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  • Queue (hairstyle)

    Queue
    Queue (hairstyle)

    Chinese men with queues c. 1880 (photographed by Lai Afong)
    Chinese name
    Traditional Chinese 辮子
    Simplified Chinese 辫子
    Transcriptions
    Standard Mandarin
    Hanyu Pinyin biànzi
    Yue: Cantonese
    Jyutping bin1 zi2
    Min names
    Traditional Chinese 頭鬃尾
    毛尾仔
    辮仔
    Transcriptions
    Southern Min
    Hokkien POJ mn̂g-bué-á/mn̂g-bé-á
    thâu-chang-bué/thâu-chang-bé
    pīⁿ-á
    Manchu name
    Manchu script ᠰᠣᠨᠴᠣᡥᠣ
    Romanization soncoho

    A queue or cue is a hairstyle historically worn by the Jurchen and Manchu peoples of Manchuria. During the Manchu-led Qing dynasty, it was required to be worn by male Han Chinese subjects.[1][2][3][4][5] The top of the scalp is shaved and the back portion of hair on the head is often grown long and is braided. The distinctive hairstyle led to its wearers being targeted during anti-Chinese riots in Australia and the United States.[6]

    The edict that Han Chinese men and others under Manchu rule give up their traditional hairstyles and wear the queue, the Tifayifu, was met with resistance, although opinions about the queue did change over time. Han women were never required to wear their hair in the traditional women’s Manchu style, liangbatou, although that too was a symbol of Manchu identity.[7]

    Predecessors and origin

    The queue hairstyle predates the Manchus. The Chinese word for queue, bian, meant plaited hair or a cord. The term bian, when used to describe the braid in the Manchu hairstyle, was originally applied by the Han dynasty to the Xiongnu. Jurchen people wore a queue like the Manchu, the Khitan people wore theirs in Tartar style and during the Tang dynasty, tribes in the west wore braids.[8][9]
    The Xianbei and Wuhuan were said to shave their heads, while Xiongnu had queues. Other evidence from Chinese histories indicate that the Tuoba or Tabgach groups of the Xianbei wore braids, since they were called “braided” by the southern Chinese. However, their hairstyle is hidden in depictions due to a hood they wore. The Liu Song dynasty’s records called them “braided caitiff”, suolu, while Southern Qi’s history said they wore their “hair hanging down the back” (pifa), and called them suotou, “braided”. A braid of hair was found at Zhalairuoer in a Tuoba grave.[10]

    Han Chinese also made the peoples they conquered undo their queues. To show submission to the Han Chinese of the Sui dynasty, the people of Turfan (Gaochang) undid their queues, as did the Göktürks upon surrendering to the Tang dynasty. Hairstyles showed affiliation to a tribal confederation or dynasty.[11] In the Western Wei cave 285 at the Mogao Caves in Dunhuang, Xianbei people are depicted with small queues hanging from their necks.[12]

    After overthrowing the Mongol Yuan dynasty, Zhu Yuanzhang, the first Ming emperor passed a law on mandatory hairstyle on 24 September 1392 mandating that all males grow their hair long and making it illegal for them to shave part of their foreheads while leaving strands of hair, which was the Mongol hairstyle. The penalty for both the barber and the person who was shaved and his sons was castration if they cut their hair and their families were to be sent to the borders for exile. This helped eradicate the partially shaved Mongol hairstyles.[13]

    The Tangut people of the Western Xia may have inherited hairstyle influences from the Tuoba. It resembled a monk’s hairstyle but was not exactly like their tonsure, it left the face to be framed on the sides and forehead by a fringe of hair by shaving the head top and leaving it bald. This made sure the Tibetans and Song Chinese could be told apart from shaved Tanguts. It was imposed by the Tangut emperor, Jingzong, threatening that their throats would be cut if they did not shave within three days. The emperor was the first one to shave.[14] Unlike the tonsure of the Tangut Western Xia, the Jurchen hairstyle of wearing the queue combined with shaving the crown was not the invention of an emperor of the dynasty but was an established Jurchen hairstyle which showed who submitted to Jin rule. This Jurchen queue and shaving hairstyle was not enforced on the Han Chinese in the Jin after an initial attempt to do so which was a rebuke to Jurchen values.[15] The Jin at first attempted to impose Jurchen hairstyle and clothes on the Han population during the Jin but the order was taken back. They also banned intermarriage.[16]

    Queue (hairstyle)
    Khitan man in tomb painting in Aohan Banner, Inner Mongolia

    Manchu Jurchen men had queues, while Mongol men swept their hair behind their ears and plaited them, Turk men wore loose hair and Xiongnu men braided their hair. Khitan males grew hair from their temples but shaved the crown of their heads. The Han Chinese men living in the Liao dynasty were not required to wear the shaved Khitan hairstyle which Khitan men wore to distinguish their ethnicity, unlike the Qing dynasty which mandated wearing of the Manchu hairstyle for men.[17] Khitan men left only two separate patches of hair on each of the forehead’s sides in front of each ear in tresses while they shaved the top of their head. Khitan wore felt hats, fur clothes and woolen cloth and the Liao emperor switched between Han and Khitan clothing.[18]
    Khitan officials used gold ornamented ribbons to found their hair locks around their foreheads, covering their heads with felt hats according to the Ye Longli’s (Yeh Lung-li) Qidan Guozhi (Ch’i-tan kuo-chih). Khitan wore the long side fringes and shaved pates.[19] Tomb murals of Khitan hairstyle show only some hair remaining near the neck and forehead with the rest of the head shaved.[20] Only at the temples were hair left while the crown was shaven.[21] The absence of Khitan clothes and hairstyles on a painting of riders previously identified as Khitan has led to experts questioning their purported identity.[22] Khitan men might have differentiate between classes by wearing different patterns on their small braids hanging off their shaved foreheads. They wore the braids occasionally with a forehead fringe with some shaving off all the forehead.[23] Some Han men adopted and mixed or combined Han clothing with Khitan clothing with Khitan boots and Han clothes or wearing Khitan clothes. Han women on the other hand did not adopt Khitan dress and continued wearing Han dress.[24][25]

    • Yelü Bei
      Yelü Bei
    • Horsemen
      Horsemen
    • Horsemen at rest
      Horsemen at rest
    • Hunters
      Hunters
    • Cooks
      Cooks
    • Boys and girls
      Boys and girls
    • Queue (hairstyle)
    • Queue (hairstyle)

    Jurchen queue

    Jurchen men, like their Manchu descendants, wore their hair in queues. In 1126, the Jurchen ordered male Han within their conquered territories to adopt the Jurchen hairstyle by shaving the front of their heads and to adopt Jurchen dress, but the order was lifted.[26] Some Han rebels impersonated Jurchen by wearing their hair in the Jurchen “pigtail” to strike fear within the Jurchen population.[27]

    The Manchu queue

    Queue (hairstyle)
    Manchu queues
    Queue (hairstyle)
    A European artist’s conception of a Manchu warrior in China – surprisingly, holding the severed head of an enemy by its queue. Later historians have noted the queue looking more like Cossack chupryna as an inconsistency in the picture. (From the cover of Martino Martini’s Regni Sinensis a Tartari devastati enarratio, 1661.)

    The queue (called soncoho in Manchu) was a specifically male hairstyle worn by the Manchu from central Manchuria and later imposed on the Han Chinese during the Qing dynasty.[28][29][30] The hair on the front of the head was shaved off above the temples every ten days and the remainder of the hair was braided into a long braid.[31]

    The Manchu hairstyle was forcefully introduced to Han Chinese and other ethnicities like the Nanai in the early 17th century during the transition from Ming to Qing. Nurhaci of the Aisin Gioro clan declared the establishment of the Later Jin dynasty, later becoming the Qing dynasty of China, after Ming dynasty forces in Liaodong defected to his side. The Ming general of Fushun, Li Yongfang, defected to Nurhaci after Nurhaci promised him rewards, titles, and Nurhaci’s own granddaughter in marriage. Other Han Chinese generals in Liaodong proceeded to defect with their armies to Nurhaci and were given women from the Aisin Gioro family in marriage. Once firmly in power, Nurhaci commanded all men in the areas he conquered to adopt the Manchu hairstyle.

    The Manchu hairstyle signified all ethnic groups submission to Qing rule, and also aided the Manchu identification of those Han who refused to accept Qing dynasty domination.

    The hairstyle was compulsory for all males and the penalty for non-compliance was execution for treason. After the fall of the Qing dynasty in 1912, the Chinese no longer had to wear the Manchu queue. While some, such as Zhang Xun, still did so as a tradition, most of them abandoned it after the last Emperor of China, Puyi, cut his queue in 1922.[32]

    The Nanais at first fought against the Nurhaci and the Manchus, led by their own Nanai Hurka chief Sosoku before surrendering to Hongtaiji in 1631. Mandatory shaving of the front of all male heads was imposed on Amur peoples like the Nanai people who were conquered by the Qing. The Amur peoples already wore the queue on the back of their heads but did not shave the front until the Qing subjected them and ordered them to shave.[33]
    The term “shaved-head people” was used to describe the Nanai people by Ulch people.[34]

    Queue order

    Queue (hairstyle)
    Chinese circus performers soon after the Manchu conquest, wearing queues. (Drawing by Johan Nieuhof, 1655–57)

    The Queue Order (simplified Chinese: 剃发令; traditional Chinese: 剃髮令; pinyin: tìfàlìng),[35][36] or tonsure decree, was a series of laws violently imposed by the Qing dynasty during the seventeenth century.[37]: 95  It was also imposed on Taiwanese indigenous peoples in 1753,[38][39] and Koreans who settled in northeast China in the late 19th century,[40][41] though the Ryukyuan people of the Ryukyu Kingdom, a tributary of China, requested and were granted an exemption from the mandate.

    Traditionally, adult Han Chinese did not cut their hair for philosophical and cultural reasons. According to the Classic of Filial Piety, Confucius said:

    We are given our body, skin and hair from our parents; which we ought not to damage. This idea is the quintessence of filial duty. (身體髮膚，受之父母，不敢毀傷，孝之始也。)[42]

    As a result of this ideology, both men and women wound their hair into a bun (a topknot) or other various hairstyles.

    Han Chinese did not object to wearing the queue braid on the back of the head as they traditionally wore all their hair long, but fiercely objected to shaving the forehead so the Qing government exclusively focused on forcing people to shave the forehead rather than wear the braid. Han rebels in the first half of the Qing who objected to Qing hairstyle wore the braid but defied orders to shave the front of the head. One person was executed for refusing to shave the front but he had willingly braided the back of his hair. It was only later that westernized revolutionaries began to view the braid as backwards and advocated adopting short-haired western styles.[43] Han rebels against the Qing like the Taiping retained their queue braids on the back but rebelled by growing hair on the front of their heads. This caused the Qing government to view shaving the front of the head as the primary sign of loyalty rather than wearing the braid on the back, which did not violate Han customs and traditional Han did not object to.[44] Koxinga criticized the Qing hairstyle by referring to the shaven pate looking like a fly.[45] Koxinga and his men objected to shaving when the Qing demanded they shave in exchange for recognizing Koxinga as a feudatory.[46] The Qing demanded that Zheng Jing and his men on Taiwan shave to receive recognition as a fiefdom. His men and Ming prince Zhu Shugui fiercely objected to shaving.[47]

    Queue (hairstyle)
    A soldier during the Boxer Rebellion with queue and conical Asian hat

    In 1644, Beijing was sacked by a coalition of rebel forces led by Li Zicheng, a minor Ming dynasty official turned leader of a peasant revolt. The Chongzhen Emperor committed suicide when the city fell, marking the official end of the Ming dynasty. The Han Chinese Ming general Wu Sangui and his army then defected to the Qing and allowed them through Shanhai pass. They then seized control of Beijing, overthrowing Li’s short-lived Shun dynasty. They then forced Han Chinese to adopt the queue as a sign of submission.[48]

    A year later, after the Qing armies reached South China, on 21 July 1645, the regent Dorgon issued an edict ordering all Han men to shave their foreheads and braid the rest of their hair into a queue identical to those worn by the Manchus.[49] Qing Manchu prince Dorgon initially canceled the order for all men in Ming territories south of the Great wall (post 1644 additions to the Qing) to shave. It was a Han official from Shandong, Sun Zhixie and Li Ruolin who voluntarily shaved their foreheads and demanded Qing Prince Dorgon impose the queue hairstyle on the entire population which led to the queue order.[50][51] The Han Chinese were given 10 days to comply or face death. Though Dorgon admitted that followers of Confucianism might have grounds for objection, most Han officials cited the Ming dynasty’s traditional System of Rites and Music as their reason for resistance. This led Dorgon to question their motives: “If officials say that people should not respect our Rites and Music, but rather follow those of the Ming, what can be their true intentions?”[48]

    In the edict, Dorgon specifically emphasized the fact that Manchus and the Qing Emperor himself all wore the queue and shaved their foreheads, so that by following the queue order, Han Chinese would look like the Manchus and the Emperor. This invoked the Confucian notion that the people were like the sons of the emperor, and should be similar in their appearance.[52][53][54]

    The slogan adopted by the Qing was “Cut the hair and keep the head, (or) keep the hair and cut the head” (Chinese: 留髮不留頭，留頭不留髮; pinyin: liú fà bù liú tóu, liú tóu bù liú fà).[55] People who resisted the order were met with deadly force. Han rebels in Shandong tortured the Qing official who suggested the queue order to Dorgon to death and killed his relatives.[56]

    The imposition of this order was not uniform; it took up to 10 years of martial enforcement for all of China to be brought into compliance, and while it was the Qing who imposed the queue hairstyle on the general population, they did not always personally execute those who did not obey. It was Han Chinese defectors who carried out massacres against people refusing to wear the queue. Li Chengdong, a Han Chinese general who had served the Ming but defected to the Qing,[57] ordered troops to carry out three separate massacres in the city of Jiading within a month, resulting in tens of thousands of deaths. The third massacre left few survivors.[58] The three massacres at Jiading District are some of the most infamous, with estimated death tolls in the tens or even hundreds of thousands.[59] Jiangyin also held out against about 10,000 Qing troops for 83 days. When the city wall was finally breached on 9 October 1645, the Qing army, led by the Han Chinese Ming defector Liu Liangzuo (劉良佐), who had been ordered to “fill the city with corpses before you sheathe your swords,” massacred the entire population, killing between 74,000 and 100,000 people.[60]

    Han Chinese soldiers in 1645 under Han General Hong Chengchou forced the queue on the people of Jiangnan, while Han people were initially paid silver to wear the queue in Fuzhou when it was first implemented.[61][62]

    The queue was the only aspect of Manchu culture that the Qing forced on the common Han population. The Qing required people serving as officials to wear Manchu clothing, but allowed other Han civilians to continue wearing Hanfu (Han clothing). Nevertheless, most Han civilian men voluntarily adopted Manchu clothing[63][64] like Changshan of their own free will. Throughout the Qing dynasty Han women continued to wear Han clothing.[65]

    However, the shaving policy was not enforced in the Tusi autonomous chiefdoms in Southwestern China where many minorities lived. There was one Han Chinese Tusi, the Chiefdom of Kokang populated by Han Kokang people.

