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60

Sixty (60) is the natural number following 59 and preceding 61, an even composite integer with the prime factorization $2^2 \times 3 \times 5.^[1] It possesses twelve positive divisors—1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60—more than any smaller positive integer, classifying it as a highly composite number and the smallest with exactly twelve divisors.^[2]^[3] This abundance of divisors made 60 particularly advantageous for ancient Mesopotamian mathematics, forming the basis of the sexagesimal (base-60) numeral system developed by the Sumerians around 3500 BCE and later refined by the Babylonians, who used it for astronomy, geometry, and time measurement.^[4]^[5] The system's legacy endures in contemporary divisions of time, with 60 seconds comprising a minute and 60 minutes forming an hour, chosen for 60's divisibility by 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, and 30 to facilitate fractional calculations without remainders.^[6] In chemistry, 60 denotes the atomic number of neodymium, a rare-earth element discovered in 1885 and essential in alloys, magnets, and lasers due to its paramagnetic properties and role in producing vibrant colors in glass.^[7] Beyond these domains, 60 holds cultural significance in various traditions, such as marking the completion of a full cycle in the Chinese zodiac's 60-year sexagenary system, symbolizing renewal and used in calendars and astrology.^[8]

Mathematics

Divisibility and abundance

Sixty has the prime factorization $60 = 2^2 \times 3 \times 5.^[9] The positive divisors of 60 are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60, giving it a total of 12 divisors.^[10] This makes 60 the smallest positive integer divisible by every integer from 1 to 6, as it equals the least common multiple of these numbers.^[11] As a highly composite number, 60 is the fourth superior highly composite number, following 2, 6, and 12 in the sequence defined by Ramanujan, where each term has more divisors than any smaller positive integer and satisfies specific exponent conditions in its prime factorization.^[12] It is also a colossally abundant number, characterized by maximizing the ratio of the sum of divisors to the number itself relative to its magnitude among numbers up to a certain bound.^[9] The sum of all divisors of 60 is 168, making it an abundant number with an abundance of 48, since the sum exceeds twice the number (120).^[13] Additionally, 60 is a unitary perfect number, as the sum of its proper unitary divisors equals 60 itself.^[14] In Roman numerals, 60 is represented as LX, combining L (50) and X (10).^[15]

Combinatorial and geometric significance

In group theory, the alternating group A_5, which consists of all even permutations of five elements, has order 60. This group is isomorphic to the rotational symmetry group of the regular icosahedron and dodecahedron, highlighting 60's significance in describing the orientation-preserving symmetries of these Platonic solids.^[16] In polyomino enumeration, there are exactly 60 distinct one-sided hexominoes, which are shapes formed by connecting six equal squares edge-to-edge, considering reflections as distinct. This count arises from combinatorial enumeration techniques that account for rotational symmetries but treat mirror images separately, providing a key example of 60 in fixed polyform catalogs.^[17] Geometrically, the truncated icosahedron, an Archimedean solid composed of 12 regular pentagons and 20 regular hexagons, features 60 vertices where three faces meet at each point. This structure exemplifies 60's role in polyhedral vertex configurations, with each vertex incident to two hexagons and one pentagon, enabling the solid's full icosahedral symmetry.^[18]

Measurement systems

Sexagesimal origins

The sexagesimal numeral system, with 60 as its base, originated in ancient Mesopotamia, where the Babylonians formalized it around 2000 BCE, inheriting and refining an earlier Sumerian framework dating back to the third millennium BCE. This development was driven by practical needs in astronomy for tracking celestial movements and in commerce for standardizing weights, measures, and transactions. The choice of 60 stemmed from its exceptional divisibility by 1, 2, 3, 4, 5, 6, 10, and 12, which facilitated accurate subdivisions and fractional representations essential for these fields.^[4] From Mesopotamia, the system disseminated through cultural and scholarly exchanges, reaching Greek mathematicians by the Hellenistic period and later documented in detail by Theon of Alexandria in the 4th century CE, who outlined Greek computational techniques using sexagesimal fractions in his commentary on Ptolemy's Almagest. It also influenced ancient Indian mathematics via Babylonian astronomical traditions.^[19]^[20]^[21] Islamic scholars, during the 8th to 13th centuries, adopted and preserved the system alongside Indian decimal methods, with figures like al-Khwarizmi integrating it into works on algebra and astronomy at the House of Wisdom in Baghdad.^[22] A key advantage of the sexagesimal base over the more common decimal system lies in its greater number of divisors, enabling more fractions—such as thirds, quarters, fifths, and sixths—to be expressed as finite terminating expansions without recurring decimals, which proved invaluable for precise computations in pre-modern eras. Despite the widespread adoption of base-10 numerals in contemporary mathematics and commerce, sexagesimal elements endure in specialized modern applications derived from these ancient traditions.^[4]

