Fact-checked by Grok 2 weeks ago

6

Six is the sixth , following five and preceding seven in the sequence of positive integers starting from one. As an even number, it is divisible into two equal parts without remainder. Six is a , meaning it has more than two distinct positive divisors, specifically 1, 2, 3, and 6. Its prime factorization is $2 \times 3. In , six holds a distinguished position as the smallest , where the sum of its proper divisors (1 + 2 + 3) equals the number itself. It is also the third , formed by the sum of the first three natural numbers (1 + 2 + 3). Geometrically, six is notable as the number of sides in a , a with equal sides and interior angles of 120 degrees each. Beyond , six appears in various scientific contexts, such as the of carbon, the essential to and life on . In physics, protons and neutrons, the main constituents of atomic nuclei, each consist of three quarks. The number's and divisibility properties also make it significant in patterns observed in nature, like the in .

Mathematics

Basic properties

Six is a that follows 5 and precedes 7 in the sequence of positive s. It is classified as an even because it is divisible by two or more positive integers other than and itself, specifically as the product of the primes 2 and . Six is the smallest , defined as a positive equal to the sum of its proper divisors (excluding itself), which for 6 are , 2, and , yielding $1 + 2 + 3 = 6. It is also the third , representing the sum of the first three s: $1 + 2 + 3 = 6, or equivalently \frac{3(3+1)}{2} = 6. Additionally, six is a , specifically $3! = 3 \times 2 \times [1](/page/1) = 6, the product of all positive integers up to . In terms of its divisors, six has exactly four positive divisors: 1, 2, , and 6, making it square-free (not divisible by any other than 1) and the smallest such composite square-free number. It is also a , since it has more positive divisors than any smaller positive .

Calculations and relations

In , fundamental operations involving 6 demonstrate basic identities and computations. For instance, with the yields $6 + [0](/page/0) = 6, preserving the value under summation with zero. Similarly, multiplication by the multiplicative identity gives $6 \times [1](/page/1) = 6, confirming 6 as unchanged when scaled by unity. Squaring 6 produces $6^2 = [36](/page/36), a result obtained by $6 \times 6. The of 6, denoted $6! = 6 \times 5 \times 4 \times [3](/page/3) \times 2 \times [1](/page/1) = 720, represents the product of all positive integers up to 6 and arises in combinatorial contexts. Key relations highlight 6's structure within . Its prime factorization is $6 = 2 \times [3](/page/3), expressing it uniquely as a product of primes. The of the first 6 numbers is 21, computed via the \sum_{k=1}^n k = \frac{n(n+1)}{2} evaluated at n=6. In the series, the partial up to 6 terms is the 6th H_6 = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \frac{1}{6} = \frac{49}{20} \approx 2.45, which approximates the natural logarithm plus the Euler-Mascheroni constant. Wilson's theorem provides an equation linking 6! to for the prime 7: (7-1)! \equiv -1 \pmod{7}, so $6! = 720 \equiv -1 \pmod{7} since $720 \div 7 = 102 remainder 6, and $6 \equiv -1 \pmod{7}. This holds as 7 divides $720 + 1 = 721 exactly, illustrating the theorem's characterization of primes. In group theory and , 6 denotes the order of the S_3, the group of all permutations of 3 distinct objects under composition, with |S_3| = 3! = 6. This equals the number of permutations of 3 items, underscoring 6's role in counting arrangements.