    The Qing dynasty required all subjects of all ethnicities to shave their foreheads and wear the queue braid including Muslims like Hui people and Salar people but some Turkic Muslim ethnicities like Uyghur and Salar people already shaved their entire heads as part of their culture and were bald so they were not able to wear the braids on the back unless they wore wigs with fake queues. According to Jonathan Neaman Lipman the Qing dynasty required Salars to wear the queue.[66][67][68] During the Qing Salar men shaved their hair bald while when they went to journey in public they put on artificial queues.[69] Uyghur men shaved their hair bald during the Qing.[70] Uyghur males at the present still shave their heads bald in the summer.[71] Chen Cheng observed that Muslim Turks in 14th–15th century Turfan and Kumul shaved their heads while non-Muslim Turks grew long hair.[72][73]

    However, after Jahangir Khoja invaded Kashgar, Turkistani Muslim begs and officials in Xinjiang eagerly fought for the “privilege” of wearing a queue to show their steadfast loyalty to the Empire. High-ranking begs were granted this right.[74]

    The purpose of the Queue Order was to demonstrate loyalty to the Qing, and refusing to shave one’s hair came to symbolize revolutionary ideals, as seen during the White Lotus Rebellion. Because of this, the members of the Taiping Rebellion were sometimes called the Long hairs (長毛) or Hair rebels (髮逆).[75]

    Resistance to the queue

    Han Chinese resistance to adopting the queue was widespread and bloody. The Chinese in the Liaodong Peninsula rebelled in 1622 and 1625 in response to the implementation of the mandatory hairstyle. The Manchus responded swiftly by killing the educated elite and instituting a stricter separation between Han Chinese and Manchus.[76]

    In 1645, the enforcement of the queue order was taken a step further by the ruling Manchus when it was decreed that any man who did not adopt the Manchu hairstyle within ten days would be executed. The intellectual Lu Xun summed up the Chinese reaction to the implementation of the mandatory Manchu hairstyle by stating, “In fact, the Chinese people in those days revolted not because the country was on the verge of ruin, but because they had to wear queues.” In 1683 Zheng Keshuang surrendered and wore a queue.[76]

    The queue became a symbol of the Qing dynasty and a custom except among Buddhist monastics.[77][78][79] Some revolutionists, supporters of the Hundred Days’ Reform or students who studied abroad cut their braids. The Xinhai Revolution in 1911 led to a complete change in hairstyle almost overnight. The queue became unpopular as it became associated with a fallen government; this is depicted in Lu Xun’s short story Storm in a Teacup and is demonstrated by the fact that Chinese citizens in Hong Kong collectively changed to short haircuts.[80]

    Cantonese outlaw bandit pirates in the Guangdong maritime frontier with Vietnam in the 17th, 18th and 19th centuries wore their hair long in defiance of the Qing laws which mandated cutting.[81]

    Many people were violating the Qing laws on hair at the end of the dynasty. Some Chinese chose to wear the queue but not to shave their crown, while those people who cut the queue off and did not shave were considered revolutionary and others maintained the state-mandated combination of the queue and shaved crown.[82]

    Exemptions

    Neither Taoist priests nor Buddhist monks were required to wear the queue by the Qing; they continued to wear their traditional hairstyles, completely shaved heads for Buddhist monks, and long hair in the traditional Chinese topknot for Taoist priests.[83][84][85]

    Foreign reaction

    Queue (hairstyle)
    Barbershop in the Qing Dynasty (1870s)

    The Manchus’ willingness to impose the queue and their dress style on the men of China was viewed as an example to emulate by some foreign observers. H. E. M. James, a British civil servant in India, wrote in 1887 that the British ought to act in a similarly decisive way when imposing their will in India. In his view, the British administration should have outlawed practises such as Sati much earlier than 1829, which James ascribed to a British unwillingness to challenge long-held Indian traditions, no matter how detrimental they were to the country.[86]

    British author Demetrius Charles Boulger in 1899 proposed that Britain form and head an alliance of “Philo-Chinese Powers” in setting up a new government for China based in Shanghai and Nanking as two capitals along the River Yangtze, to counter the interests of other powers in the region like the Russians due to what he believed was the imminent collapse of the Qing dynasty. The Yangtze valley was controlled by Qing officials such as Liu Kunyi and Zhang Zhidong, who were not under Beijing’s influence and whom Boulger believed Britain could work with to stabilize China. He proposed that at Nanjing and Hankou a force of Chinese soldiers trained by the British be deployed and in Hong Kong, Weihaiwei and the Yangtze valley and it would have no allegiance to the Qing, and as such they in his idea would forgo the queue and be made to grow their hair long as a symbolic measure to “increasing the confidence of the Chinese in the advent of a new era”.[87][88] Boulger stated he could not discern from the Chinese he spoke to on whether the queue was invented by Nurhaci to impose on the Chinese as a symbol of loyalty or whether it was an already established Manchu custom as no one seemed to know the origin of it from his or other sinologists’ inquiries.[89]

    English adventurer Augustus Frederick Lindley wrote that the beardless, youthful long haired Han Chinese rebels from Hunan in the Taiping armies who grew all their hair long while fighting against the Qing dynasty were among the most beautiful men in the world unlike, in his mind, the Han Chinese who wore the queue, with Lindley calling the shaved part “a disfigurement”.[90]

    Vietnam

    After Nguyễn Huệ defeated the Later Lê dynasty, high ranking Lê loyalists and the last Lê emperor Lê Chiêu Thống fled Vietnam for asylum in Qing China. They went to Beijing where Lê Chiêu Thống was appointed a Chinese mandarin of the fourth rank in the Han Yellow Bordered Banner, while lower ranking loyalists were sent to cultivate government land and join the Green Standard Army in Sichuan and Zhejiang. They adopted Qing clothing and adopt the queue hairstyle, effectively becoming naturalized subjects of the Qing dynasty affording them protection against Vietnamese demands for extradition. Some Lê loyalists were also sent to Central Asia in Urumqi.[91][92] Modern descendants of the Lê monarch can be traced to southern Vietnam and Urumqi, Xinjiang.[93][94][95]

    Other queues

    Queue (hairstyle)
    Maximilien Robespierre (1758–1794) wearing a powdered wig tied in a queue that was a common piece of men’s dress by c. 1795.
    Queue (hairstyle)
    A Spanish soldier wearing a queue (1761)

    In the 18th century, European soldiers styled their traditionally long hair into a queue called the “soldier’s queue.” The 18th century custom of tying long powdered curly wigs (which normally reached down the back and chest) behind the neck began at first among soldiers and hunters, as seen as early as 1678 in a depiction of King of France Louis XIV hunting with his long hair tied back.[96][97] By the 1730s, the queue had spread from the military and became widespread also among civilians.[98] A 1697 depiction of a royal guard during the wedding of the Louis, Duke of Burgundy shows the sporting of this hairstyle, which came to influence civilian fashions due to the frequent wars France engaged in during Louis’ reign.[99]

    The queue, worn with the wig either curled, known as a tyewig (a wig tied into a queue) or covered with a silk bag, known as a bag wig, gradually replaced the unwieldy big wigs.[99] Wigs that did not feature a queue such as the bob wig were favoured by those who could not afford a long wig. The type of wig became an indicator of one’s rank, occupation and political leanings.[100] Powdered wigs tied in a queue remained important to men’s fashion until the change of dress in the 1790s which was affected both by the French Revolution (1789–1799) and the Pitt’s hair powder tax in 1795 in Britain[99] although formal court dress of European monarchies still required a powdered wig or long powdered hair tied in a queue until the accession of Napoleon Bonaparte to the throne as emperor in 1804.

    The first western army in which the wearing of a queue by its soldiers was forbidden was the Russian army in the 1780s, before the outbreak of the French Revolution in 1789. Grigory Potemkin, a Russian statesman and favourite of Catherine the Great, abhorred the tight uniforms and uncomfortable powdered wigs tied in a queue worn by soldiers of the Russian Army and instigated a complete revision of both. Along with comfortable, practical, well-fitting uniforms, his reforms introduced short, natural hairstyles for all, without wigs and long powdered hair tied in a queue.[101]

    Soldiers of all other western armies adopted the wearing of short hairstyle without a queue only after the outbreak of the French Revolution during the French Revolutionary Wars and Napoleonic Wars (1792–1815).

    The French army plaited their wigs into a short queue (the French word for “tail”) tied with a ribbon in the back, while the British military used the Ramillies wig, which featured a very long queue tied with two black ribbons, one at the neck and one at the tail end.[102] The French army continued keeping queues until the French Consulate period (1799-1804), when Napoleon Bonaparte and other officers promoted close cropped hair, known as à la Titus. Napoleon himself, initially wearing long hair tied in a queue, changed his hairstyle and cut his hair short while in Egypt in 1798.[103] However, hair policy in the French army was not uniform; some regiments such as the Imperial Guard foot grenadiers stuck to queues long afterwards, while the 2nd Line Infantry kept their queues as late as 1812. Short hair only became mandated at the end of the First Empire with the ordinance of 25 September 1815.[104] Marshal Jean Lannes notably stood out due to his refusal to cut his queue.[105][106]

    British soldiers and sailors during the 18th century also wore their hair in a queue. While not always braided, the hair was pulled back very tight into a single tail, wrapped around a piece of leather and tied down with a ribbon. The hair was often greased and powdered in a fashion similar to powdered wigs, or tarred in the case of sailors. It was said that the soldiers’ hair was pulled back so tightly that they had difficulty closing their eyes afterwards. The use of white hair powder in the British Army was discontinued in 1796 and queues were ordered to be cut off four years later.[107] They continued to be worn in the Royal Navy for a while longer, where they were known as “pigtails“. Officers wore pigtails until 1805 and other ranks continued to wear them until about 1820.[108]

    In the Prussian Army and those of several other states within the Holy Roman Empire, the wearing of soldier’s queue was mandatory under the reign of Frederick William I of Prussia. An artificial or “patent” queue was issued to recruits whose hair was too short to plait. The style was abolished in the Prussian Army by reforms of Gerhard von Scharnhorst in 1807.[109]

    Queue (hairstyle)
    United States Army officers (future U.S.President William Henry Harrison among them) wearing powdered wigs tied in a queue during the Treaty of Greenville negotiations in 1795

    In the United States Army, queues or pigtails had been worn since the Revolutionary War (1775–1783) until 1801. The order to remove all queues was issued on 30 April 1801 by Major General James Wilkinson. The order was highly unpopular with both officers and men, leading to several desertions and threats of resignation. One senior officer, Lieutenant Colonel Thomas Butler, was eventually court-martialled in 1803 for failing to cut his hair.[110]
    In 1817 during a period known as the Era of Good Feelings, president James Monroe (1758–1831), the last U.S. president who was a Revolutionary War veteran, went on a country-wide Great Goodwill Tour. On the tour he donned a Revolutionary War officer’s uniform and tied his long, powdered hair in a queue according to the old-fashioned style of the 18th century.[111]

    See also

    • Beard and haircut laws by country
    • Braid (hairstyle)
    • Cheongsam
    • Chupryna
    • Foot binding
    • Hanfu
    • List of hairstyles
    • Mohawk
    • Pigtail
    • Pigtail Ordinance
    • Rattail (haircut)
    • Sikha
    • Tonsure