Time and angular divisions

In timekeeping, the division of the hour into 60 minutes and the minute into 60 seconds originates from the sexagesimal system employed in ancient civilizations, which facilitated precise measurements using devices such as water clocks and sundials.^[23] Water clocks, or clepsydrae, and sundials were instrumental in tracking these subdivisions during both day and night, with the Babylonians and Egyptians adapting the 60-based units to divide the 12 temporal hours of daylight and darkness.^[23] This structure allowed for consistent fractional divisions, though the formal naming of minutes (prima minuta) and seconds (secunda minuta) emerged later with mechanical clocks in the 14th and 17th centuries, respectively.^[23] In angular measurement, a degree is subdivided into 60 arcminutes (also called minutes of arc), with each arcminute further divided into 60 arcseconds, a convention inherited from Mesopotamian astronomy and applied in navigation and celestial observations.^[24] These units enable fine-grained angular precision essential for determining positions of stars and planets, as well as for maritime navigation where small differences in angle correspond to significant distances on Earth.^[24] The full circle encompasses 360 degrees, equivalent to 6 × 60, a division rooted in Babylonian astronomy that approximated the solar year's length of about 360 days and aligned with the zodiac's 12 constellations, each spanning 30 degrees.^[24] This 360-degree framework standardized the measurement of celestial rotations and zodiacal paths, providing a cohesive system for tracking annual solar motion.^[25] The sexagesimal basis of these divisions supports exact fractional representations, enhancing accuracy in both time and angular calculations.^[24]

Science and technology

Atomic and chemical properties

Neodymium (Nd) is a chemical element with atomic number 60, classified as a rare-earth metal within the lanthanide series.^[26] It appears as a soft, silvery-white metal that tarnishes slowly in air and reacts with water to form neodymium hydroxide.^[26] Neodymium's high magnetic susceptibility makes it essential for producing powerful permanent magnets, particularly in neodymium-iron-boron (NdFeB) alloys, which are used in electric motors for hybrid vehicles, wind turbines, and computer hard drives.^[27] Additionally, neodymium is doped into yttrium aluminum garnet (Nd:YAG) crystals to create solid-state lasers employed in medical procedures, industrial cutting, and welding applications.^[27] Cobalt-60 (^{60}Co) is a radioactive isotope of cobalt with a half-life of 5.27 years, produced by neutron irradiation of stable cobalt-59 in nuclear reactors.^[28] It decays via beta emission followed by two gamma rays, making it a potent source of high-energy gamma radiation.^[28] In medicine, cobalt-60 has been widely used in external beam radiotherapy, such as gamma knife radiosurgery—which treats over 30,000 patients annually worldwide, typically as outpatients—to target and treat brain tumors and other cancers.^[28] Beyond oncology, it serves as a key agent for sterilizing heat-sensitive medical supplies like syringes and surgical dressings through gamma irradiation, a process that ensures sterility without compromising material integrity.^[28] Buckminsterfullerene (C_{60}), an allotrope of carbon consisting of 60 atoms arranged in a truncated icosahedron—a soccer ball-like structure with 12 pentagonal and 20 hexagonal faces—was discovered in 1985 by a team at Rice University using laser vaporization of graphite.^[29] This hollow, cage-like molecule exhibits remarkable stability due to its symmetric bonding, with no dangling bonds, and its fullerene family has since revolutionized carbon chemistry.^[30] In nanotechnology, C_{60} finds applications as a building block for carbon nanotubes and in lubricants, where its spherical shape enables low-friction coatings, while ongoing research explores its potential in drug delivery and superconductors like alkali-metal-doped variants.^[29]^[30] The truncated icosahedral symmetry of C_{60} underpins its unique electronic properties, facilitating phenomena such as wave-particle duality observed in quantum experiments.^[30]