Numeral and writing systems

Historical evolution

The earliest representations of the number 6 appear in cuneiform script around 3000 BCE, where it was depicted as six vertical wedge impressions on clay tablets, reflecting an additive system derived from for counting goods in early Mesopotamian accounting. This wedge-based form, part of a (base-60) system inherited by the Babylonians, emphasized practical enumeration rather than abstract symbolism, with the vertical wedges grouped to denote units up to 59 before shifting place values. In , circa 3000 BCE, the hieroglyph for 6 consisted of six vertical strokes or reed-like lines in an additive system, as seen in inscriptions from onward, such as those on tomb walls and administrative . This straightforward tally evolved into more cursive forms by the , simplifying the strokes for quicker writing on while maintaining the base-10 structure for measurements and records. Phoenician numerals, emerging around 1000 BCE in the , represented 6 through an additive system using a vertical stroke with a or six simple lines, akin to earlier tallies, though sometimes adapted from the letter waw in alphabetic notation for numerical values. This form, resembling a backwards 9 in some inscriptions on goods and stelae, facilitated across the Mediterranean and influenced subsequent scripts like . The digit 6 took a more distinctive shape in the of from the BCE, appearing as a vertical stroke with horizontal accents or a simple hook, as evidenced in Ashoka's edicts, marking the shift toward a place-value decimal system. By the Gupta period (4th–6th centuries CE), it evolved into a curved, single-stroke form resembling a lowercase "b" or elongated hook, spread through imperial inscriptions and mathematical texts like those of , laying the groundwork for modern variants. In the world by the , the Gupta-derived numeral for 6 adopted a looped or teardrop shape in Eastern forms, refined in Western Ghubar script with a closed oval and tail, as documented in treatises by and al-Uqlidisi for astronomical and commercial calculations. This looped design, emphasizing fluidity for dust-board writing, was transmitted to via routes and scholarly exchanges. The Italian mathematician introduced the Hindu-Arabic numeral 6 to in his 1202 work , depicting it as an open loop similar to the modern form, promoting its use over for efficiency in his examples of and . By the , Gutenberg's standardized the open-bottom 6 in Gothic and italic typefaces across European texts, solidifying its graphical consistency in manuscripts and early printed books. Key milestones in other traditions include the Chinese rod numerals from the 2nd century BCE, where 6 was formed by a horizontal (denoting 5) crossed with a vertical (1) on counting boards for and in classical . Similarly, the Maya vigesimal system, developed by 300 BCE, represented 6 as a (5) surmounted by a (1) in codices and stelae, integrating it into calendrical and astronomical computations.

Typographic variants

The digit 6 in the standard Western numeral system is commonly rendered with a closed upper and a lower curve or , but handwriting styles often feature an open-bottom variant where the lower portion forms a simple baseline curve without enclosing a , allowing for a fluid single-stroke motion. The closed-bottom form, incorporating a small enclosed at the base, is more typical in printed typography to ensure distinctiveness from similar characters like the lowercase 'g'. In other scripts, the representation of 6 varies significantly. The Devanagari numeral for 6 is ६, a symbol with a horizontal bar at the top and a curved form below, used in languages such as and . In the Cyrillic script, the digit 6 is identical to its Latin counterpart, facilitating shared usage across European languages that employ Cyrillic alphabets. For Hebrew, the value 6 in —the traditional Jewish numerological system—is represented by the letter vav (ו), a vertical line with a small hook, rather than a positional numeral. Typeface designs introduce further stylistic diversity for the digit 6. typefaces like feature a distinctive descending tail on the lower curve, adding elegance and readability in body text. fonts such as present a rounded, minimalist form without serifs, emphasizing clean geometry suitable for modern interfaces. Decorative styles, including or variants, render 6 with angular lines and intricate flourishes, evoking medieval manuscripts while maintaining numerical clarity. The 6 is encoded in as U+0036 for the basic Latin form, ensuring consistent rendering across digital platforms. Additional variants include the fullwidth form ｶ (U+FF16) for East Asian compatibility and mathematical 𝟔 (U+1D7DE) for use in equations and technical documents. In some standard fonts, the 6 closely resembles the lowercase 'b', potentially causing confusion in quick reading or for individuals with ; designs, such as , modify shapes with bottom-heavy weighting to reduce such ambiguities and improve legibility.