    References

    1. ↑ Evans, Thammy (2006). Great Wall of China: Beijing & Northern China. Bradt Travel Guide Great Wall of China. Bradt Travel Guides. p. 41. ISBN 1841621587.
    2. ↑ University of Hawaii at Manoa Art Gallery; Chazen Museum of Art; Museum of International Folk Art (N.M.); Evergrand Art Museum (Taoyuan, Taiwan) (2009). Writing with Thread: Traditional Textiles of South Minorities : a Special Exhibition from the Collection of Huang Ying Feng and the Evergrand Art Museum in Taoyuan, Taiwan. University of Hawai’i Art Gallery. p. 118. ISBN 978-0982033210.
    3. ↑ Ebrey, Patricia Buckley; Walthall, Anne; Palais, James B. (2006). Pre-modern East Asia: To 1800: A Cultural, Social, and Political History. Houghton Mifflin. p. 370. ISBN 0618133860.
    4. ↑ Millward, James (1998). Beyond the Pass: Economy, Ethnicity, and Empire in Qing Central Asia, 1759–1864 (illustrated ed.). Stanford University Press. p. 305. ISBN 0804729336.
    5. ↑ Bromber, Katrin; Krawietz, Birgit; Maguire, Joseph, eds. (2013). Sport Across Asia: Politics, Cultures, and Identities. Vol. 21 (illustrated ed.). Routledge. p. 53. ISBN 978-0415884389.
    6. ↑ Rubo, Han; Liu, Hatty (23 July 2019). “Hairy History”. The World of Chinese. Retrieved 12 September 2024.
    7. ↑ Pyun, Kyunghee; Wong, Aida Yuen (2018). Fashion, Identity, and Power in Modern Asia. Springer. ISBN 978-3319971995.
    8. ↑ East Asian History. 1994. p. 54.
    9. ↑ Godley, Michael R. (December 1994). “The End of the Queue: Hair as a Symbol in Chinese History” (PDF). East Asian History (8).
    10. ↑ Kuwayama, George (1991). Ancient Mortuary Traditions of China: Papers on Chinese Ceramic Funerary Sculptures. Far Eastern Art Council, Los Angeles County Museum of Art. p. 59. ISBN 978-0-87587-157-8.
    11. ↑ Abramson, Marc Samuel (2005). “4 Deep Eyes and High Noses : Physiognomy and the depiction of barbarians in Tang China”. In Di Cosmo, Nicola; Wyatt, Don J (eds.). Political Frontiers, Ethnic Boundaries and Human Geographies in Chinese History (illustrated ed.). Routledge. pp. 126, 127. ISBN 1135790957.
    12. ↑ Duan, Wenjie; Kapila Vatsyayan (1994). Tan, Chung (ed.). Dunhuang Art: Through the Eyes of Duan Wenjie. Indira Gandhi National Centre for the Arts. p. 118. ISBN 8170173132.
    13. ↑ Chan, Hok-Lam (2009). “Ming Taizu’s ‘Placards’ on Harsh Regulations and Punishments Revealed in Gu Qiyuan’s ‘Kezuo Zhuiyu.’“. Asia Major. 22 (1): 28. JSTOR 41649963. Retrieved 16 April 2021.
    14. ↑ Keay, John (2011). China: A History. Basic Books. p. 312. ISBN 978-0-465-02518-3.
    15. ↑ Keay, John (2011). China: A History. Basic Books. p. 335. ISBN 978-0-465-02518-3.
    16. ↑ Franke, Herbert; Twitchett, Denis C. (1994). The Cambridge History of China: Volume 6, Alien Regimes and Border States, 907–1368. Cambridge University Press. ISBN 978-0-521-24331-5.
    17. ↑ Tackett, Nicolas (2017). The Origins of the Chinese Nation: Song China and the Forging of an East Asian World Order. Cambridge University Press. p. 241. ISBN 978-1108186926.
    18. ↑ Zhu, Ruixi; Zhang, Bangwei; Liu, Fusheng; Cai, Chongbang; Wang, Zengyu (2016). A Social History of Medieval China. The Cambridge China Library. Bang Qian Zhu (translator), Peter Ditmanson (contributor) (illustrated ed.). Cambridge University Press. p. 34. ISBN 978-1107167865.
    19. ↑ Liu, Cary Yee-Wei; Ching, Dora C. Y., eds. (1999). Arts of the Sung and Yüan: Ritual, Ethnicity, and Style in Painting. contributors: Metropolitan Museum of Art (New York, N.Y.), Princeton University. Art Museum, Kuo li ku kung po wu yüan, Guo li gu gong bo wu yuan. Art Museum, Princeton University. pp. 168, 159, 179. ISBN 0943012309.
    20. ↑ Shen, Hsueh-man; Asia Society, eds. (2006). Gilded Splendor: Treasures of China’s Liao Empire. contributors: Asia Society Museum, Museum für Ostasiatische Kunst (Berlin, Germany), Museum Rietberg (illustrated ed.). Harry N. Abrams. p. 114. ISBN 8874393326.
    21. ↑ Zhou, Xun; Gao, Chunming; Shang-hai shih hsi chʿü hsüeh hsiao.; Chung-kuo fu chuang shih yen chiu tsu (1987). 5000 years of Chinese costumes (reprint ed.). Commercial Press. p. 130. ISBN 9620750551.
    22. ↑ Tung, W. U.; Tung, Wu; Museum of Fine Arts, Boston (1997). Tales from the Land of Dragons: 1000 Years of Chinese Painting (illustrated ed.). Museum of Fine Arts. p. 136. ISBN 0878464395.
    23. ↑ Johnson, Linda Cooke (2011). Women of the Conquest Dynasties: Gender and Identity in Liao and Jin China. UPCC book collections on Project MUSE (illustrated ed.). University of Hawaiʻi Press. pp. 53, 190. ISBN 978-0824834043.
    24. ↑ Johnson, Linda Cooke (2011). Women of the Conquest Dynasties: Gender and Identity in Liao and Jin China. UPCC book collections on Project MUSE (illustrated ed.). University of Hawaiʻi Press. pp. 48, 52, 53. ISBN 978-0824834043.
    25. ↑ “China Review International, Volume 19”. China Review International. 19. University of Hawaiʻi, Center for Chinese Studies and University of Hawaii Press: 101. 2012.
    26. ↑ 张博泉 (Zhang Boquan) (1984), pp. 97–98.
    27. ↑ Sinor 1990, p. 417.
    28. ↑ Jia Sheng (贾笙), 宋金时代的“留发不留头” Archived 16 November 2010 at the Wayback Machine (“Keep the hair, lose the head” in the Song-Jin era)
    29. ↑ “身体的争夺与展示” (PDF). Archived from the original (PDF) on 7 July 2011.
    30. ↑ Zi Yunju (紫雲居). 中國的髮爪與接觸巫術 [Hair, nails, and magic of China]. Archived from the original on 21 July 2011.
    31. ↑ Li Ziming (李子明). 剃头的故事晚清出国人员生活小记 [Barber’s tale: notes from the life of Chinese abroad in the late Qing era]. Archived from the original on 11 July 2011.
    32. ↑ Lao Lu (Chinese: 老鲁), Chinese: 彻底改变两百年官定习俗 民国初年剪辫轶话 Archived 1 June 2009 at the Wayback Machine (Thoroughly changing the customs officially established for 200 years: the story of queue-cutting in the early Republic of China period)
    33. ↑ Forsyth, James (1994). A History of the Peoples of Siberia: Russia’s North Asian Colony 1581–1990 (illustrated, reprint, revised ed.). Cambridge University Press. p. 214. ISBN 0521477719.
    34. ↑ Majewicz, Alfred F., ed. (2011). Materials for the Study of Tungusic Languages and Folklore (illustrated, reprint ed.). Walter de Gruyter. p. 21. ISBN 978-3110221053. Volume 15, Issue 4 of Trends in Linguistics. Documentation [TiLDOC]
    35. ↑ Paolo Santangelo (2013). Zibuyu, “What The Master Would Not Discuss”, according to Yuan Mei (1716–1798): A Collection of Supernatural Stories (2 vols). Brill. pp. 829–. ISBN 978-90-04-21628-0.
    36. ↑ Evelyn S. Rawski (1998). The Last Emperors: A Social History of Qing Imperial Institutions. University of California Press. pp. 41–. ISBN 978-0-520-92679-0. tifa queue.
    37. ↑ Tu, Hang (2025). Sentimental Republic: Chinese Intellectuals and the Maoist Past. Harvard University Asia Center. ISBN 9780674297579.
    38. ↑ 清朝乾隆23年清政府令平埔族人學清俗 Archived 9 July 2009 at the Wayback Machine
    39. ↑ “清朝之剃髮結辮” (PDF). Archived from the original (PDF) on 12 June 2009.
    40. ↑ “On Qing Government’s Land Policy towards Chaoxian Reclamation People of Break Probibition”.
    41. ↑ “论清朝对图们江以北朝鲜垦民的”薙发易服”政策”. ltgx.net. Archived from the original on 23 March 2020. Retrieved 9 January 2010.
    42. ↑ De Bary; William T. (1999). Sources of Chinese Tradition. Columbia University Press. p. 326.
    43. ↑ Godley, Michael R. (September 2011). “The End of the Queue: Hair as Symbol in Chinese History”. China Heritage Quarterly (27). China Heritage Project, ANU College of Asia & the Pacific (CAP), The Australian National University. ISSN 1833-8461.
    44. ↑ Meyer-Fong, Tobie (2013). What Remains: Coming to Terms with Civil War in 19th Century China (illustrated ed.). Stanford University Press. p. 83. ISBN 978-0804785594.
    45. ↑ Hang, Xing (2016). Conflict and Commerce in Maritime East Asia: The Zheng Family and the Shaping of the Modern World, c. 1620–1720. Cambridge University Press. p. 77. ISBN 978-1316453841.
    46. ↑ Hang, Xing (2016). Conflict and Commerce in Maritime East Asia: The Zheng Family and the Shaping of the Modern World, c. 1620–1720. Cambridge University Press. p. 86. ISBN 978-1316453841.
    47. ↑ Hang, Xing (2016). Conflict and Commerce in Maritime East Asia: The Zheng Family and the Shaping of the Modern World, c. 1620–1720. Cambridge University Press. p. 187. ISBN 978-1316453841.
    48. 1 2 Kuhn, Philip A. (1990). Soulstealers: The Chinese Sorcery Scare of 1768. Harvard University Press. pp. 53–54.
    49. ↑ Wakeman (1985), p. 647; Struve (1988), p. 662; Dennerline (2002), p. 87 which calls this edict “the most untimely promulgation of [Dorgon’s] career.”
    50. ↑ Wakeman 1985, p. 868.
    51. ↑ Lui, Adam Yuen-chung (1989). Two Rulers in One Reign: Dorgon and Shun-chih, 1644–1660. Faculty of Asian Studies monographs // The Australian National University (illustrated ed.). Faculty of Asian Studies, Australian National University. p. 37. ISBN 0731506545. Dorgon did not want to see anything go wrong in a province and this might be the main reason why the government … When the Chinese were ordered to wear the queue, Sun and Li took the initiative in changing their Ming hairstyle to …
    52. ↑ Cheng, Weikun (1998). “6 politics of the queue: agitation and resistance in the beginning and end of qing china”. In Hiltebeitel, Alf; Miller, Barbara D. (eds.). Hair: Its Power and Meaning in Asian Cultures (illustrated ed.). SUNY Press. p. 125. ISBN 0791437418.
    53. ↑ Hang, Xing (2016). “2 From smuggler-pirates to loyal Confucians”. Conflict and Commerce in Maritime East Asia: The Zheng Family and the Shaping of the Modern World, c. 1620–1720. Cambridge University Press. p. 40. ISBN 978-1316453841.
    54. ↑ Wakeman (1985), pp. 647, 650.
    55. ↑ Chee Kiong Tong; et al. (2001). Alternate Identities: The Chinese of Contemporary Thailand. Brill Publishers. p. 44. ISBN 978-981-210-142-6. keep your hair and lose your head.
    56. ↑ 研堂見聞雜記
    57. ↑ Faure (2007), p. 164.
    58. ↑ Ebrey (1993), p. .
    59. ↑ Ebrey (1993), p. 271.
    60. ↑ Wakeman 1975b, p. 83
    61. ↑ Godley, Michael R. (September 2011). “The End of the Queue: Hair as Symbol in Chinese History”. China Heritage Quarterly (27). China Heritage Project, The Australian National University. ISSN 1833-8461.
    62. ↑ Justus Doolittle (1876). Social Life of the Chinese: With Some Account of Their Religious, Governmental, Educational, and Business Customs and Opinions. With Special But Not Exclusive Reference to Fuhchau. Harpers. pp. 242–.
    63. ↑ Rhoads 2011, p. 61.
    64. ↑ Twitchett, Denis; Fairbank, John K. (2008) Cambridge History of China Volume 9 Part 1 The Ch’ing Empire to 1800, pp. 87–88
    65. ↑ 周锡保. 《中国古代服饰史》. 中国戏剧出版社. 2002: 449. ISBN 978-7104003595.
    66. ↑ Lipman, Jonathan N. (1997). “4 / Strategies of Resistance Integration by Violence”. Familiar Strangers : A History of Muslims in Northwest China. University of Washington Press. p. 105. ISBN 0-295-97644-6.
    67. ↑ Lipman, Jonathan N. (2011). Familiar Strangers: A History of Muslims in Northwest China. University of Washington Press. ISBN 978-0295800554.
    68. ↑ Lipman, Jonathan N. (1997). Familiar Strangers: A History of Muslims in Northwest China. University of Washington Press. ISBN 978-0-295-97644-0. JSTOR j.ctvbtzmb8.
    69. ↑ Arienne M. Dwyer (2007). Salar. Otto Harrassowitz Verlag. p. 22. ISBN 978-3-447-04091-4.
    70. ↑ Pamela Kyle Crossley; Helen F. Siu; Donald S. Sutton (2006). Empire at the Margins: Culture, Ethnicity, and Frontier in Early Modern China. University of California Press. p. 127. ISBN 978-0-520-23015-6.
    71. ↑ Chen, Yangbin (2008). Muslim Uyghur Students in a Chinese Boarding School: Social Recapitalization as a Response to Ethnic Integration. Emerging Perspectives on Education in China. Lexington Books. p. 165. ISBN 978-1461633846.
    72. ↑ Yakup, Abdurishid (2005). The Turfan Dialect of Uyghur. Vol. 63 of Turcologica Series (illustrated ed.). Otto Harrassowitz Verlag. p. 187. ISBN 3447052333.
    73. ↑ Brophy, David (2016). Uyghur Nation: Reform and Revolution on the Russia-China Frontier. Harvard University Press. ISBN 978-0674970465.
    74. ↑ James Millward (1998). Beyond the Pass: Economy, Ethnicity, and Empire in Qing Central Asia, 1759–1864. Stanford University Press. pp. 204–. ISBN 978-0-8047-9792-4.
    75. ↑ Hiltebeitel, Alf, ed. (1998). Hair: Its Power and Meaning in Asian Cultures. State University of New York Press. p. 128.
    76. 1 2 “鄭氏王朝的滅亡”. taiwanus.net. 21 February 2024.
    77. ↑ ““cutting tail”reflected the jiang nan social in guang xu dynasty two years” (PDF). Archived from the original (PDF) on 14 August 2011. Retrieved 26 September 2009.
    78. ↑ “清代妖术恐慌及政府的对策：以两次剪辫谣言为例_历史千年”. lsqn.cn. 24 April 2013.{{cite web}}: CS1 maint: deprecated archival service (link)
    79. ↑ 頭可斷辮子不可剪 清朝留學生剪辮=偷了情 Archived 3 October 2011 at the Wayback Machine
    80. ↑ Wiltshire, Trea. [First published 1987] (republished & reduced 2003). Old Hong Kong – Volume One. Central, Hong Kong: Text Form Asia books Ltd. ISBN 962-7283-59-2 (Vol. One)
    81. ↑ Anthony, Robert J. (2014). “Violence and Predation on the Sino-Vietnamese Maritime Frontier, 1450–1850”. Asia Major. 27 (2): 104. JSTOR 44740552. Retrieved 16 April 2021.
    82. ↑ Modern Chinese Literature and Culture, Volume 19, Issue 1. Foreign Language Publications. 2007. p. 175.
    83. ↑ Rhoads 2011, p. 60.
    84. ↑ The Museum Journal. The Museum. 1921. pp. 102–.
    85. ↑ George Cockburn (1896). John Chinaman: His Ways and Notions. J. Gardner Hitt. pp. 86–.
    86. ↑ James, Henry Evan Murchison (1888). The Long White Mountain, or, A journey in Manchuria: with some account of the history, people, administration and religion of that country. Longmans, Green, and Co. pp. 110–112.
    87. ↑ Boulger, Demetrius C. (1899). “The Dissolution of the Chinese Empire”. The North American Review. 168 (508): 264. JSTOR 25119153.
    88. ↑ Boulger, Demetrius C. (1900). “America’s Share in a Partition of China”. The North American Review. 171 (525): 171–181. JSTOR 25105038.
    89. ↑ Boulger, Demetrius Charles (28 April 1893). “Chapter IX”. China (PDF). p. 76. Archived from the original (PDF) on 24 May 2020. Retrieved 24 May 2020.
    90. ↑ Meyer-Fong, Tobie (2013). What Remains: Coming to Terms with Civil War in 19th Century China (illustrated ed.). Stanford University Press. p. 94. ISBN 978-0804785594.
    91. ↑ Anderson, James A.; Whitmore, John K. (2014). China’s Encounters on the South and Southwest: Reforging the Fiery Frontier Over Two Millennia. Handbook of Oriental Studies. Section 3 Southeast Asia (reprint, revised ed.). Brill. p. 309. ISBN 978-9004282483.
    92. ↑ Annam and its Minor Currency, chapter 16.
    93. ↑ “新疆曾有大批越南皇室后裔，乾隆时期投靠中国并前往乌鲁木齐开荒” (in Chinese). 27 May 2018.
    94. ↑ Thái Mỹ (24 April 2019). “Con trai vua Lê Thế Tông ở đất Thanh Châu” (in Vietnamese).
    95. ↑ Lê Tiên Long (9 December 2018). “After Minh Mang reigned Nguyen Dynasty, why he deported Le royal descendants to the Southern Vietnam?” (in Vietnamese).
    96. ↑ Joan Nunn (2000). Fashion in Costume, 1200–2000. New Amsterdam Books. p. 62. ISBN 9781566632799.
    97. ↑ Francis Michael Kelly, Randolph Schwabe (2002). European Costume and Fashion, 1490–1790. Dover Publications. p. 168. ISBN 9780486423227.
    98. ↑ Vaughan Lee, Heather; Blanco, José F.; Doering, Mary; Hunt-Hurst, Patricia Kay (2015). Clothing and Fashion [4 Volumes] American Fashion from Head to Toe [4 Volumes]. ABC-CLIO. p. 135. ISBN 9781610693103.
    99. 1 2 3 Mark Ledbury, Robert Wellington (2020). The Versailles Effect: Objects, Lives, and Afterlives of the Domaine. Bloomsbury Publishing USA. p. 98. ISBN 9781501357763.
    100. ↑ Lynn Festa (2005). “Personal Effects: Wigs and Possessive Individualism in the Long Eighteenth Century”. Eighteenth-Century Life. 29 (2): 59.
    101. ↑ Bruce W. Menning, David Schimmelpenninck van der Oye (2004). Reforming the Tsar’s Army. Military Innovation in Imperial Russia from Peter the Great to the Revolution. Cambridge University Press. p. 279. ISBN 9780521819886.
    102. ↑ Beverly Chico (2013). Hats and Headwear Around the World A Cultural Encyclopedia. ABC-CLIO. p. 484. ISBN 9781610690638.
    103. ↑ Philip Dwyer (2007). Napoleon, The Path to Power 1769 – 1799. Yale University Press. p. 472. ISBN 978-0-300-13754-5.
    104. ↑ Terry Crowdy (2015). Napoleon’s Infantry Handbook An Essential Guide to Life in the Grand Army. Pen & Sword Military. p. 78. ISBN 9781783462957.
    105. ↑ R. F. Delderfield (2002). Napoleon’s Marshals. Cooper Square Press. p. 74. ISBN 9781461661191.
    106. ↑ Janice Best (2023). Power and Propaganda in French Second Empire Theatre Playing Napoleon. Cambridge Scholars Publishing. p. 237. ISBN 9781527500914.
    107. ↑ Stocqueler, Joachim Hayward (1871) A Familiar History of the British Army, from the Restoration in 1660 to the Present Time, Edward Stanford, London (pp. 103–104)
    108. ↑ Wilkinson-Latham, Robert (1977) The Royal Navy, 1790–1970 Osprey Publishing, ISBN 0-85045-248-1 (p. 34)
    109. ↑ Hudson, Elizabeth Harriot (1878), (The Life And Times of Louisa, Queen of Prussia With an Introductory Sketch of Prussian History: Volume II reprinted by Adamant Media Corporation (13 September 2001) (pp. 214–215)
    110. ↑ Borch, Fred L. (March 2012). “Lore of the Corps – The True Story of a Colonel’s Pigtail and a Court-Martial”. The Army Lawyer (Special Edition): 1–2. Retrieved 20 May 2022.
    111. ↑ Wilentz, Sean (2008). The Rise of American Democracy: Jefferson to Lincoln. Horton New York. p. 202.

