|Divisors||1, 2, 4|
|Roman numeral (unicode)||Ⅳ, ⅳ|
- 1 In mathematics
- 2 List of basic calculations
- 3 Evolution of the glyph
- 4 In religion
- 5 In politics
- 6 In computing
- 7 In science
- 8 In logic and philosophy
- 9 In technology
- 10 In transport
- 11 In sports
- 12 In other fields
- 13 In music
- 14 Groups of four
- 15 References
- 16 External links
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4 is the smallest squared prime (p2) and the only even number in this form. 4 is also the only square one more than a prime number.
A number is a multiple of 4 if its last two digits are a multiple of 4. For example, 1092 is a multiple of 4 because 92 = 4 × 23.
In addition, 2 + 2 = 2 × 2 = 22 = 4. Continuing the pattern in Knuth's up-arrow notation, 2 ↑↑ 2 = 2 ↑↑↑ 2 = 4, and so on, for any number of up arrows. (That is, 2 [n] 2 = 4 for every positive integer n, where a [n] b is the hyperoperation.)
A four-sided plane figure is a quadrilateral (quadrangle) which include kites, rhombi, rectangles and squares, sometimes also called a tetragon. A circle divided by 4 makes right angles and four quadrants. Because of it, four (4) is the base number of the plane (mathematics). Four cardinal directions, four seasons, the duodecimal system, and the vigesimal system are based on four.
A solid figure with four faces as well as four vertices is a tetrahedron, and 4 is the smallest possible number of faces (as well as vertices) of a polyhedron. The regular tetrahedron is the simplest Platonic solid. A tetrahedron, which can also be called a 3-simplex, has four triangular faces and four vertices. It is the only self-dual regular polyhedron.
Four-dimensional space is the highest-dimensional space featuring more than three convex regular figures:
- Two-dimensional: infinitely many convex regular polygons.
- Three-dimensional: five convex regular polyhedra (the five Platonic solids).
- Four-dimensional: six convex regular polychora.
- Five-dimensional and every higher-dimensional: three regular convex polytopes (regular simplexes, hypercubes, cross-polytopes).
Four is the only integer n for which the (non trivial) alternating group An is not simple.
The four-color theorem states that a planar graph (or, equivalently, a flat map of two-dimensional regions such as countries) can be colored using four colors, so that adjacent vertices (or regions) are always different colors. Three colors are not, in general, sufficient to guarantee this. The largest planar complete graph has four vertices.
Lagrange's four-square theorem states that every positive integer can be written as the sum of at most four square numbers. Three are not always sufficient; 7 for instance cannot be written as the sum of three squares.
Each natural number divisible by 4 is a difference of squares of two natural numbers, i.e. 4x = y2 − z2.
List of basic calculations
|4 × x||4||8||12||16||20||24||28||32||36||40||44||48||52||56||60||64||68||72||76||80||84||88||92||96||100||200||400||4000|
|4 ÷ x||4||2||1.3||1||0.8||0.6||0.571428||0.5||0.4||0.4||0.36||0.3||0.307692||0.285714||0.26||0.25|
|x ÷ 4||0.25||0.5||0.75||1||1.25||1.5||1.75||2||2.25||2.5||2.75||3||3.25||3.5||3.75||4|
Evolution of the glyph
Representing 1, 2 and 3 in as many lines as the number represented worked well. The Brahmin Indians simplified 4 by joining its four lines into a cross that looks like the modern plus sign. The Shunga would add a horizontal line on top of the numeral, and the Kshatrapa and Pallava evolved the numeral to a point where the speed of writing was a secondary concern. The Arabs' 4 still had the early concept of the cross, but for the sake of efficiency, was made in one stroke by connecting the "western" end to the "northern" end; the "eastern" end was finished off with a curve. The Europeans dropped the finishing curve and gradually made the numeral less cursive, ending up with a glyph very close to the original Brahmin cross.
Television stations that operate on channel 4 have occasionally made use of another variation of the "open 4", with the open portion being on the side, rather than the top. This version resembles the Canadian Aboriginal syllabics letter ᔦ or the Coptic letter Ϥ. The magnetic ink character recognition "CMC-7" font also uses this variety of "4".