Etymology

Greek roots

The prefix hexa- originates from Ancient Greek ἑξα- (hexa-), the combining form of ἕξ (héx), meaning "six," which traces back to Proto-Indo-European \swéḱs. This root underwent phonetic shifts in Greek, where the initial s developed into an sound, represented by the (῾), resulting in héx. The word ἕξ is first attested in the of , dating to the 8th century BCE, where it appears in contexts describing quantities, such as groups of warriors or days. Key compounds derived from ἕξ include hexágōnon (ἑξάγωνον), denoting a , which Euclid formalized in his Elements around 300 BCE as part of geometric constructions, such as inscribing a regular in a circle (Book IV, Proposition 15). Similarly, hexámetros (ἑξάμετρος) refers to a line of verse consisting of six metrical feet, primarily dactylic, forming the basis of epic poetry like Homer's Iliad and Odyssey. These terms highlight the early integration of the root into mathematical and literary domains during the Classical period. In the Hellenistic era, the use of hexa- expanded in mathematical treatises, with serving as a foundational shape in due to its regular properties and relation to . This classical foundation influenced later scientific nomenclature, such as (introduced in the late ), a containing six carbon atoms, like glucose, reflecting the enduring legacy of numerical prefixes in chemistry. Notably, ἕξ itself was not employed as a numeral in ; instead, the acrophonic system used the archaic letter (Ϝ) for 6, later evolving into the ligature (ϛ) in medieval manuscripts to denote the value without phonetic use.

Latin and Romance influences

The Latin numeral for six, sex, derives from the s(w)éḱs, which denoted the number six across several ancient languages. This form appears in compounds such as sexaginta ("sixty," literally "six tens"), a term used by in his De Amicitia (ca. 44 BCE) to refer to periods of sixty years, illustrating its classical usage in numerical and temporal contexts. In English, the prefix sex- borrowed from Latin sex entered the language in the 16th century, forming terms like sextant (an instrument measuring a sixth of a circle, first attested in 1620) and sexennial (occurring every six years, from Latin sexennis, appearing in English by 1646). These borrowings highlight the prefix's role in technical and periodic nomenclature, distinct from its numerical base. In , evolved through phonetic simplification, particularly the reduction of the intervocalic /ks/ cluster to /s/, yielding forms like six (from Old sis), sei, and seis. This change reflects broader developments, where consonant clusters were lightened for ease of pronunciation, as seen in the progression from /seks/ to /sis/ in early Romance varieties. (Note: While parallel to hexa-, the Latin path emphasizes post-classical adaptations.) Latin employed alternative forms like sen- in certain compounds, such as senarius ("of six each," from distributive sēnī), possibly to avoid direct repetition of sex in repetitive or distributive contexts, as in senary systems denoting base-six groupings. Additionally, the numeral influenced Roman calendrical naming, with the sixth month originally termed Sextilis (from sextus, "sixth"), later renamed Augustus in 8 BCE to honor Emperor Augustus. Among derivations, emerged in 19th-century musical terminology to describe a composition or ensemble for six performers, adapted from Italian sestetto via Latin roots.

Germanic and English

The English word "six" derives from Old English (also spelled siȝe), from Proto-Germanic *sehs, ultimately from the same swéḱs. This Germanic branch preserved the initial *s- sound, unlike the Greek development to *h-. Cognates include zes, sechs, and sex. The word entered as six, maintaining its form to the present day. This direct Germanic inheritance forms the basis for the numeral as used in , distinct from Latin borrowings like sextet.