    Works cited

    • Dennerline, Jerry (2002). “The Shun-chih Reign”. In Peterson, Willard J. (ed.). Cambridge History of China, Vol. 9, Part 1: The Ch’ing Dynasty to 1800. Cambridge: Cambridge University Press. pp. 73–119. ISBN 0-521-24334-3.
    • Ebrey, Patricia (1993). Chinese Civilization: A Sourcebook. Simon and Schuster.
    • Faure, David (2007). Emperor and Ancestor: State and Lineage in South China. Stanford University Press. ISBN 978-0-8047-5318-0.
    • Nguyễn Khắc Thuần (2005), Danh tướng Việt Nam, Nhà Xuất bản Giáo dục.
    • Rhoads, Edward J. M. (2011). Manchus and Han: Ethnic Relations and Political Power in Late Qing and Early Republican China, 1861–1928. University of Washington Press. ISBN 978-0-295-80412-5. Retrieved 10 March 2014.
    • Sinor, Denis, ed. (1990). The Cambridge History of Early Inner Asia, Volume 1 (illustrated, reprint ed.). Cambridge University Press. ISBN 0-521-24304-1. Retrieved 10 March 2014.
    • Struve, Lynn (1988). “The Southern Ming”. In Frederic W. Mote; Denis Twitchett; John King Fairbank (eds.). Cambridge History of China, Volume 7, The Ming Dynasty, 1368–1644. Cambridge: Cambridge University Press. pp. 641–725. ISBN 0-521-24332-7.
    • Wakeman, Frederic (1975b), “Localism and Loyalism During the Ch’ing Conquest of Kiangnan: The Tragedy of Chiang-yin”, in Frederic Wakeman Jr.; Carolyn Grant (eds.), Conflict and Control in Late Imperial China, Berkeley: Center of Chinese Studies, University of California, Berkeley, pp. 43–85, ISBN 978-0520025974.
    • Wakeman, Frederic (1985). The Great Enterprise: The Manchu Reconstruction of Imperial Order in Seventeenth-Century China. University of California Press. ISBN 0-520-04804-0. In two volumes. [Volume 1. University of California Press. 31 March 1986. ISBN 9780520048041.]
    • 张博泉 (Zhang Boquan) (1984). 《金史简编》. 辽宁人民出版社.

    • Also mentioned in “Dragonwings”, by Laurence Yep, Chapter 4

    Further reading

    • Struve, Lynn A (1998), Voices from the Ming-Qing Cataclysm: China in Tigers’ Jaws, Yale University Press, ISBN 978-0-300-07553-3. (312 pages).

    External links

    This article is adapted from “Queue (hairstyle)” on Wikipedia, written by its contributors, and used under CC BY-SA 4.0. Text on this page is available under the same licence. Source snapshot: Wikipedia via Kiwix, 2026-07-20.

  • Borromean rings

    Borromean rings
    Borromean rings

    L6a4
    Crossing no. 6
    Hyperbolic volume 7.327724753
    Stick no. 9
    Conway notation .1
    A–B notation 63
    2
    Thistlethwaite L6a4
    Other
    alternating, hyperbolic

    In mathematics, the Borromean rings[a] are three simple closed curves in three-dimensional space that are topologically linked and cannot be separated from each other, but that break apart into two unknotted and unlinked loops when any one of the three is cut or removed. Most commonly, these rings are drawn as three circles in the plane, in the pattern of a Venn diagram, alternatingly crossing over and under each other at the points where they cross. Other triples of curves are said to form the Borromean rings as long as they are topologically equivalent to the curves depicted in this drawing.

    The Borromean rings are named after the Italian House of Borromeo, who used the circular form of these rings as an element of their coat of arms, but designs based on the Borromean rings have been used in many cultures, including by the Norsemen and in Japan. They have been used in Christian symbolism as a sign of the Trinity, and in modern commerce as the logo of Ballantine beer, giving them the alternative name Ballantine rings. Physical instances of the Borromean rings have been made from linked DNA or other molecules, and they have analogues in the Efimov state and Borromean nuclei, both of which have three components bound to each other although no two of them are bound.

    Geometrically, the Borromean rings may be realized by linked ellipses, or (using the vertices of a regular icosahedron) by linked golden rectangles. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In knot theory, the Borromean rings can be proved to be linked by counting their Fox n-colorings. As links, they are Brunnian, alternating, algebraic, and hyperbolic. In arithmetic topology, certain triples of prime numbers have analogous linking properties to the Borromean rings.

    Definition and notation

    It is common in mathematics publications that define the Borromean rings to do so as a link diagram, a drawing of curves in the plane with crossings marked to indicate which curve or part of a curve passes above or below at each crossing. Such a drawing can be transformed into a system of curves in three-dimensional space by embedding the plane into space and deforming the curves drawn on it above or below the embedded plane at each crossing, as indicated in the diagram. The commonly used diagram for the Borromean rings consists of three equal circles centered at the points of an equilateral triangle, close enough together that their interiors have a common intersection (such as in a Venn diagram or the three circles used to define the Reuleaux triangle). Its crossings alternate between above and below when considered in consecutive order around each circle;[2][3][4] another equivalent way to describe the over-under relation between the three circles is that each circle passes over a second circle at both of their crossings, and under the third circle at both of their crossings.[5] Two links are said to be equivalent if there is a continuous deformation of space (an ambient isotopy) taking one to another, and the Borromean rings may refer to any link that is equivalent in this sense to the standard diagram for this link.[4]

    In The Knot Atlas, the Borromean rings are denoted with the code “L6a4”; the notation means that this is a link with six crossings and an alternating diagram, the fourth of five alternating 6-crossing links identified by Morwen Thistlethwaite in a list of all prime links with up to 13 crossings.[6] In the tables of knots and links in Dale Rolfsen’s 1976 book Knots and Links, extending earlier listings in the 1920s by Alexander and Briggs, the Borromean rings were given the Alexander–Briggs notation “63
    2    “, meaning that this is the second of three 6-crossing 3-component links to be listed.[6][7] The Conway notation for the Borromean rings, “.1”, is an abbreviated description of the standard link diagram for this link.[8]

    History and symbolism

    Borromean rings
    Valknut on Stora Hammars I stone
    Borromean rings
    Symbol of the Christian Trinity, adapted from a 13th-century manuscript
    Borromean rings
    Linked triangles in the Marundeeswarar Temple

    The name “Borromean rings” comes from the use of these rings, in the form of three linked circles, in the coat of arms of the aristocratic Borromeo family in Northern Italy.[9][10] The link itself is much older and has appeared in the form of the valknut, three linked equilateral triangles with parallel sides, on Norse image stones dating back to the 7th century.[11] The Ōmiwa Shrine in Japan is also decorated with a motif of the Borromean rings, in their conventional circular form.[2] A stone pillar in the 6th-century Marundeeswarar Temple in India shows three equilateral triangles rotated from each other to form a regular enneagram; like the Borromean rings these three triangles are linked and not pairwise linked,[12] but this crossing pattern describes a different link than the Borromean rings.[13]

    Borromean rings
    A Seifert surface of the Borromean rings

    The Borromean rings have been used in different contexts to indicate strength in unity.[14] In particular, some have used the design to symbolize the Trinity.[3] A 13th-century French manuscript depicting the Borromean rings labeled as unity in trinity was lost in a fire in the 1940s, but reproduced in an 1843 book by Adolphe Napoléon Didron. Didron and others have speculated that the description of the Trinity as three equal circles in canto 33 of Dante’s Paradiso was inspired by similar images, although Dante does not detail the geometric arrangement of these circles.[15][16] The psychoanalyst Jacques Lacan found inspiration in the Borromean rings as a model for his topology of human subjectivity, with each ring representing a fundamental Lacanian component of reality (the “real”, the “imaginary”, and the “symbolic”).[17]

    The rings were used as the logo of Ballantine beer, and are still used by the Ballantine brand beer, now distributed by the current brand owner, the Pabst Brewing Company.[18][19] For this reason they have sometimes been called the “Ballantine rings”.[3][18]

    The first work of knot theory to include the Borromean rings was a catalog of knots and links compiled in 1876 by Peter Tait.[3] In recreational mathematics, the Borromean rings were popularized by Martin Gardner, who featured Seifert surfaces for the Borromean rings in his September 1961 “Mathematical Games” column in Scientific American.[19] In 2006, the International Mathematical Union decided at the 25th International Congress of Mathematicians in Madrid, Spain to use a new logo based on the Borromean rings.[2]

    Partial and multiple rings

    In medieval and renaissance Europe, a number of visual signs consist of three elements interlaced together in the same way that the Borromean rings are shown interlaced (in their conventional two-dimensional depiction), but with individual elements that are not closed loops. Examples of such symbols are the Snoldelev stone horns[20] and the Diana of Poitiers crescents.[3]

    Some knot-theoretic links contain multiple Borromean rings configurations; one five-loop link of this type is used as a symbol in Discordianism, based on a depiction in the Principia Discordia.[21]

    Mathematical properties

    Linkedness

    Borromean rings
    Algebraic link diagram for the Borromean rings. The vertical dotted black midline is a Conway sphere separating the diagram into 2-tangles.