- Four Noble Truths – Dukkha, Samudaya, Nirodha, Magga
- Four sights – observations which affected Prince Siddhartha deeply and made him realize the sufferings of all beings, and compelled him to begin his spiritual journey—an old man, a sick man, a dead man, and an ascetic
- Four Great Elements – earth, water, fire, and wind
- Four Heavenly Kings
- Four Foundations of Mindfulness – contemplation of the body, contemplation of feelings, contemplation of mind, contemplation of mental objects
- Four Right Exertions
- Four Bases of Power
- Four jhānas
- Four arūpajhānas
- Four Divine Abidings – loving-kindness, compassion, sympathetic joy, and equanimity
- Four stages of enlightenment – stream-enterer, once-returner, non-returner, and arahant
- Four main pilgrimage sites – Lumbini, Bodh Gaya, Sarnath, and Kusinara
- The Tetragrammaton is the four-letter name of God.
- Ezekiel has a vision of four living creatures: a man, a lion, an ox, and an eagle.
- The four Matriarchs (foremothers) of Judaism are Sarah, Rebekah, Leah, and Rachel.
- The Four Species (lulav, hadass, aravah and etrog) are taken as one of the mitzvot on the Jewish holiday of Sukkot. (Judaism)
- The Four Cups of Wine to drink on the Jewish holiday of Passover. (Judaism)
- The Four Questions to be asked on the Jewish holiday of Passover. (Judaism)
- The Four Sons to be dealt with on the Jewish holiday of Passover. (Judaism)
- The Four Expressions of Redemption to be said on the Jewish holiday of Passover. (Judaism)
- The four Gospels: Matthew, Mark, Luke, and John. (Christianity)
- The Four Horsemen of the Apocalypse ride in the Book of Revelation. (Christianity)
- The Four Holy Cities of Judaism
- There are four Vedas: Rigveda, Samaveda, Yajurveda and Atharvaveda.
- In Puruṣārtha, there are four aims of human life: Dharma, Artha, Kāma, Moksha.
- The four stages of life Brahmacharya (student life), Grihastha (household life), Vanaprastha (retired life) and Sannyasa (renunciation).
- The four primary castes or strata of society: Brahmana (priest/teacher), Kshatriya (warrior/politician), Vaishya (landowner/entrepreneur) and Shudra (servant/manual laborer).
- The swastika symbol is traditionally used in Hindu religions as a sign of good luck and signifies good from all four directions.
- The god Brahma has four faces.
- Eid al-Adha lasts for four days, from the 10th to the 14th of Dhul Hijja.
- There are four Rashidun or Rightly Guided Caliphs: Abu Bakr, Umar ibn al-Khattab, Uthman ibn Affan and Ali ibn Abi Talib.
- The Four Arch Angels in Islam are: Jibraeel (Gabriel), Mikaeel (Michael), Izraeel (Azrael), and Israfil (Raphael)
- There are four months in which war is not permitted: Muharram, Rajab, Dhu al-Qi'dah and Dhu al-Hijjah.
- There are four Sunni schools of fiqh: Hanafi, Shafi`i, Maliki and Hanbali.
- There are four major Sunni Imams: Abū Ḥanīfa, Muhammad ibn Idris ash-Shafi`i, Malik ibn Anas and Ahmad ibn Hanbal.
- There are four books in Islam: Torah, Zaboor, Injeel, Quran.
- Waiting for four months is ordained for those who take an oath for abstention from their wives.
- The waiting period of the woman whose husband dies, is four months and ten days.
- When Abraham said: “My Lord, show me how You give life to the dead,” Allah said: “Why! Do you have no faith?” Abraham replied: “Yes, but in order that my heart be at rest.” He said: “Then take four birds, and tame them to yourself, then put a part of them on every hill, and summon them; they will come to you flying. [Al-Baqara 2:260]
- The respite of four months was granted to give time to the mushriks in Surah At-Tawba so that they should consider their position carefully and decide whether to make preparation for war or to emigrate from the country or to accept Islam.
- Those who accuse honorable women (of unchastity) but do not produce four witnesses, flog them with eighty lashes, and do not admit their testimony ever after. They are indeed transgressors. [An-Noor 24:4]
- In a more general sense, numerous mythological and cosmogonical systems consider Four corners of the world as essentially corresponding to the four points of the compass.
- Four is the sacred number of the Zia, an indigenous tribe located in the U.S. state of New Mexico.
