.

• On most phones, the 6 key is associated with the lettersM,N,andO,but on theBlackBerry Pearlit is the key forJandK,and on the BlackBerry 8700 series andCurve 8900with full keyboard, it is the key forF
• The "6-meter band"inamateur radioincludes the frequencies from 50 to 54 MHz
• 6 is theresin identification codeused in recycling to identifypolystyrene

In calendars

• In the ancientRoman calendar,Sextiliswas the sixth month. After theJulian reform,Junebecame the sixth month and Sextilis was renamed August
• Sextidi was the sixth day of thedécadein theFrench Revolutionary calendar

In the arts and entertainment

Games

• The number of sides on a cube, hence the highest number on a standarddie
• The six-sided tiles on a hex grid are used in many tabletop andboard games.
• The highest number on one end of a standarddomino

Literature

• The Power of Sixis a book written by Pittacus Lore, and the second in theLorien Legaciesseries.
• Number 6 is a character in the book seriesLorien Legacies.

TV

• Number Six(Tricia Helfer) is a family of fictional characters from the reimagined science fiction television series,Battlestar Galactica.
• Number 6,the main protagonist inThe Prisonerplayed byPatrick McGoohan,and portrayed byJim Caviezelinthe remake.
• Six is a character in the television seriesBlossomplayed byJenna von Oÿ.
• Six is the nickname of Kal Varrik, a central character in the television seriesDark Matter,played byRoger Cross.
• Sixis aHistorychannel series that chronicles the operations and daily lives of SEAL Team Six.
• Six Feet Under,anHBOseries that ran from 2005 to 2011.

Movies

• Number 6 (Teresa Palmer) is a character in the movieI Am Number Four(2011).
• The 6th Day(2000), starringArnold Schwarzenegger.
• The Sixth Sense(1999), written and directed byM. Night Shyamalanand starringHaley Joel OsmentandBruce Willis.
• Girl 6(1996), directed bySpike Lee.

Musicals

• Sixis a modern retelling of the lives of the six wives ofHenry VIIIpresented as a pop concert.

Anthropology

• The name of the smallest group ofCub Scoutsand Guiding's equivalentBrownies,traditionally consisting of six people and is led by a "sixer".
• Acoffinis traditionally buried six feet under the ground; thus, the phrase "six feet under" means that a person (or thing, or concept) is dead
• There are said to be no more thansix degrees of separationbetween any two people on Earth.
• In Western astrology,Virgois the 6th astrological sign of the Zodiac
• TheSix Dynastiesform part of Chinese history
• Six is a luckynumber in Chinese culture.
• TheBirmingham Sixwere a British miscarriage of justice, held in prison for 16 years.
• "Six" is used as an informal slang term for the BritishSecret Intelligence Service,MI6.

In other fields

International maritime signal flagfor 6

• Six-packis a common form of packaging for six bottles or cans of drink (especially beer), and by extension, other assemblages of six items.Also, six is half adozen.
• The maximum number of dots in abraillecell.
• Extrasensory perceptionis sometimes called the "sixth sense".
• Six Flagsis an American company running amusement parks and theme parks in the U.S., Canada, and Mexico.
• In theU.S. Army"Six" as part of aradio call signis used by the commanding officer of a unit, while subordinate platoon leaders usually go by "One".(For a similar example see also:Rainbow Six.)

See also

• List of highways numbered 6
• Six degrees (disambiguation).

References

• The Odd Number 6,JA Todd, Math. Proc. Camb. Phil. Soc. 41 (1945) 66–68
• A Property of the Number Six,Chapter 6, P Cameron, JH v. Lint,Designs, Graphs, Codes and their LinksISBN0-521-42385-6
• Wells, D.The Penguin Dictionary of Curious and Interesting NumbersLondon: Penguin Group. (1987): 67 - 69

External links

• The Number 6
• The Positive Integer 6
• Prime curiosities: 6