    In knot theory, the Borromean rings are a simple example of a Brunnian link, a link that cannot be separated but that falls apart into separate unknotted loops as soon as any one of its components is removed. There are infinitely many Brunnian links, and infinitely many three-curve Brunnian links, of which the Borromean rings are the simplest.[13][22]

    There are a number of ways of seeing that the Borromean rings are linked. One is to use Fox n-colorings, colorings of the arcs of a link diagram with the integers modulo n so that at each crossing, the two colors at the undercrossing have the same average (modulo n) as the color of the overcrossing arc, and so that at least two colors are used. The number of colorings meeting these conditions is a knot invariant, independent of the diagram chosen for the link. A trivial link with three components has n 3 − n {\displaystyle n^{3}-n} {\displaystyle n^{3}-n} colorings, obtained from its standard diagram by choosing a color independently for each component and discarding the n {\displaystyle n} {\displaystyle n} colorings that only use one color. For standard diagram of the Borromean rings, on the other hand, the same pairs of arcs meet at two undercrossings, forcing the arcs that cross over them to have the same color as each other, from which it follows that the only colorings that meet the crossing conditions violate the condition of using more than one color. Because the trivial link has many valid colorings and the Borromean rings have none, they cannot be equivalent.[4][23]

    The Borromean rings are an alternating link, as their conventional link diagram has crossings that alternate between passing over and under each curve, in order along the curve. They are also an algebraic link, a link that can be decomposed by Conway spheres into 2-tangles. They are the simplest alternating algebraic link which does not have a diagram that is simultaneously alternating and algebraic.[24] It follows from the Tait conjectures that the crossing number of the Borromean rings (the fewest crossings in any of their link diagrams) is 6, the number of crossings in their alternating diagram.[4]

    Ring shape

    Borromean rings
    One possible 3D arrangement of Borromean rings: if they are circular when viewed from above, then they need to curve outside of a single plane.[4]

    The Borromean rings are typically drawn with their rings projecting to circles in the plane of the drawing, but three-dimensional circular Borromean rings are an impossible object: it is not possible to form the Borromean rings from circles in three-dimensional space.[4]

    More generally Michael H. Freedman and Richard Skora (1987) proved using four-dimensional hyperbolic geometry that no Brunnian link can be exactly circular.[25] For three rings in their conventional Borromean arrangement, this can be seen from considering the link diagram. If one assumes that two of the circles touch at their two crossing points, then they lie in either a plane or a sphere. In either case, the third circle must pass through this plane or sphere four times, without lying in it, which is impossible.[26] Another argument for the impossibility of circular realizations, by Helge Tverberg, uses inversive geometry to transform any three circles so that one of them becomes a line, making it easier to argue that the other two circles do not link with it to form the Borromean rings.[27]

    Borromean rings
    Realization of Borromean rings using ellipses
    Borromean rings
    Three linked golden rectangles in a regular icosahedron

    However, the Borromean rings can be realized using ellipses.[2] These may be taken to be of arbitrarily small eccentricity: no matter how close to being circular their shape may be, as long as they are not perfectly circular, they can form Borromean links if suitably positioned. A realization of the Borromean rings by three mutually perpendicular golden rectangles can be found within a regular icosahedron by connecting three opposite pairs of its edges.[2] Every three unknotted polygons in Euclidean space may be combined, after a suitable scaling transformation, to form the Borromean rings. If all three polygons are planar, then scaling is not needed.[28] In particular, because the Borromean rings can be realized by three triangles, the minimum number of sides possible for each of its loops, the stick number of the Borromean rings is nine.[29]


    Unsolved problem in mathematics
    Are there three unknotted curves, not all circles, that cannot form the Borromean rings?
    More unsolved problems in mathematics

    More generally, Matthew Cook has conjectured that any three unknotted simple closed curves in space, not all circles, can be combined without scaling to form the Borromean rings. After Jason Cantarella suggested a possible counterexample, Hugh Nelson Howards weakened the conjecture to apply to any three planar curves that are not all circles. On the other hand, although there are infinitely many Brunnian links with three links, the Borromean rings are the only one that can be formed from three convex curves.[28]

    Ropelength

    Borromean rings
    Logo of the International Mathematical Union

    In knot theory, the ropelength of a knot or link is the shortest length of flexible rope (of radius one) that can realize it. Mathematically, such a realization can be described by a smooth curve whose radius-one tubular neighborhood avoids self-intersections. The minimum ropelength of the Borromean rings has not been proven, but the smallest value that has been attained is realized by three copies of a 2-lobed planar curve.[2][30] Although it resembles an earlier candidate for minimum ropelength, constructed from four circular arcs of radius two,[31] it is slightly modified from that shape, and is composed from 42 smooth pieces defined by elliptic integrals, making it shorter by a fraction of a percent than the piecewise-circular realization. It is this realization, conjectured to minimize ropelength, that was used for the International Mathematical Union logo. Its length is ≈ 58.006 {\displaystyle \approx 58.006} {\displaystyle \approx 58.006}, while the best proven lower bound on the length is 12 π ≈ 37.699 {\displaystyle 12\pi \approx 37.699} {\displaystyle 12\pi \approx 37.699}.[2][30]

    For a discrete analogue of ropelength, the shortest representation using only edges of the integer lattice, the minimum length for the Borromean rings is exactly 36 {\displaystyle 36} {\displaystyle 36}. This is the length of a representation using three 2 × 4 {\displaystyle 2\times 4} {\displaystyle 2\times 4} integer rectangles, inscribed in Jessen’s icosahedron in the same way that the representation by golden rectangles is inscribed in the regular icosahedron.[32]

    Hyperbolic geometry

    Borromean rings
    The complement of the Borromean rings, a hyperbolic manifold formed from two ideal octahedra, seen repeatedly in this view. The rings are infinitely far away, at the octahedron vertices.

    The Borromean rings are a hyperbolic link: the space surrounding the Borromean rings (their link complement) admits a complete hyperbolic metric of finite volume. Although hyperbolic links are now considered plentiful, the Borromean rings were one of the earliest examples to be proved hyperbolic, in the 1970s,[33][34] and this link complement was a central example in the video Not Knot, produced in 1991 by the Geometry Center.[35]

    Hyperbolic manifolds can be decomposed in a canonical way into gluings of hyperbolic polyhedra (the Epstein–Penner decomposition) and for the Borromean complement this decomposition consists of two ideal regular octahedra.[34][36] The volume of the Borromean complement is 16 Λ ( π / 4 ) = 8 G ≈ 7.32772 … {\displaystyle 16\Lambda (\pi /4)=8G\approx 7.32772\dots } {\displaystyle 16\Lambda (\pi /4)=8G\approx 7.32772\dots } where Λ {\displaystyle \Lambda } {\displaystyle \Lambda } is the Lobachevsky function and G {\displaystyle G} {\displaystyle G} is Catalan’s constant.[36] The complement of the Borromean rings is universal, in the sense that every closed 3-manifold is a branched cover over this space.[37]

    Number theory

    In arithmetic topology, there is an analogy between knots and prime numbers in which one considers links between primes. The triple of primes (13, 61, 937) are linked modulo 2 (the Rédei symbol is −1) but are pairwise unlinked modulo 2 (the Legendre symbols are all 1). Therefore, these primes have been called a “proper Borromean triple modulo 2”[38] or “mod 2 Borromean primes”.[39]

    Physical realizations

    Borromean rings
    Borromean ring knitting project by knot theorist Laura Taalman
    Borromean rings
    Molecular Borromean rings[40]

    A monkey’s fist knot is essentially a 3-dimensional representation of the Borromean rings, albeit with three layers, in most cases.[41] Sculptor John Robinson has made artworks with three equilateral triangles made out of sheet metal, linked to form Borromean rings and resembling a three-dimensional version of the valknut.[13][29] A common design for a folding wooden tripod consists of three pieces carved from a single piece of wood, with each piece consisting of two lengths of wood, the legs and upper sides of the tripod, connected by two segments of wood that surround an elongated central hole in the piece. Another of the three pieces passes through each of these holes, linking the three pieces together in the Borromean rings pattern. Tripods of this form have been described as coming from Indian or African hand crafts.[42][43]

    In chemistry, molecular Borromean rings are the molecular counterparts of Borromean rings, which are mechanically-interlocked molecular architectures. In 1997, biologist Chengde Mao and coworkers of New York University succeeded in constructing a set of rings from DNA.[44] In 2003, chemist Fraser Stoddart and coworkers at UCLA utilised coordination chemistry to construct a set of rings in one step from 18 components.[40] Borromean ring structures have been used to describe noble metal clusters shielded by a surface layer of thiolate ligands.[45] A library of Borromean networks has been synthesized by design by Giuseppe Resnati and coworkers via halogen bond driven self-assembly.[46] In order to access the molecular Borromean ring consisting of three unequal cycles a step-by-step synthesis was proposed by Jay S. Siegel and coworkers.[47]

    In physics, a quantum-mechanical analog of Borromean rings is called a halo state or an Efimov state, and consists of three bound particles that are not pairwise bound. The existence of such states was predicted by physicist Vitaly Efimov in 1970, and confirmed by multiple experiments beginning in 2006.[48][49] This phenomenon is closely related to a Borromean nucleus, a stable atomic nucleus consisting of three groups of particles that would be unstable in pairs.[50] Another analog of the Borromean rings in quantum information theory involves the entanglement of three qubits in the Greenberger–Horne–Zeilinger state.[14]

    Notes

    1. ↑ Pronounced /bɒroʊˈmiːən/[1]