- The Chinese, Vietnamese, the Koreans, and the Japanese are superstitious about the number four because it is a homonym for "death" in their languages.
- In Slavic mythology, the god Svetovid has four heads.
- Four Freedoms: four fundamental freedoms that Franklin D. Roosevelt declared ought to be enjoyed by everyone in the world: Freedom of Speech, Freedom of Religion, Freedom from Want, Freedom from Fear.
- Gang of Four: Popular name for four Chinese Communist Party leaders who rose to prominence during China's Cultural Revolution, but were ousted in 1976 following the death of Chairman Mao Zedong. Among the four was Mao's widow, Jiang Qing. Since then, many other political factions headed by four people have been called "Gangs of Four".
- 0x04 is the ASCII code of the character End of Transmission, which is abbreviated to EOT.
- Four bits (half a byte) are called a nybble
- A tetramer is a thing formed out of four sub-units.
- Four terrestrial (or rocky) planets in the Solar system: Mercury, Venus, Earth, and Mars.
- Four giant gas/ice planets in the Solar system: Jupiter, Saturn, Uranus, and Neptune.
- Four of Jupiter's moons (the Galilean moons) are readily visible from Earth.
- Messier object M4, a magnitude 7.5 globular cluster in the constellation Scorpius.
- The New General Catalogue object NGC 4, a faint galaxy in the constellation Pisces
- The Roman numeral IV stands for subgiant in the Yerkes spectral classification scheme.
- The Roman numeral IV (usually) stands for the fourth-discovered satellite of a planet or minor planet (e.g. Jupiter IV)
- Four is the number of nucleobase types in DNA and RNA – adenine, guanine, cytosine, thymine (uracil in RNA).
- Many chordates have four feet, legs or leglike appendages (tetrapods).
- The mammalian heart consists of four chambers.
- Many mammals (Carnivora, Ungulata) use four fingers for movement.
- All insects with wings except flies have four wings.
- Insects of the superorder Endopterygota, also known as Holometabola, such as butterflies, ants, bees, beetles, fleas, flies, moths, and wasps, undergo holometabolism—complete metamorphism in four stages—from (1) embryo (ovum, egg), to (2) larva (such as grub, caterpillar), then (3) pupa (such as the chrysalis), and finally (4) the imago.
- In the common ABO blood group system, there are 4 blood types (A, B, O, AB).
- Humans have four canines and four wisdom teeth.
- The cow's stomach is divided in four digestive compartments: reticulum, rumen, omasum and abomasum.
- Valency of carbon (that is basis of life on the Earth) is four. Also because of its tetrahedral crystal bond structure, diamond (one of the natural allotropes of carbon) is the hardest known naturally occurring material. It is also the valence of silicon, whose compounds form the majority of the mass of the Earth's crust.
- The atomic number of beryllium
- There are four basic states of matter: solid, liquid, gas, and plasma.
- Special relativity and general relativity treat nature as four-dimensional: 3D regular space and one-dimensional time are treated together and called spacetime. Also, any event E has a light cone composed of four zones of possible communication and cause and effect (outside the light cone is strictly incommunicado).
- There are four fundamental forces (electromagnetism, gravitation, the weak nuclear force, and the strong nuclear force).
- In statistical mechanics, the four functions inequality is an inequality for four functions on a finite distributive lattice.
In logic and philosophy
- The symbolic meanings of the number four are linked to those of the cross and the square. "Almost from prehistoric times, the number four was employed to signify what was solid, what could be touched and felt. Its relationship to the cross (four points) made it an outstanding symbol of wholeness and universality, a symbol which drew all to itself". Where lines of latitude and longitude intersect, they divide the earth into four proportions. Throughout the world kings and chieftains have been called "lord of the four suns" or "lord of the four quarters of the earth", which is understood to refer to the extent of their powers both territorially and in terms of total control of their subjects' doings.
- The Square of Opposition, in both its Aristotelian version and its Boolean version, consists of four forms: A ("All S is R"), I ("Some S is R"), E ("No S is R"), and O ("Some S is not R").
- In regard to whether two given propositions can have the same truth value, there are four separate logical possibilities: the propositions are subalterns (possibly both are true, and possibly both are false); subcontraries (it is possible that both are true, but not that both are false); contraries (it is possible that both are false, but not that both are true); or contradictories (it is not possible that both are true, and it is not possible that both are false).