Sciences

Chemistry

Carbon, with 6, possesses four electrons in its outermost , conferring tetravalency that enables it to form covalent bonds central to . This property allows carbon atoms to arrange into hexagonal rings, as exemplified by (C₆H₆), a fundamental aromatic consisting of six carbon atoms in a planar hexagonal structure with alternating double bonds. was first isolated in 1825 by from compressed illuminating gas, marking an early milestone in understanding cyclic carbon compounds. Hexagonal arrangements appear prominently in crystal structures involving carbon and water. Graphite, an allotrope of carbon, features layers of carbon atoms arranged in hexagonal lattices, where each atom bonds to three others in a planar sp² hybridized network, stacked via weak van der Waals forces. Similarly, snowflakes exhibit sixfold symmetry due to the hexagonal crystal structure of ice (Ih), in which water molecules form hydrogen-bonded hexagonal rings that dictate the growth pattern of ice crystals. Several key compounds incorporate six carbon atoms, highlighting the number's prevalence in chemical nomenclature and structure. (C₆H₁₄) is the sixth member of the series, a straight-chain saturated used as a and in fuel formulations. (C₆H₁₂O₆), an aldohexose , features a six-carbon chain with an group, serving as a source in biological systems. In the periodic table, Group 6 comprises transition metals including (Cr), (Mo), tungsten (W), and seaborgium (Sg), each exhibiting six valence electrons (ns¹(n-1)d⁵ configuration) that influence their variable oxidation states and catalytic properties. In coordination chemistry, a of six is ubiquitous, resulting in octahedral complexes where a central metal bonds to six s at the vertices of an , as seen in compounds like [Cr(H₂O)₆]³⁺, providing stability through symmetric ligand field effects. Recent advancements in explore small carbon clusters, such as C₆, and hypothetical smaller s, though stable fullerenes typically begin at C₂₀, with ongoing research focusing on their potential in energy storage via computational modeling of sub-C₆₀ structures.

Physics and other natural sciences

In , the of incorporates six distinct types, or "flavors," of —up, down, , strange, , and —which serve as fundamental building blocks for protons, neutrons, and other hadrons. The underlying this framework was first proposed in 1964 by and , initially with three flavors, and later extended to six as experimental evidence from particle accelerators confirmed the heavier , , and quarks in the 1970s and 1990s. Hexagonal geometries appear recurrently in natural physical processes due to their in packing and . For instance, bee honeycombs exhibit hexagonal cells because this arrangement maximizes space utilization while minimizing wax usage, achieving a packing density of approximately 74% in close-packed structures, as demonstrated by mathematical models of . Similarly, snowflakes display six-fold arising from the hexagonal of , where molecules bond in a way that favors six-sided growth patterns during vapor deposition. This same manifests in geological formations like the basalt columns of Northern Ireland's , where cooling lava contracts and fractures along hexagonal planes to relieve stress, resulting in over 40,000 interlocking polygonal pillars. In biology, the number six is prominent in arthropod anatomy, with insects belonging to the subphylum Hexapoda, characterized by six legs (three pairs attached to the thorax) that enable diverse locomotion and sensory functions across over a million species. Additionally, in molecular biology, the purine bases adenine and guanine in DNA feature a fused ring system including a six-membered pyrimidine ring, which contributes to the stability and base-pairing specificity essential for genetic encoding. Beyond these areas, six dimensions play a key role in , particularly in , where the extra spatial dimensions beyond the familiar four (three space plus time) are often compactified into six-dimensional Calabi-Yau manifolds to reconcile with , a concept central to developments in the . In quantum computing, recent 2020s milestones include the 2025 demonstration of a six-qubit photonic quantum processor by the , enabling scalable entanglement and error-corrected operations for advancing practical quantum algorithms. Fundamentally, classical mechanics describes a rigid body in as having : three for translational motion along the x, y, and z axes, and three for rotational motion about those axes.