    References

    1. ↑ Mackey & Mackay 1922 The Pronunciation of 10,000 Proper Names
    2. 1 2 3 4 5 6 7 Gunn, Charles; Sullivan, John M. (2008), “The Borromean Rings: A video about the New IMU logo”, in Sarhangi, Reza; Séquin, Carlo H. (eds.), Bridges Leeuwarden: Mathematics, Music, Art, Architecture, Culture, London: Tarquin Publications, pp. 63–70, ISBN 978-0-9665201-9-4; see the video itself at “The Borromean Rings: A new logo for the IMU Archived 2021-03-08 at the Wayback Machine” [w/video], International Mathematical Union
    3. 1 2 3 4 5 Cromwell, Peter; Beltrami, Elisabetta; Rampichini, Marta (March 1998), “The Borromean rings”, The mathematical tourist, The Mathematical Intelligencer, 20 (1): 53–62, doi:10.1007/bf03024401, S2CID 189888135
    4. 1 2 3 4 5 6 Aigner, Martin; Ziegler, Günter M. (2018), “Chapter 15: The Borromean Rings Don’t Exist”, Proofs from THE BOOK (6th ed.), Springer, pp. 99–106, doi:10.1007/978-3-662-57265-8_15, ISBN 978-3-662-57265-8
    5. ↑ Chamberland, Marc; Herman, Eugene A. (2015), “Rock-paper-scissors meets Borromean rings”, The Mathematical Intelligencer, 37 (2): 20–25, doi:10.1007/s00283-014-9499-4, MR 3356112, S2CID 558993
    6. 1 2 “Borromean rings“, The Knot Atlas.
    7. ↑ Rolfsen, Dale (1990), Knots and Links, Mathematics Lecture Series, vol. 7 (2nd ed.), Publish or Perish, Inc., Houston, TX, p. 425, ISBN 0-914098-16-0, MR 1277811
    8. ↑ Conway, J. H. (1970), “An enumeration of knots and links, and some of their algebraic properties”, Computational Problems in Abstract Algebra (Proc. Conf., Oxford, 1967), Oxford: Pergamon, pp. 329–358, MR 0258014; see description of notation, pp. 332–333, and second line of table, p. 348.
    9. ↑ Crum Brown, Alexander (December 1885), “On a case of interlacing surfaces”, Proceedings of the Royal Society of Edinburgh, 13: 382–386
    10. ↑ Schoeck, Richard J. (Spring 1968), “Mathematics and the languages of literary criticism”, The Journal of Aesthetics and Art Criticism, 26 (3): 367–376, doi:10.2307/429121, JSTOR 429121
    11. ↑ Bruns, Carson J.; Stoddart, J. Fraser (2011), “The mechanical bond: A work of art”, in Fabbrizzi, L. (ed.), Beauty in Chemistry, Topics in Current Chemistry, vol. 323, Springer, pp. 19–72, doi:10.1007/128_2011_296, PMID 22183145
    12. ↑ Lakshminarayan, Arul (May 2007), “Borromean triangles and prime knots in an ancient temple”, Resonance, 12 (5): 41–47, doi:10.1007/s12045-007-0049-7, S2CID 120259064
    13. 1 2 3 Jablan, Slavik V. (1999), “Are Borromean links so rare?”, Proceedings of the 2nd International Katachi U Symmetry Symposium, Part 1 (Tsukuba, 1999), Forma, 14 (4): 269–277, MR 1770213
    14. 1 2 Aravind, P. K. (1997), “Borromean entanglement of the GHZ state” (PDF), in Cohen, R. S.; Horne, M.; Stachel, J. (eds.), Potentiality, Entanglement and Passion-at-a-Distance, Boston Studies in the Philosophy of Science, Springer, pp. 53–59, doi:10.1007/978-94-017-2732-7_4, MR 1739812
    15. ↑ Didron, Adolphe Napoléon (1843), Iconographie Chrétienne (in French), Paris: Imprimerie Royale, pp. 568–569
    16. ↑ Saiber, Arielle; Mbirika, aBa (2013), “The Three Giri of Paradiso 33″ (PDF), Dante Studies (131): 237–272, JSTOR 43490498
    17. ↑ Ragland-Sullivan, Ellie; Milovanovic, Dragan (2004), “Introduction: Topologically Speaking”, Lacan: Topologically Speaking, Other Press, ISBN 978-1-892746-76-4
    18. 1 2 Glick, Ned (September 1999), “The 3-ring symbol of Ballantine Beer”, The mathematical tourist, The Mathematical Intelligencer, 21 (4): 15–16, doi:10.1007/bf03025332, S2CID 123311380
    19. 1 2 Gardner, Martin (September 1961), “Surfaces with edges linked in the same way as the three rings of a well-known design”, Mathematical Games, Scientific American, reprinted as Gardner, Martin (1991), “Knots and Borromean Rings”, The Unexpected Hanging and Other Mathematical Diversions, University of Chicago Press, pp. 24–33; see also Gardner, Martin (September 1978), “The Toroids of Dr. Klonefake”, Asimov’s Science Fiction, vol. 2, no. 5, p. 29
    20. ↑ Baird, Joseph L. (1970), “Unferth the þyle“, Medium Ævum, 39 (1): 1–12, doi:10.2307/43631234, JSTOR 43631234, the stone bears also representations of three horns interlaced
    21. ↑ “Mandala”, Principia Discordia (4th ed.), March 1970, p. 43
    22. ↑ Bai, Sheng; Wang, Weibiao (2020), “New criteria and constructions of Brunnian links”, Journal of Knot Theory and Its Ramifications, 29 (13): 2043008, 27, arXiv:2006.10290, doi:10.1142/S0218216520430087, MR 4213076, S2CID 219792382
    23. ↑ Nanyes, Ollie (October 1993), “An elementary proof that the Borromean rings are non-splittable”, American Mathematical Monthly, 100 (8): 786–789, doi:10.2307/2324788, JSTOR 2324788
    24. ↑ Thistlethwaite, Morwen B. (1991), “On the algebraic part of an alternating link”, Pacific Journal of Mathematics, 151 (2): 317–333, doi:10.2140/pjm.1991.151.317, MR 1132393
    25. ↑ Freedman, Michael H.; Skora, Richard (1987), “Strange actions of groups on spheres”, Journal of Differential Geometry, 25: 75–98, doi:10.4310/jdg/1214440725; see in particular Lemma 3.2, p. 89
    26. ↑ Lindström, Bernt; Zetterström, Hans-Olov (1991), “Borromean circles are impossible”, American Mathematical Monthly, 98 (4): 340–341, doi:10.2307/2323803, JSTOR 2323803. Note however that Gunn & Sullivan (2008) write that this reference “seems to incorrectly deal only with the case that the three-dimensional configuration has a projection homeomorphic to” the conventional three-circle drawing of the link.
    27. ↑ Tverberg, Helge (2010), “On Borromean rings” (PDF), The Mathematical Scientist, 35 (1): 57–60, MR 2668444, archived from the original (PDF) on 2021-03-16, retrieved 2021-03-16
    28. 1 2 Howards, Hugh Nelson (2013), “Forming the Borromean rings out of arbitrary polygonal unknots”, Journal of Knot Theory and Its Ramifications, 22 (14): 1350083, 15, arXiv:1406.3370, doi:10.1142/S0218216513500831, MR 3190121, S2CID 119674622
    29. 1 2 Burgiel, H.; Franzblau, D. S.; Gutschera, K. R. (1996), “The mystery of the linked triangles”, Mathematics Magazine, 69 (2): 94–102, doi:10.1080/0025570x.1996.11996399, JSTOR 2690662, MR 1394792
    30. 1 2 Cantarella, Jason; Fu, Joseph H. G.; Kusner, Rob; Sullivan, John M.; Wrinkle, Nancy C. (2006), “Criticality for the Gehring link problem” (PDF), Geometry & Topology, 10 (4): 2055–2116, arXiv:math/0402212, doi:10.2140/gt.2006.10.2055, MR 2284052
    31. ↑ Cantarella, Jason; Kusner, Robert B.; Sullivan, John M. (2002), “On the minimum ropelength of knots and links” (PDF), Inventiones Mathematicae, 150 (2): 257–286, arXiv:math/0103224, Bibcode:2002InMat.150..257C, doi:10.1007/s00222-002-0234-y, MR 1933586, S2CID 730891
    32. ↑ Uberti, R.; Janse van Rensburg, E. J.; Orlandini, E.; Tesi, M. C.; Whittington, S. G. (1998), “Minimal links in the cubic lattice”, in Whittington, Stuart G.; Sumners, Witt De; Lodge, Timothy (eds.), Topology and Geometry in Polymer Science, IMA Volumes in Mathematics and its Applications, vol. 103, New York: Springer, pp. 89–100, doi:10.1007/978-1-4612-1712-1_9, MR 1655039; see Table 2, p. 97
    33. ↑ Riley, Robert (1979), “An elliptical path from parabolic representations to hyperbolic structures”, in Fenn, Roger (ed.), Topology of Low-Dimensional Manifolds: Proceedings of the Second Sussex Conference, 1977, Lecture Notes in Mathematics, vol. 722, Springer, pp. 99–133, doi:10.1007/BFb0063194, ISBN 978-3-540-09506-4, MR 0547459
    34. 1 2 Ratcliffe, John G. (2006), “The Borromean rings complement”, Foundations of Hyperbolic Manifolds, Graduate Texts in Mathematics, vol. 149 (2nd ed.), Springer, pp. 459–461, ISBN 978-0-387-33197-3, MR 2249478
    35. ↑ Abbott, Steve (July 1997), “Review of Not Knot and Supplement to Not Knot“, The Mathematical Gazette, 81 (491): 340–342, doi:10.2307/3619248, JSTOR 3619248, S2CID 64589738
    36. 1 2 William Thurston (March 2002), “7. Computation of volume”, The Geometry and Topology of Three-Manifolds, p. 165, archived from the original (PDF) on 2020-07-27, retrieved 2012-01-17
    37. ↑ Hilden, Hugh M.; Lozano, María Teresa; Montesinos, José María (1983), “The Whitehead link, the Borromean rings and the knot 946 are universal”, Seminario Matemático de Barcelona, 34 (1): 19–28, MR 0747855
    38. ↑ Vogel, Denis (2005), Masseyprodukte in der Galoiskohomologie von Zahlkörpern [Massey products in the Galois cohomology of number fields], Mathematisches Institut, Georg-August-Universität Göttingen: Seminars Winter Term 2004/2005, Göttingen: Universitätsdrucke Göttingen, pp. 93–98, doi:10.11588/heidok.00004418, MR 2206880
    39. ↑ Morishita, Masanori (2010), “Analogies between knots and primes, 3-manifolds and number rings”, Sugaku Expositions, 23 (1): 1–30, arXiv:0904.3399, MR 2605747
    40. 1 2 Chichak, Kelly S.; Cantrill, Stuart J.; Pease, Anthony R.; Chiu, Sheng-Hsien; Cave, Gareth W. V.; Atwood, Jerry L.; Stoddart, J. Fraser (May 28, 2004), “Molecular Borromean rings” (PDF), Science, 304 (5675): 1308–1312, Bibcode:2004Sci…304.1308C, doi:10.1126/science.1096914, PMID 15166376, S2CID 45191675
    41. ↑ Ashley, Clifford Warren (1993) [1944], The Ashley Book of Knots, Doubleday, p. 354, ISBN 978-0-385-04025-9
    42. ↑ Freeman, Jim (2015), “Gathering clues from Margot’s extraordinary objects”, Tewkesbury Historical Society Bulletin, 24
    43. ↑ “African Borromean Rings”, Mathematics and Knots, Centre for the Popularisation of Maths, University of Wales, 2002, retrieved 2021-02-12
    44. ↑ Mao, C.; Sun, W.; Seeman, N. C. (1997), “Assembly of Borromean rings from DNA”, Nature, 386 (6621): 137–138, Bibcode:1997Natur.386..137M, doi:10.1038/386137b0, PMID 9062186, S2CID 4321733
    45. ↑ Natarajan, Ganapati; Mathew, Ammu; Negishi, Yuichi; Whetten, Robert L.; Pradeep, Thalappil (2015-12-02), “A unified framework for understanding the structure and modifications of atomically precise monolayer protected gold clusters”, The Journal of Physical Chemistry C, 119 (49): 27768–27785, doi:10.1021/acs.jpcc.5b08193, ISSN 1932-7447
    46. ↑ Kumar, Vijith; Pilati, Tullio; Terraneo, Giancarlo; Meyer, Franck; Metrangolo, Pierangelo; Resnati, Giuseppe (2017), “Halogen bonded Borromean networks by design: topology invariance and metric tuning in a library of multi-component systems”, Chemical Science, 8 (3): 1801–1810, doi:10.1039/C6SC04478F, PMC 5477818, PMID 28694953
    47. ↑ Veliks, Janis; Seifert, Helen M.; Frantz, Derik K.; Klosterman, Jeremy K.; Tseng, Jui-Chang; Linden, Anthony; Siegel, Jay S. (2016), “Towards the molecular Borromean link with three unequal rings: double-threaded ruthenium(ii) ring-in-ring complexes”, Organic Chemistry Frontiers, 3 (6): 667–672, doi:10.1039/c6qo00025h
    48. ↑ Kraemer, T.; Mark, M.; Waldburger, P.; Danzl, J. G.; Chin, C.; Engeser, B.; Lange, A. D.; Pilch, K.; Jaakkola, A.; Nägerl, H.-C.; Grimm, R. (2006), “Evidence for Efimov quantum states in an ultracold gas of caesium atoms”, Nature, 440 (7082): 315–318, arXiv:cond-mat/0512394, Bibcode:2006Natur.440..315K, doi:10.1038/nature04626, PMID 16541068, S2CID 4379828
    49. ↑ Moskowitz, Clara (December 16, 2009), “Strange physical theory proved after nearly 40 years”, Live Science
    50. ↑ Tanaka, K. (2010), “Observation of a Large Reaction Cross Section in the Drip-Line Nucleus 22C”, Physical Review Letters, 104 (6) 062701, Bibcode:2010PhRvL.104f2701T, doi:10.1103/PhysRevLett.104.062701, PMID 20366816, S2CID 7951719

    External links


    This article is adapted from “Borromean rings” on Wikipedia, written by its contributors, and used under CC BY-SA 4.0. Text on this page is available under the same licence. Source snapshot: Wikipedia via Kiwix, 2026-07-20.

  • Boom hitch

    Boom hitch
    Boom hitch
    Category Hitch
    Efficiency high
    Origin Described by Ashley in 1944
    Typical use robust attachment to a foundation
    ABoK #1687

    The boom hitch is a type of knot. It is a rather robust and secure method of attaching a line, or rope to a fixed object like a pipe, post, or sail boom.[1]

    It can be finished with a slip, that is, a bight tucked under rather than the whole line pulled through in the last step. This will make it easier to untie.

    The slack must be worked out of all the loops around the foundation. One way is to vigorously wiggle and tug on both ends at the same time. The hitch should not spread out along the foundation, keep the passes around the foundation bunched together snugly. Once tight, this hitch is resistant to sliding along the foundation even if the surface is smooth such as on a steel pipe.

    See also

    References

    1. ↑ Geoffrey Budworth, The Ultimate Encyclopedia of Knots & Ropework (Anness Publishing Ltd., 1999, 2007), 96.

    This article is adapted from “Boom hitch” on Wikipedia, written by its contributors, and used under CC BY-SA 4.0. Text on this page is available under the same licence. Source snapshot: Wikipedia via Kiwix, 2026-07-20.

  • Quantum invariant

    In the mathematical field of knot theory, a quantum knot invariant or quantum invariant of a knot or link is a linear sum of colored Jones polynomial of surgery presentations of the knot complement.[1][2][3]

    List of invariants

    • Finite type invariant
    • Kontsevich invariant
    • Kashaev’s invariant
    • Witten–Reshetikhin–Turaev invariant (Chern–Simons)
    • Invariant differential operator[4]
    • Rozansky–Witten invariant
    • Vassiliev knot invariant
    • Dehn invariant
    • LMO invariant[5]
    • Turaev–Viro invariant
    • Dijkgraaf–Witten invariant[6]
    • Reshetikhin–Turaev invariant
    • Tau-invariant
    • I-Invariant
    • Klein J-invariant
    • Quantum isotopy invariant[7]
    • Ermakov–Lewis invariant
    • Hermitian invariant
    • Goussarov–Habiro theory of finite-type invariant
    • Linear quantum invariant (orthogonal function invariant)
    • Murakami–Ohtsuki TQFT
    • Generalized Casson invariant
    • Casson-Walker invariant
    • Khovanov–Rozansky invariant
    • HOMFLY polynomial
    • K-theory invariants
    • Atiyah–Patodi–Singer eta invariant
    • Link invariant[1]
    • Casson invariant
    • Seiberg–Witten invariants
    • Gromov–Witten invariant
    • Arf invariant
    • Hopf invariant

    See also

    References

    1. 1 2 Reshetikhin, N.; Turaev, V. G. (1991). “Invariants of 3-manifolds via link polynomials and quantum groups”. Inventiones Mathematicae. 103 (3): 547–597. doi:10.1007/BF01239527. MR 1091619.
    2. ↑
      Kontsevich, Maxim (1993). “Vassiliev’s knot invariants”. Adv. Soviet Math. 16: 137.
    3. ↑
      Watanabe, Tadayuki (2007). “Knotted trivalent graphs and construction of the LMO invariant from triangulations”. Osaka J. Math. 44 (2): 351. Retrieved 4 December 2012.
    4. ↑ Letzter, Gail (2004). “Invariant differential operators for quantum symmetric spaces, II”. arXiv:math/0406194.
    5. ↑ Sawon, Justin (2000). “Topological quantum field theory and hyperkähler geometry”. arXiv:math/0009222.
    6. ↑ Petit, Jerome (1999). “The invariant of Turaev-Viro from Group category” (PDF). hal.archives-ouvertes.fr. Retrieved 2019-11-04.
    7. ↑ Lawton, Sean (June 28, 2007). “Generators of SL ⁡ ( 2 , C ) {\displaystyle \operatorname {SL} (2,\mathbb {C} )} {\displaystyle \operatorname {SL} (2,\mathbb {C} )}-Character Varieties of Arbitrary Rank Free Groups” (PDF). The 7th KAIST Geometric Topology Fair. Archived from the original (PDF) on 20 July 2007. Retrieved 13 January 2022.

    Further reading

    • Freedman, Michael H. (1990). Topology of 4-manifolds. Princeton, N.J: Princeton University Press. ISBN 978-0691085777. OL 2220094M.
    • Ohtsuki, Tomotada (December 2001). Quantum Invariants. World Scientific Publishing Company. ISBN 9789810246754. OL 9195378M.

    External links

    This article is adapted from “Quantum invariant” on Wikipedia, written by its contributors, and used under CC BY-SA 4.0. Text on this page is available under the same licence. Source snapshot: Wikipedia via Kiwix, 2026-07-20.

  • Boa knot

    Boa knot
    Boa knot
    Category Binding
    Origin Peter Collingwood in 1996
    Related strangle knot, double constrictor knot
    Typical use Best used for securing objects in cylindrical loads

    The boa knot[1][2][3][4] is a modern binding knot invented by weaver Peter Collingwood in 1996. His intention was to develop a knot that would hold well when the constricted object was cut close to the winds of the knot.

    The boa knot is related to the strangle knot and the double constrictor knot. It combines both the structure and qualities of these other two knots. The boa knot can be very difficult to untie and is inappropriate when frequent or fast untying is needed. The knotted part needs to lie over a convex surface to hold.

    The boa knot is best used for securing objects in cylindrical loads. Said knot is hard to move around.

    Tying

    • Start with making a loop counter-clockwise
      Start with making a loop counter-clockwise
    • Place another loop in the same direction over the first loop
      Place another loop in the same direction over the first loop
    • Twist these loops, by turning the right side in clockwise direction, so you get a figure-eight
      Twist these loops, by turning the right side in clockwise direction, so you get a figure-eight
    • Take a cylindrical object and place it from under above in the first loops
      Take a cylindrical object and place it from under above in the first loops
    • Then put the object through the other loops
      Then put the object through the other loops
    • Pull the loose ends away from it, carefully shaping the knot. Make sure that the strands lined up as at the start. So they shouldn't slip out.
      Pull the loose ends away from it, carefully shaping the knot. Make sure that the strands lined up as at the start. So they shouldn’t slip out.
    • This should tighten the loops to the point where they cling firmly to the desired object, yielding the boa knot. (Frontside)
      This should tighten the loops to the point where they cling firmly to the desired object, yielding the boa knot. (Frontside)
    • Boa-knot. (Backside)
      Boa-knot. (Backside)

    Alternative

    Boa knot
    Strangle knot
    Boa knot
    Double constrictor knot

    See also

    References

    1. ↑ Handbook of Knots by Des Pawson — ISBN 1-4053-0467-7
    2. ↑ Knots — Andrew Adamides — ISBN 978-0-572-03329-3
    3. ↑ The Complete Guide to Knots and Knot Tying — Geoffrey Budworth — p.164 — ISBN 0-7548-0422-4
    4. ↑ Knotting Matters — issue 55 — p19

    External links

    This article is adapted from “Boa knot” on Wikipedia, written by its contributors, and used under CC BY-SA 4.0. Text on this page is available under the same licence. Source snapshot: Wikipedia via Kiwix, 2026-07-20.