- Aristotle held that there are basically four causes in nature: the efficient cause, the matter, the end, and the form.
- The Stoics held with four basic categories, all viewed as bodies (substantial and insubstantial): (1) substance in the sense of substrate, primary formless matter; (2) quality, matter's organization to differentiate and individualize something, and coming down to a physical ingredient such as pneuma, breath; (3) somehow holding (or disposed), as in a posture, state, shape, size, action, and (4) somehow holding (or disposed) toward something, as in relative location, familial relation, and so forth.
- Immanuel Kant expounded a table of judgments involving four three-way alternatives, in regard to (1) Quantity, (2) Quality, (3) Relation, (4) Modality, and, based thereupon, a table of four categories, named by the terms just listed, and each with three subcategories.
- Arthur Schopenhauer's doctoral thesis was On the Fourfold Root of the Principle of Sufficient Reason.
- Franz Brentano held that any major philosophical period has four phases: (1) Creative and rapidly progressing with scientific interest and results; then declining through the remaining phases, (2) practical, (3) increasingly skeptical, and (4) literary, mystical, and scientifically worthless—until philosophy is renewed through a new period's first phase. (See Brentano's essay "The Four Phases of Philosophy and Its Current State" 1895, tr. by Mezei and Smith 1998.)
- C. S. Peirce, usually a trichotomist, discussed four methods for overcoming troublesome uncertainties and achieving secure beliefs: (1) the method of tenacity (policy of sticking to initial belief), (2) the method of authority, (3) the method of congruity (following a fashionable paradigm), and (4) the fallibilistic, self-correcting method of science (see "The Fixation of Belief", 1877); and four barriers to inquiry, barriers refused by the fallibilist: (1) assertion of absolute certainty; (2) maintaining that something is absolutely unknowable; (3) maintaining that something is absolutely inexplicable because absolutely basic or ultimate; (4) holding that perfect exactitude is possible, especially such as to quite preclude unusual and anomalous phenomena (see "F.R.L." [First Rule of Logic], 1899).
- Paul Weiss built a system involving four modes of being: Actualities (substances in the sense of substantial, spatio-temporally finite beings), Ideality or Possibility (pure normative form), Existence (the dynamic field), and God (unity). (See Weiss's Modes of Being, 1958).
- Karl Popper outlined a tetradic schema to describe the growth of theories and, via generalization, also the emergence of new behaviors and living organisms: (1) problem, (2) tentative theory, (3) (attempted) error-elimination (especially by way of critical discussion), and (4) new problem(s). (See Popper's Objective Knowledge, 1972, revised 1979.)
- John Boyd (military strategist) made his key concept the decision cycle or OODA loop, consisting of four stages: (1) observation (data intake through the senses), (2) orientation (analysis and synthesis of data), (3) decision, and (4) action. Boyd held that his decision cycle has philosophical generality, though for strategists the point remains that, through swift decisions, one can disrupt an opponent's decision cycle.
- Richard McKeon outlined four classes (each with four subclasses) of modes of philosophical inquiry: (1) Modes of Being (Being); (2) Modes of Thought (That which is); (3) Modes of Fact (Existence); (4) Modes of Simplicity (Experience)—and, corresponding to them, four classes (each with four subclasses) of philosophical semantics: Principles, Methods, Interpretations, and Selections. (See McKeon's "Philosophic Semantics and Philosophic Inquiry" in Freedom and History and Other Essays, 1989.)
- Jonathan Lowe (E.J. Lowe) argues in The Four-Category Ontology, 2006, for four categories: kinds (substantial universals), attributes (relational universals and property-universals), objects (substantial particulars), and modes (relational particulars and property-particulars, also known as "tropes"). (See Lowe's "Recent Advances in Metaphysics," 2001, Eprint)
- Four opposed camps of the morality and nature of evil: moral absolutism, amoralism, moral relativism, and moral universalism.
- The resin identification code used in recycling to identify low-density polyethylene.
- Most furniture has four legs – tables, chairs, etc.
- The four color process (CMYK) is used for printing.
- Wide use of rectangles (with four angles and four sides) because they have effective form and capability for close adjacency to each other (houses, rooms, tables, bricks, sheets of paper, screens, film frames).
- In the Rich Text Format specification, language code 4 is for the Chinese language. Codes for regional variants of Chinese are congruent to 4 mod 256.