Engineering and computing applications

In electrical engineering, 60 Hz serves as the standard frequency for alternating current (AC) power systems in North America, including the United States and Canada, as well as parts of Central and South America such as Mexico, Brazil, and Colombia, and certain Asian countries like Japan (western regions), South Korea, and Taiwan.^[31] This choice originated in 1891 when Westinghouse engineers in Pittsburgh selected 60 Hz for its balance of efficiency in transmission and motor performance, contrasting with the 50 Hz standard adopted by AEG in Europe and prevalent in most of Africa, Asia, Australia, and Europe today. The 60 Hz frequency enables synchronous operation of generators and loads, minimizing losses in long-distance power distribution while supporting appliances designed for higher rotational speeds in motors.^[32] In computing, the exbibyte (EiB) represents a unit of digital information storage equivalent to $2^{60} bytes, or precisely 1,152,921,504,606,846,976 bytes, as defined in the International Electrotechnical Commission (IEC) standard 60027-2 Amendment 2 from 1998, which introduced binary prefixes to distinguish powers of two from decimal multiples. This standardization addresses ambiguities in data measurement, where traditional terms like "exabyte" could refer to either $10^{18} or $2^{60} bytes; the exbibyte specifically supports binary-aligned storage in systems like hard drives and memory, facilitating precise capacity reporting in high-performance computing environments.^[33] Beyond power and data, 60 appears in other engineering standards, such as nominal -60 V DC power distribution in some telecommunications networks, where it provides a safe, low-voltage alternative to higher systems while powering equipment like switches and transmission gear, though -48 V remains more common globally.^[34] In audio engineering history, early phonograph records from the late 19th century operated at speeds around 60 rpm, allowing for short playback durations of about two minutes on 7-inch discs made of hard rubber or shellac, before standardization shifted to 78 rpm by the 1920s for improved sound quality and consistency.

Cultural and historical uses

Ancient numeral systems

While the sexagesimal (base-60) numeral system originated in ancient Mesopotamia with the Babylonians, who employed it for astronomical and administrative purposes, its conceptual influence extended to diverse cultures through trade and migration, adapting 60 as a practical unit in non-positional counting methods.^[35] Among the Ekagi (also known as Kapauku) people of the highlands in what is now Papua, Indonesia, a base-60 counting system was traditionally used for tallying goods such as shells and pigs, structured around body-part enumeration that progressed from fingers and arms to other anatomical points, reaching 60 before cycling or doubling. This system, documented in ethnographic studies, emphasized 30 as a subunit within the 60-base framework, facilitating trade in resource-scarce environments without reliance on written notation.^[36] In medieval Europe, 60 appeared in vernacular counting terms across Germanic and Slavic traditions, reflecting practical groupings for commerce and measurement. The English term "threescore," meaning three times twenty or 60, was commonly used in the King James Bible to denote quantities, such as in Psalm 90:10 ("The days of our years are threescore years and ten"), preserving an older Indo-European practice of scoring by twenties that indirectly evoked sexagesimal divisibility. Similarly, in Polish and Lithuanian contexts of the Polish-Lithuanian Commonwealth, "kopa" designated a bundle or unit of 60 items, such as groszy coins, serving as a standard in monetary and trade accounting from the 14th century onward.^[37]^[38]

Calendars and cycles

The sexagenary cycle in the traditional Chinese calendar, known as the ganzhi system, forms a recurring 60-year loop by pairing the 10 heavenly stems (representing yin-yang and five elements) with the 12 earthly branches (associated with the zodiac animals).^[39] This combination yields 60 unique pairs, each designating a year, month, day, or hour, and has been used since the Shang dynasty (c. 1600–1046 BCE) for dating, astrology, and imperial records.^[40] The cycle's structure reflects the least common multiple of 10 and 12, enabling a balanced integration of celestial and terrestrial influences in Chinese cosmology.^[41] In Hindu lunisolar calendars, such as the Vikram Samvat and Saka eras, a parallel 60-year cycle called the Samvatsara chakra organizes time through named years derived from the relative positions of Jupiter and Saturn.^[42] Each Samvatsara, spanning one Jupiter year (approximately 12 lunar months), carries a poetic name like Prabhava or Vibhava, influencing festivals, agriculture, and astrological predictions; the cycle repeats indefinitely, symbolizing cosmic renewal.^[43] This system, rooted in Vedic astronomy, aligns lunar months with solar years, with the 60-year period approximating five Jupiter orbits.^[44] In Germanic and Slavic agricultural traditions, the term "Schock" (from Middle High German schoc, meaning a bundle or shock of sheaves) denotes a unit of 60, commonly applied to harvest counts like sheaves of grain or market batches of eggs, apples, or other produce.^[45] This grouping facilitated practical tallying in pre-modern rural economies, as seen in Hanseatic League commerce and village inventories from the 14th to 17th centuries, where payments or quotas were often reckoned in schocks.^[46] Similar usages appear in Slavic languages, such as Czech "šok" for 60 items, tracing to shared Indo-European practices of bundling for trade and seasonal labor.^[47] The prevalence of 60 in these cycles stems from its high divisibility, harmonizing decimal and duodecimal elements across diverse cultural timekeeping.^[48]