  • Quadrisecant

    Quadrisecant
    Three quadrisecants of a trefoil knot[1]

    In geometry, a quadrisecant or quadrisecant line of a space curve is a line that passes through four points of the curve. This is the largest possible number of intersections that a generic space curve can have with a line, and for such curves the quadrisecants form a discrete set of lines. Quadrisecants have been studied for curves of several types:

    • Knots and links in knot theory, when nontrivial, always have quadrisecants, and the existence and number of quadrisecants has been studied in connection with knot invariants including the minimum total curvature and the ropelength of a knot.
    • The number of quadrisecants of a non-singular algebraic curve in complex projective space can be computed by a formula derived by Arthur Cayley.
    • Quadrisecants of arrangements of skew lines touch subsets of four lines from the arrangement. They are associated with ruled surfaces and the Schläfli double six configuration.

    Definition and motivation

    A quadrisecant is a line that intersects a curve, surface, or other set in four distinct points. It is analogous to a secant line, a line that intersects a curve or surface in two points; and a trisecant, a line that intersects a curve or surface in three points.[2]

    Compared to secants and trisecants, quadrisecants are especially relevant for space curves, because they have the largest possible number of intersection points of a line with a generic curve. In the plane, a generic curve can be crossed arbitrarily many times by a line; for instance, small generic perturbations of the sine curve are crossed infinitely often by the horizontal axis. In contrast, if an arbitrary space curve is perturbed by a small distance to make it generic, there will be no lines through five or more points of the perturbed curve. Nevertheless, any quadrisecants of the original space curve will remain present nearby in its perturbation.[3] For generic space curves, the quadrisecants form a discrete set of lines. In contrast, when trisecants occur, they form continuous families of lines.[4]

    One explanation for this phenomenon is visual: looking at a space curve from far away, the space of such points of view can be described as a two-dimensional sphere, one point corresponding to each direction. Pairs of strands of the curve may appear to cross from all of these points of view, or from a two-dimensional subset of them. Three strands will form a triple crossing when the point of view lies on a trisecant, and four strands will form a quadruple crossing from a point of view on a quadrisecant. Each constraint that the crossing of a pair of strands lies on another strand reduces the number of degrees of freedom by one (for a generic curve), so the points of view on trisecants form a one-dimensional (continuously infinite) subset of the sphere, while the points of view on quadrisecants form a zero-dimensional (discrete) subset. C. T. C. Wall writes that the fact that generic space curves are crossed at most four times by lines is “one of the simplest theorems of the kind”, a model case for analogous theorems on higher-dimensional transversals.[3]

    Depending on the properties of the curve, it may have no quadrisecants, finitely many, or infinitely many. These considerations make it of interest to determine conditions for the existence of quadrisecants, or to find bounds on their number in various special cases, such as knotted curves,[5][6] algebraic curves,[7] or arrangements of lines.[8]

    For special classes of curves

    Knots and links

    In three-dimensional Euclidean space, every nontrivial tame knot or link has a quadrisecant. Originally established in the case of knotted polygons and smooth knots by Erika Pannwitz,[5]
    this result was extended to knots in suitably general position and links with nonzero linking number,[6]
    and later to all nontrivial tame knots and links.[9]

    Pannwitz proved more strongly that, for a locally flat disk having the knot as its boundary, the number of singularities of the disk can be used to construct a lower bound on the number of distinct quadrisecants. The existence of at least one quadrisecant follows from the fact that any such disk must have at least one singularity.[5][10] Morton & Mond (1982) conjectured that the number of distinct quadrisecants of a given knot is always at least n ( n − 1 ) / 2 {\displaystyle n(n-1)/2} {\displaystyle n(n-1)/2}, where n {\displaystyle n} {\displaystyle n} is the crossing number of the knot.[6][10] Counterexamples to this conjecture have since been discovered.[10]

    Two-component links have quadrisecants in which the points on the quadrisecant appear in alternating order between the two components,[6] and nontrivial knots have quadrisecants in which the four points, ordered cyclically as a b c d {\displaystyle abcd} {\displaystyle abcd} on the knot, appear in order a c b d {\displaystyle acbd} {\displaystyle acbd} along the quadrisecant.[11] The existence of these alternating quadrisecants can be used to derive the Fáry–Milnor theorem, a lower bound on the total curvature of a nontrivial knot.[11] Quadrisecants have also been used to find lower bounds on the ropelength of knots.[12]

    G. T. Jin and H. S. Kim conjectured that, when a knotted curve K {\displaystyle K} {\displaystyle K} has finitely many quadrisecants, K {\displaystyle K} {\displaystyle K} can be approximated with an equivalent polygonal knot with its vertices at the points where the quadrisecants intersect K {\displaystyle K} {\displaystyle K}, in the same order as they appear on K {\displaystyle K} {\displaystyle K}. However, their conjecture is false: in fact, for every knot type, there is a realization for which this construction leads to a self-intersecting polygon, and another realization where this construction produces a knot of a different type.[13]

    Unsolved problem in mathematics
    Does every wild knot have infinitely many quadrisecants?
    More unsolved problems in mathematics

    It has been conjectured that every wild knot has an infinite number of quadrisecants.[9]

    Algebraic curves

    Arthur Cayley derived a formula for the number of quadrisecants of an algebraic curve in three-dimensional complex projective space, as a function of its degree and genus.[7] For a curve of degree d {\displaystyle d} {\displaystyle d} and genus g {\displaystyle g} {\displaystyle g}, the number of quadrisecants is[14]
    ( d − 2 ) ( d − 3 ) 2 ( d − 4 ) 12 − g ( d 2 − 7 d + 13 − g ) 2 . {\displaystyle {\frac {(d-2)(d-3)^{2}(d-4)}{12}}-{\frac {g(d^{2}-7d+13-g)}{2}}.} {\displaystyle {\frac {(d-2)(d-3)^{2}(d-4)}{12}}-{\frac {g(d^{2}-7d+13-g)}{2}}.}
    This formula assumes that the given curve is non-singular; adjustments may be necessary if it has singular points.[15][16]

    Skew lines

    Quadrisecant
    The Schläfli double six

    In three-dimensional Euclidean space, every set of four skew lines in general position has either two quadrisecants (also in this context called transversals) or none. Any three of the four lines determine a hyperboloid, a doubly ruled surface in which one of the two sets of ruled lines contains the three given lines, and the other ruling consists of trisecants to the given lines. If the fourth of the given lines pierces this surface, it has two points of intersection, because the hyperboloid is defined by a quadratic equation. The two trisecants of the ruled surface, through these two points, form two quadrisecants of the given four lines. On the other hand, if the fourth line is disjoint from the hyperboloid, then there are no quadrisecants.[17] In spaces with complex number coordinates rather than real coordinates, four skew lines always have exactly two quadrisecants.[8]

    The quadrisecants of sets of lines play an important role in the construction of the Schläfli double six, a configuration of twelve lines intersecting each other in 30 crossings. If five lines a i {\displaystyle a_{i}} {\displaystyle a_{i}} (for i = 1 , 2 , 3 , 4 , 5 {\displaystyle i=1,2,3,4,5} {\displaystyle i=1,2,3,4,5}) are given in three-dimensional space, such that all five are intersected by a common line b 6 {\displaystyle b_{6}} {\displaystyle b_{6}} but are otherwise in general position, then each of the five quadruples of the lines a i {\displaystyle a_{i}} {\displaystyle a_{i}} has a second quadrisecant b i {\displaystyle b_{i}} {\displaystyle b_{i}}, and the five lines b i {\displaystyle b_{i}} {\displaystyle b_{i}} formed in this way are all intersected by a common line a 6 {\displaystyle a_{6}} {\displaystyle a_{6}}. These twelve lines and the 30 intersection points a i b j {\displaystyle a_{i}b_{j}} {\displaystyle a_{i}b_{j}} form the double six.[18][19]

    An arrangement of n {\displaystyle n} {\displaystyle n} complex lines with a given number of pairwise intersections and otherwise skew may be interpreted as an algebraic curve with degree n {\displaystyle n} {\displaystyle n} and with genus determined from its number of intersections, and Cayley’s aforementioned formula used to count its quadrisecants. The same result as this formula can also be obtained by classifying the quadruples of lines by their intersections, counting the number of quadrisecants for each type of quadruple, and summing over all quadruples of lines in the given set.[8]

    References

    1. ↑ Jin, Gyo Taek (December 2017), “Polygonal approximation of unknots by quadrisecants”, in Reiter, Philipp; Blatt, Simon; Schikorra, Armin (eds.), New Directions in Geometric and Applied Knot Theory, De Gruyter Open, pp. 159–175, doi:10.1515/9783110571493-008, ISBN 978-3-11-057149-3
    2. ↑ Eisenbud, David; Harris, Joe (2016), 3264 and All That: A second course in algebraic geometry, Cambridge, UK: Cambridge University Press, p. 377, doi:10.1017/CBO9781139062046, ISBN 978-1-107-60272-4, MR 3617981
    3. 1 2 Wall, C. T. C. (1977), “Geometric properties of generic differentiable manifolds”, in Palis, Jacob; do Carmo, Manfredo (eds.), Geometry and Topology: Proceedings of the Latin American School of Mathematics (ELAM III) held at the Instituto de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, July 1976, Lecture Notes in Mathematics, vol. 597, pp. 707–774, doi:10.1007/BFb0085382, ISBN 978-3-540-08345-0, MR 0494233
    4. ↑ Denne, Elizabeth (2018), “Quadrisecants and essential secants of knots”, in Blatt, Simon; Reiter, Philipp; Schikorra, Armin (eds.), New directions in geometric and applied knot theory, Partial Differential Equations and Measure Theory, De Gruyter, Berlin, pp. 138–158, doi:10.1515/9783110571493-006, MR 3915943, S2CID 128222971
    5. 1 2 3 Pannwitz, Erika (1933), “Eine elementargeometrische Eigenschaft von Verschlingungen und Knoten”, Mathematische Annalen, 108 (1): 629–672, doi:10.1007/BF01452857, S2CID 123026724
    6. 1 2 3 4 Morton, Hugh R.; Mond, David M. Q. (1982), “Closed curves with no quadrisecants”, Topology, 21 (3): 235–243, doi:10.1016/0040-9383(82)90007-6, MR 0649756
    7. 1 2 Cayley, Arthur (1863), Philosophical Transactions of the Royal Society of London, vol. 153, The Royal Society, pp. 453–483, JSTOR 108806
    8. 1 2 3 Wong, B. C. (1934), “Enumerative properties of r {\displaystyle r} {\displaystyle r}-space curves”, Bulletin of the American Mathematical Society, 40 (4): 291–296, doi:10.1090/S0002-9904-1934-05854-3, MR 1562839
    9. 1 2 Kuperberg, Greg (1994), “Quadrisecants of knots and links”, Journal of Knot Theory and Its Ramifications, 3: 41–50, arXiv:math/9712205, doi:10.1142/S021821659400006X, MR 1265452, S2CID 6103528
    10. 1 2 3 Jin, Gyo Taek (2005), “Quadrisecants of knots with small crossing number”, Physical and numerical models in knot theory (PDF), Ser. Knots Everything, vol. 36, Singapore: World Scientific Publishing, pp. 507–523, doi:10.1142/9789812703460_0025, ISBN 978-981-256-187-9, MR 2197955
    11. 1 2 Denne, Elizabeth Jane (2004), Alternating quadrisecants of knots, Ph.D. thesis, University of Illinois at Urbana-Champaign, arXiv:math/0510561, Bibcode:2005math…..10561D
    12. ↑ Denne, Elizabeth; Diao, Yuanan; Sullivan, John M. (2006), “Quadrisecants give new lower bounds for the ropelength of a knot”, Geometry & Topology, 10: 1–26, arXiv:math/0408026, doi:10.2140/gt.2006.10.1, MR 2207788
    13. ↑ Bai, Sheng; Wang, Chao; Wang, Jiajun (2018), “Counterexamples to the quadrisecant approximation conjecture”, Journal of Knot Theory and Its Ramifications, 27 (2), 1850022, arXiv:1605.00538, doi:10.1142/S0218216518500220, MR 3770471, S2CID 119601013
    14. ↑ Griffiths, Phillip; Harris, Joseph (2011), Principles of Algebraic Geometry, Wiley Classics Library, vol. 52, John Wiley & Sons, p. 296, ISBN 9781118030776
    15. ↑ Welchman, W. G. (April 1932), “Note on the trisecants and quadrisecants of a space curve”, Mathematical Proceedings of the Cambridge Philosophical Society, 28 (2): 206–208, Bibcode:1932PCPS…28..206W, doi:10.1017/s0305004100010872, S2CID 120725025
    16. ↑ Maxwell, Edwin A. (July 1935), “Note on the formula for the number of quadrisecants of a curve in space of three dimensions”, Mathematical Proceedings of the Cambridge Philosophical Society, 31 (3): 324–326, Bibcode:1935PCPS…31..324M, doi:10.1017/s0305004100013086, S2CID 122279811
    17. ↑ Hilbert, David; Cohn-Vossen, Stephan (1952), Geometry and the Imagination, New York: Chelsea, p. 164; reprinted 1990, ISBN 978-0-8284-1087-8
    18. ↑ Schläfli, Ludwig (1858), Cayley, Arthur (ed.), “An attempt to determine the twenty-seven lines upon a surface of the third order, and to derive such surfaces in species, in reference to the reality of the lines upon the surface”, Quarterly Journal of Pure and Applied Mathematics, 2: 55–65, 110–120
    19. ↑ Coxeter, H. S. M. (2006), “An absolute property of four mutually tangent circles”, Non-Euclidean geometries, Math. Appl. (N. Y.), vol. 581, New York: Springer, pp. 109–114, doi:10.1007/0-387-29555-0_5, ISBN 978-0-387-29554-1, MR 2191243; Coxeter repeats Schläfli’s construction, and provides several references to simplified proofs of its correctness

    This article is adapted from “Quadrisecant” on Wikipedia, written by its contributors, and used under CC BY-SA 4.0. Text on this page is available under the same licence. Source snapshot: Wikipedia via Kiwix, 2026-07-20.