- Credit card machines have four-twelve function keys.
- On most phones, the 4 key is associated with the letters G, H, and I, but on the BlackBerry cellular phone, it is the key for D and F.
- On many computer keyboards, the "4" key may also be used to type the dollar sign ($) if the shift key is held down.
- It is the number of bits in a nibble, equivalent to half a byte
- In internet slang, "4" can replace the word "for" (as "four" and "for" are pronounced similarly). For example, typing "4u" instead of "for you".
- In Leetspeak, "4" may be used to replace the letter "A".
- The TCP/IP stack consists of four layers.
- Many internal combustion engines are called four-stroke engines because they complete one thermodynamic cycle in four distinct steps: Intake, compression, power, and exhaust.
- Most vehicles, including motor vehicles, and particularly cars/automobiles and light commercial vehicles have four road wheels.
- "Quattro", meaning four in the Italian language, is used by Audi as a trademark to indicate that four-wheel drive (4WD) technologies are used on Audi-branded cars. The word "Quattro" was initially used by Audi in 1980 in its original 4WD coupé, the Audi Quattro. Audi also has a privately held subsidiary company called quattro GmbH.
- List of highways numbered 4
- In the Australian Football League, the top level of Australian rules football, each team is allowed 4 "interchanges" (substitute players), who can be freely substituted at any time, subject to a limit on the total number of substitutions.
- In baseball:
- There are four bases in the game: first base, second base, third base, and home plate; to score a run, an offensive player must complete, in the sequence shown, a circuit of those four bases.
- When a batter receives four pitches that the umpire declares to be "balls" in a single at-bat, a base on balls, informally known as a "walk", is awarded, with the batter sent to first base.
- For scoring, number 4 is assigned to the second baseman.
- Four is the most runs that can be scored on any single at bat, whereby all three baserunners and the batter score (the most common being via a grand slam).
- In basketball, the number four is used to designate the power forward position, often referred to as "the four spot" or "the four".
- In cricket, a four is a specific type of scoring event, whereby the ball crosses the boundary after touching the ground at least one time, scoring four runs. Taking four wickets in four consecutive balls is typically referred to as a double hat trick (two consecutive, overlapping hat tricks).
- In rowing, a four refers to a boat for four rowers, with or without coxswain. In rowing nomenclature, 4− represents a coxless four and 4+ represents a coxed four.
- In rugby league:
- In rugby union:
In other fields
- The phrase "four-letter word" is used to describe many swear words in the English language.
- Four is the only number whose name in English has the same number of letters as its value.
- Four (四, formal writing: 肆, pinyin sì) is considered an unlucky number in Chinese, Korean, Vietnamese and Japanese cultures mostly in Eastern Asia because it sounds like the word "death" (死, pinyin sǐ). To avoid complaints from people with tetraphobia, many numbered product lines skip the "four": e.g. Nokia cell phones (there is no series beginning with a 4), Palm PDAs, etc. Some buildings skip floor 4 or replace the number with the letter "F", particularly in heavily Asian areas. See tetraphobia and Numbers in Chinese culture.
- The number of characters in a canonical four-character idiom.
- In the NATO phonetic alphabet, the digit 4 is called "fower".
- In Astrology, Cancer is the 4th astrological sign of the Zodiac.
- In Tarot, The Emperor is the fourth trump or Major Arcana card.
- In Tetris, a game named for the Greek word for 4, every shape in the game is formed of 4 blocks each.
- 4 represents the number of Justices on the Supreme Court of the United States necessary to grant a writ of certiorari (i.e., agree to hear a case; it is one less than the number necessary to render a majority decision).
- Number Four is a character in the book series Lorien Legacies
- In written music, common time is constructed of four beats per measure and a quarter note receives one beat.
- In popular or modern music, the most common time signature is also founded on four beats, i.e., 4/4 having four quarter note beats.
- The common major scale is built on two sets of four notes (e.g., CDEF, GABC), where the first and last notes create an octave interval (a pair-of-four relationship).
- The interval of a perfect fourth is a foundational element of many genres of music, represented in music theory as the tonic and subdominant relationship. Four is also embodied within the circle of fifths (also known as circle of fourths), which reveals the interval of four in more active harmonic contexts.
- The typical number of movements in a symphony.
- The number of completed, numbered symphonies by Johannes Brahms.