Religion and symbolism

Hindu traditions

In Hinduism, the number 60 symbolizes completion and renewal, most prominently through the Sashti Poorthi, a major ritual celebrating an individual's 60th birthday as a pivotal life milestone. This ceremony, derived from Sanskrit terms meaning "completion of sixty years," involves elaborate family rituals including Vedic chants, offerings to deities like Lord Vishnu, and the symbolic renewal of marital vows between husband and wife, signifying gratitude for past blessings and vows for spiritual growth in the remaining years.^[49]^[50] Performed typically by Brahmin priests, it features homas (fire rituals) and feasts shared with relatives, emphasizing the transition to a phase of wisdom and detachment from worldly pursuits.^[51] In some Vaishnava traditions, such as Gaudiya Vaishnavism, 60 denotes the count of transcendental qualities (gunas) inherent to Lord Narayana, the supreme form of Vishnu, representing divine attributes such as omnipresence, eternal youth, and boundless compassion that underpin cosmic preservation. These 60 gunas form the foundational excellences shared across Vishnu's manifestations, with Lord Krishna embodying all 60 plus four unique qualities like unparalleled beauty and flute mastery, elevating him to the supreme personality in that philosophy.^[52]^[53] This numerical significance highlights 60 as a multiplier of divine perfection in certain Hindu scriptural enumerations. The association of 60 with longevity stems from its alignment with the full Jupiter cycle (Brihaspati varshas) in Hindu astrology, where Jupiter's 12-year orbital period completes five revolutions in approximately 60 years, marking a complete astrological epoch that ancient texts viewed as encompassing a primary phase of human life before elder reflection. In traditional views, this cycle underscores 60 years as a threshold of maturity and renewal, often invoked in rituals like Sashti Poorthi to pray for extended vitality toward the ideal human lifespan of 100 years.^[54] This Jovian framework also briefly informs broader Hindu calendar cycles, such as the 60-year Samvatsara sequence naming annual epochs.^[55]

Other religious contexts

In the Hebrew Bible and Christian Old Testament, the number 60, often rendered as "threescore," symbolizes completeness or a significant milestone in human life and events. For instance, Psalm 90:10 describes the average human lifespan as "threescore years and ten," equating to 70 years, emphasizing the brevity and toil of earthly existence.^[56] Similarly, Genesis 25:26 records that Isaac was 60 years old when his wife Rebekah gave birth to the twins Esau and Jacob, marking a pivotal moment in the patriarchal lineage.^[57] In Islamic tradition, the number 60 appears in hadith literature concerning the creation of Adam, who is described as being 60 cubits tall—approximately 90 feet—upon his formation by Allah, with subsequent generations decreasing in stature.^[58] Buddhist texts reference 60 in contexts of enlightenment and cosmic time. Shortly after his awakening, the Buddha dispatched 60 arahants—fully enlightened disciples—to propagate the Dhamma across the world, initiating the monastic missionary tradition with the exhortation to teach for the welfare of many.^[59] In Mahayana scriptures like the Lotus Sutra, the disciple Shariputra is depicted as having cultivated the bodhisattva path through almsgiving and other perfections over vast eons, illustrating the immense temporal scale required for spiritual progress before his realization. Theravada monastic codes, such as the Patimokkha, include the 60th rule prohibiting the hiding of a fellow monk's belongings even in jest, as part of the 227 precepts governing communal harmony.

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