Culture and society

Anthropology

In , the holds significance in the structural analysis of human , particularly in systems. George P. Murdock's seminal classification identifies six basic types of terminologies observed across global societies: (lineal, distinguishing parents and siblings distinctly), (generational, merging relatives by generation), (bifurcate merging, combining same-sex siblings' children), (matrilineal skew, fusing certain maternal kin), Omaha (patrilineal skew, similar but for paternal lines), and Sudanese (descriptive, using unique terms for each relative). These systems illustrate how cultures cognitively map familial relations, with lineal types emphasizing distinctions and generational types prioritizing age-based groupings, influencing , , and social obligations. Rituals in various societies incorporate six as a structuring element, often symbolizing completeness in spatial or transitional frameworks. For instance, certain Native American cosmologies, such as among the Zuni, recognize six directions—, north, south, above, and below—each associated with colors (white, blue, yellow, red, speckled, and black, respectively) and integrated into initiation rites, origin myths, and ceremonial orientations to align human actions with cosmic harmony. This expands the four cardinal directions to encompass vertical axes, reflecting a holistic where rituals invoke all six for balance and protection during life transitions. Economic anthropology highlights six's role in pre-monetary exchange systems, notably through the ancient Mesopotamian sexagesimal (base-60) numeral system, where 6 served as a fundamental divisor (60 = 6 × 10), enabling precise divisions in barter and resource allocation. The shekel, a key unit of silver weight equivalent to 1/60 of a mina, facilitated trade in commodities like barley and wool, with the system's divisibility by 6 supporting equitable sharing in temple and palace economies. Ethnographic studies of hunter-gatherers further reveal that small bands or foraging parties of around 5–6 individuals represent an optimal size for cooperation, resource pooling, and conflict avoidance, as seen in groups like the Hill Pandaram, where such units balance mobility and social cohesion without exceeding cognitive limits akin to Dunbar's layered networks. Cross-culturally, attitudes toward six vary, with some societies viewing it positively for its associations with harmony. In Chinese ethnography, 6 (liù) is auspicious due to its phonetic resemblance to liú ("flow" or "smooth"), evoking prosperity and ease in social and economic flows, though less emphasized than 8 and without the avoidance tied to 4 (sì, sounding like "death").

Religion and symbolism

In Abrahamic traditions, the number six holds significant theological weight, particularly in Judaism and Christianity. According to the Book of Genesis, God created the heavens and the earth over six days, culminating in the creation of humanity on the sixth day, before resting on the seventh to establish the Sabbath. This sequence underscores six as emblematic of the material world and human labor, often contrasted with seven as divine completion. In Christianity, the New Testament's Book of Revelation (13:18) identifies 666 as the "number of the beast," symbolizing ultimate human imperfection and opposition to God, as six falls short of the sacred seven, representing a tripled form of incompleteness associated with the Antichrist. In , six features prominently in core doctrines and practices. The six articles of faith (arkan al-iman) form the foundation of belief: affirmation of God (), angels, divine scriptures, prophets, the Day of Judgment, and divine predestination. Additionally, certain voluntary prayers (salat al-nafl) incorporate six rak'ahs, such as Salat ul-Awabeen, performed between Maghrib and Isha to seek repentance and spiritual purification. Hinduism integrates six into its philosophical and calendrical systems, drawing from ancient Vedic texts around 1500 BCE. The six orthodox schools of philosophy, or darshanas—Nyaya (logic), Vaisheshika (atomism), Samkhya (enumeration), Yoga (discipline), Purva Mimamsa (ritual interpretation), and Uttara Mimamsa (Vedanta, or non-dualism)—provide comprehensive frameworks for understanding reality, ethics, and liberation (). Complementing this, the traditional divides the year into six seasons (ritus): Vasanta (spring), Grishma (summer), Varsha (), Sharad (autumn), Hemanta (pre-winter), and Shishira (winter), each spanning two months and influencing rituals, , and cosmology as described in Vedic literature. In Buddhism, six denotes the realms of cyclic existence (samsara), where sentient beings reincarnate based on karma. These include the realms of gods (deva), demigods (asura), humans, animals, hungry ghosts (preta), and hell beings (naraka), illustrated in the Wheel of Life (bhavachakra) as interdependent states driven by ignorance and attachment, with the goal of escaping via enlightenment. Across these traditions, six often symbolizes imperfection or limitation, positioned one short of seven—the latter evoking divine wholeness in Judeo-Christian contexts and completeness in broader numerological interpretations. This recurring motif highlights humanity's earthly constraints, contrasting with transcendent ideals.