- The number of strings on a violin, a viola, a cello, double bass, a cuatro and a ukulele, and the number of string pairs on a mandolin.
- "Four calling birds" is the gift on the fourth day of Christmas in the carol "The Twelve Days of Christmas".
- Four, British rock band Bloc Party's fourth full-length album.
- Four, British-Irish boy band One Direction's fourth full-length album.
- 4, American artist Beyoncé's fourth album.
- 4, British–American band Foreigner's fourth studio album.
Groups of four
- Big Four (disambiguation)
- Four rules: addition, subtraction, multiplication, division.
- Greek classical elements (fire, air, water, earth).
- Four seasons: spring, summer, autumn, winter.
- The Four Seasons (disambiguation)
- A leap year occurs every four years.
- Four is the minimal number of contemporary continents: Americas, Eurafrasia, Sahul, Antarctica.
- Approximately four weeks (4 times 7 days) to a lunar month (synodic month = 29.53 days). Thus the number four is universally an integral part of primitive sacred calendars.
- Four weeks of Advent (and four Advent candles on the Advent wreath).
- Four cardinal directions: north, south, east, west.
- Four Temperaments: sanguine, choleric, melancholic, phlegmatic.
- Four Humors: blood, yellow bile, black bile, phlegm.
- Four Great Ancient Capitals of China.
- Four-corner method.
- Cardinal principles.
- Four cardinal virtues: justice, prudence, temperance, fortitude.
- Four suits of playing cards: hearts, diamonds, clubs, spades.
- Four nations of the United Kingdom: England, Wales, Scotland, Northern Ireland.
- Four provinces of Ireland: Munster, Ulster, Leinster, Connacht.
- Four Noble Truths in the Buddhist religion.
- Four estates: politics, administration, judiciary, journalism. Especially in the expression "Fourth Estate", which means journalism.
- Four Corners is the only location in the United States where four states come together at a single point: Colorado, Utah, New Mexico, and Arizona.
- Four Evangelists – Matthew, Mark, Luke, and John
- Four Doctors of Western Church – Saint Gregory the Great, Saint Ambrose, Saint Augustine, and Saint Jerome
- Four Doctors of Eastern Church – Saint John Chrysostom, Saint Basil the Great, and Gregory of Nazianzus and Saint Athanasius
- Four Galilean moons of Jupiter – Io, Europa, Ganymede, and Callisto
- The Gang of Four was a Chinese communist political faction.
- The Fantastic Four: Mr. Fantastic, The Invisible Woman, The Human Torch, The Thing.
- The Teenage Mutant Ninja Turtles: Leonardo, Michelangelo, Donatello, Raphael
- The Interesting Four: Seiko, The Human Stapler, Weather Woman, and Mr. Wonderful. Parody superhero team featured on NBC's Saturday Night Live.
- The Beatles were also known as the "Fab Four": John Lennon, Ringo Starr, George Harrison, Paul McCartney.
- Gang of Four is a British post-punk rock band formed in the late 1970s.
- Four rivers in the Garden of Eden (Genesis 2:10–14): Pishon (perhaps the Jaxartes or Syr Darya), Gihon (perhaps the Oxus or Amu Darya), Hiddekel (Tigris), and P'rat (Euphrates).
- There are also four years in a single Olympiad (duration between the Olympic Games). Many major international sports competitions follow this cycle, among them the FIFA World Cup and its women's version, the FIBA World Championships for men and women, and the Rugby World Cup.
- There are four limbs on the human body.
- Four Houses of Hogwarts in the Harry Potter series: Gryffindor, Hufflepuff, Ravenclaw, Slytherin.
- Four known continents of the world in the A Song of Ice and Fire series: Westeros, Essos, Sothoryos, Ulthos.
- Bryan Bunch, The Kingdom of Infinite Number. New York: W. H. Freeman & Company (2000): 48
- Georges Ifrah, The Universal History of Numbers: From Prehistory to the Invention of the Computer transl. David Bellos et al. London: The Harvill Press (1998): 394, Fig. 24.64
- Chevalier, Jean and Gheerbrant, Alain (1994), The Dictionary of Symbols. The quote beginning "Almost from prehistoric times..." is on p. 402.
- Wells, D. The Penguin Dictionary of Curious and Interesting Numbers London: Penguin Group. (1987): 55–58
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