  • Blood knot

    Blood knot
    Blood knot
    Names Blood knot, Barrel knot
    Category Bend
    Efficiency 80%
    Origin Unknown
    Related Improved clinch knot
    Releasing Jamming
    Typical use Tying fishing line, joining monofilament nylon line
    ABoK #295, #345, #1413

    A blood knot (barrel knot) is a bend knot most usefully employed for joining sections of monofilament nylon line while maintaining a high portion of the line’s inherent strength. Other knots used for this purpose can cause a substantial loss of strength. In fly fishing, this serves to build a leader of gradually decreasing diameter with the castable fly line attached at the large diameter end and the fly or hook at the small diameter end. The principal drawback to the blood knot is the dexterity required to tie it. It is also likely to jam, which is not a concern in fishing line, which is no great loss to cut, but may pose a concern in normal rope. The term “blood knot” can also refer to “a double overhand knot tied in a cat-o’-nine-tails”.[1]: 82 

    The barrel knot, called blood knot by Keith Rollo, is the best bend there is for small, stiff or slippery line. The ends may be trimmed short and the knot offers the least resistance possible when drawn through water.

    — The Ashley Book of Knots[1]: 259 

    A half blood knot (also clinch knot) is a knot that is used for securing a fishing line to a fishing lure, snap or swivel. When two half blood knots are used to join two lines, they are considered as one knot and called a blood knot. A half blood knot is one of the strongest knots for tying a medium-size hook to a medium-size line such as hooksize 4 to 4/0 onto line size 6–30 lb (2.7–13.6 kg).[2]

    The half blood knot has found use in surgery for securing slippery monofilament sutures. [3]

    Tying the knot

    In tying the blood knot, the two lines to be joined are overlapped for 6–8 cm (2.4–3.1 in) with the short ends of the two lines in opposite directions. The short end of one line is then wrapped 4–6 times around the second line and the remaining portion of the first short end brought back and passed between the lines at the beginning of the wraps.
    The short end of the second line is then wrapped 4–6 times around the first line and the end of this line brought back and passed through what is now an oval space between the first wrap of each set.[4][5]

    Blood knot
    Blood knot step by step

    The above method has been called by Stanley Barnes “outcoil”, and is contrasted with the method that resembles the finished knot from the start, “incoil”.[6] In fishing line, and in other material if not deliberately set snug and maybe re-set after some initial tensioning, the outcoil form will transform into the incoil form.

    The knot is tightened by moistening it and pulling on the long ends of the line. This causes the wraps to tighten and compress, creating two short sections of “barrel”, which look much like a hangman’s knot, that slide together. The short ends of the line are then trimmed close to the wraps, or one of the ends may be left intact to be used for a second fly or lure, called a “dropper”.

    See also

    References

    1. 1 2 Ashley, Clifford W. (June 21, 1944). The Ashley Book of Knots. Garden City, New York: Doubleday & Company, Inc. OCLC 634638. OL 7436849M.
    2. ↑ Wilson, Geoff (2003). Encyclopedia of Fishing Knots & Rigs. Croydon, Victoria: Australian Fishing Network. p. 9. ISBN 1865130400. OCLC 55590004. OL 8648495M.
    3. ↑ https://www.youtube.com/watch?v=I0_d2fVDfHc
    4. ↑ Kreh, Lefty (August 10, 2007). Fishing Knots: Proven to Work for Light Tackle and Fly Fishing. Mechanicsburg, Pennsylvania: Stackpole Books. ISBN 978-0-8117-3407-3. OCLC 83609812. OL 17850673M.
    5. ↑ “Blood Knot”. www.animatedknots.com. Retrieved 2025-04-29.
    6. ↑ Barnes, Stanley (1951). Anglers’ Knots in Gut & Nylon (2nd ed.). Birmingham, England: Cornish Brothers Ltd. p. 112. OCLC 1405159. OL 26913460M. We may for convenience call Chaytor’s method of coiling from the centre outwards the outcoil method, as opposed to the incoil method where the first coil is made at the outermost limit of the knot, and the last is near the point at the centre.

    External links


    This article is adapted from “Blood knot” on Wikipedia, written by its contributors, and used under CC BY-SA 4.0. Text on this page is available under the same licence. Source snapshot: Wikipedia via Kiwix, 2026-07-20.

  • Prusik

    Prusik hitch
    Prusik
    Names Prusik hitch, Prusik knot
    Category Hitch
    Origin Karl Prusik, 1931
    Related Bachmann knot, Blake’s hitch, Klemheist knot, Cow hitch
    Releasing Non-jamming
    Typical use Climbing
    ABoK #1763

    A Prusik (/ˈprʌsɪk/ PRUSS-ik) is a friction hitch or knot used to attach a loop of cord around a rope, applied in climbing, canyoneering, mountaineering, caving, rope rescue, ziplining, and by arborists. The term Prusik is a name for both the loops of cord used to tie the hitch and the hitch itself, and the verb is “to prusik” or “prusiking” (i.e. using a Prusik to ascend).[1][2][3] More casually, the term is used for any friction hitch or device that can grab a rope (see autoblock). Due to the pronunciation, the word is often misspelled Prussik, Prussick, or Prussic.

    The Prusik hitch is named after its putative inventor, the Austrian mountaineer Karl Prusik. It was shown in a 1931 Austrian mountaineering manual for rope ascending. It was used on several mountaineering routes of the era to ascend the final summit, where a rope could be thrown over the top and anchored so that climbers could attain the summit by prusiking up the other side of the rope.

    A Prusik made from cord does little or no damage to the rope it is attached to, whereas some mechanical ascenders (not Prusiks) can cause damage from normal use, and especially if the device slips during climbing or is heavily loaded or shock-loaded.

    Advantages

    Prusik
    Climber using loops attached to prusik knots to move up a fixed rope

    Climbers carry Prusik cords mainly for emergency use, as they are lighter than other options. Prusiks are fast to place on a rope, and with practice can be placed with one hand. The loops of cord can be used as slings, and are thus multi-functional in a climbing environment.

    Prusiks will work around two ropes, even two ropes of different diameters. Prusiks provide a strong attachment that will not damage or break the rope, and so are used in some rope-rescue techniques. Prusiks are good to use in hauling systems where multiple rope-grabs may be needed, and where mechanical rope-grabs are not available.

    Prusiks are far less likely to damage the main rope than mechanical rope-grabs such as a jumar. An overloaded Prusik will initially slip, causing no damage. If loaded to great excess, the worst result is that it slides until the heat of friction causes physical failure of the Prusik cord, rather than the rope. Mechanical rope-grabs when overloaded will sometimes damage the sheath of the rope, or in extreme cases sever the rope entirely.

    Depending on which variant is used, Prusik hitches have the advantage of working in both directions. Most mechanical rope-grabs work like a ratchet, moving freely up the rope, but grabbing when a load is placed down on them. Traditional Prusiks (such as those shown below) will grab when pulled by the tail, either up or down, and will slide either way when pushed by the barrel.

    Although the Prusik Climb technique may be seen as outdated by some, the US Army still includes it in its annual Best Ranger competition. Rangers in the competition routinely make it up a 65-foot (19.8 metre) rope in under a minute.

    Disadvantages

    Prusiks are ineffective upon frozen wet ropes. This is due to the necessity of friction for the Prusik to function. Mechanical devices (such as jumars) to grab the rope are available that are easier and faster to use, but heavier, more expensive, and bulkier.

    After being put under a great deal of weight, the Prusik can be quite constricted and difficult to untie. This varies, depending on the relative diameter of the ropes.

    Related hitches and equipment

    Although Prusik can be used in a general way, the Prusik hitch is a specific hitch. The two main alternatives are the Bachmann knot and the Klemheist knot (see also the Tarbuck knot). Each has its advantages and disadvantages, mainly in how easy they are to use for climbing a rope. Another variation is the autoblock or French Prusik, used by some people as a backup knot while rappelling.

    A Purcell Prusik is a related cord popular among cavers and rope-rescue people. A somewhat longer loop than the normal Prusik is used around the rope, then a second Prusik is used around the cord loop itself to form a foot loop. The foot loop is then easily adjusted in length and position.

    A Prusik-Minding-Pulley is common in rope rescue. The rope to be pulled is passed through a pulley, and a Prusik is tied on the loaded side. When the rope is pulled, the Prusik rides against the pulley, and the rope slides through it; but when the rope is relaxed, the Prusik slides away from the pulley and grabs the rope. Thus, the combination acts as a ratchet (or Progress Capture Device (PCD)).

    The Farrimond friction hitch is a kind of taut-line hitch that is similar in form to a Prusik.

    Equipment

    A Prusik loop is made of narrow but strong nylon accessory cord tied into a loop using a double fisherman’s knot.[4] A sling or Prusik-dedicated sewn loop can also be used. A short piece of rope spliced to form a circle is called a becket.[5] Note that Dyneema/Spectra has a very low melting point and should not be used in Prusik hitches unless the cord or sling is specifically engineered for it (as seen in some sheathed constructions). The length of this loop depends on the application. For instance, the loop used for an Auto-Bloc might only be 20 cm, whereas the foot loop for climbing a rope might work better with a length of 100 cm or more. As a general rule, longer loops are preferable over shorter ones, as a loop can always be shortened by tying a knot in it.

    The effectiveness of the Prusik hitch relies on the surface area between the hitch and the main line, and the diameter of the cord used. Normally, the greater the difference between the diameter of the cord used for the hitch and the main line, the greater the ability for the hitch to hold. However, the smaller the diameter of the cord used, the lower its safe working load. In addition, smaller diameter cords often jam too tight when placed under load, and are hard to handle when wearing gloves.

    Tying

    The Prusik is tied by wrapping the “tail” of the Prusik loop around the rope a number of times, usually 2-4 times depending on the materials, (each time, through the other (bow) end), forming a barrel around the rope with a tail hanging out from the middle. When the tail is weighted, the turns tighten and make a slight bend in the rope. When weight is removed, the loop can be moved along the rope by placing a hand directly on the barrel or pushing it from “behind”. Breaking the Prusik free from the rope after it has been weighted can be difficult, and is most easily done by pushing the bow, (the loop of cord which runs along the barrel, from the top wrap to the bottom wrap), along the tail a little. This unwinds the barrel wraps to loosen the grip of the hitch, and makes movement easier.
    Note: Step 2 is also called a girth hitch (a single wrap) and is not a good Prusik (yet).
    Note: Step 4 has a twist in the webbing that is inconsistent with the preceding images, but will not affect use except to make it slightly more difficult to loosen after a heavy loading. (Of course, this is not an issue when using normal round cord for the Prusik.)

    • Step 1.
      Step 1.
    • Step 2.
      Step 2.
    • Step 3.
      Step 3.
    • Step 4.
      Step 4.
    • Locked while holding tension.
      Locked while holding tension.
    • Slides readily without tension.
      Slides readily without tension.
    • A variation of the Prusik knot with additional friction.
      A variation of the Prusik knot with additional friction.

    Many materials may be used to tie a Prusik. The Prusik knot illustrated above is made with webbing, however webbing is not recommended for heavier loads and/or securing a person since it has been shown to slip.[6] Adding more wraps increases the grip.

    A double strap or sling for hoisting a spar at middle length. One bight is rove through the other and a tackle is hooked to the single bight.

    Applications

    In addition to being a useful rope-grab for rope-rescue applications, Prusiks are popular for:

    • Prusik
      A Prusik (left) and autoblock (right). Both are used as rappel backups.

      Rappel backup/self-belay below the device: A Prusik is placed below the descender and controlled with the brake hand. It acts as an automatic ‘dead man’s handle’ should the climber be incapacitated or need to use both hands. Careful setup of the rappel backup is critical. An ‘autoblock‘ or ‘French Prusik’ knot is most widely used in this application.[8]

    • Rappel backup/self-belay above the device: A Prusik is placed above the descender and controlled with the hand not being used as the brake hand. This configuration allows an easier and faster transition from rappelling to climbing the rope, but can also result in the Prusik locking tight as the amount of friction required to hold the load at that point is far higher than that experienced by a self-belay below the device.
    • Prusiking or ascending the line: Two Prusiks used in tandem can be used to climb a fixed rope. One Prusik is attached to the belay loop sewn onto the front of a harness, and the other attached below that is a longer length of cord reaching to one foot. The climber can then stand up in the foot loop, slide the Prusik hitch of the waist loop further up the rope and then “sit” down on it. Once in the sitting position, the climber can slide the foot loop up the rope and repeat the process.
    • Escaping the belay: In a lead-climbing situation, should the climber become incapacitated in a position where they cannot be safely lowered to the ground, the belayer must escape the belay in order to effect rescue. After locking the rope in the belay device with one hand, the belayer can tie a Prusik to the rope with the other hand, and then use the Prusik loop to transfer the load to a fixed anchor. The belayer can then go to effect rescue or get help.
    • Rescue applications: Rope rescue teams, such as in swiftwater rescue or in high-angle technical rescue, use a Prusik hitch as a ‘ratchet’ or progress capture device. A Prusik with a Prusik minding pulley is used to hold a load while tensioning a line. The pulley advances the Prusik up the line and prevents it from going back out. This can be used to raise a patient or tension a highline for a Tyrolean traverse, or in boat-on-tether and similar rescue operations.
    • Handcuffs
      Prusik
      Bosun’s or Prusik handcuffs

      A length of rope that has been tied so that the Prusik knot is tied around itself leaving two large loops can be quickly used as handcuffs by slipping the loops around the detainee’s hands and pulling the running ends tight and securing them with a square knot. When the detainee attempts to pull his hands apart, the Prusik tightens in the same way as when it is tied to another rope. To create Prusik handcuffs, tie a loose Prusik around one of your fingers and then slip it off, leaving the knot shape intact. Then slip the free ends of the rope through the “hole” in the knot where your finger used to be. Alternatively, use a handcuff knot, which is the more usual knot to accomplish this task.

    • Testing rigs for tensile strength and pull force: RepRap researchers have used the Prusik knot to secure a fiber in order to measure the pull force of an extruder mechanism and estimate the tensile strength of a fiber.[9]

    When to carry (climbing, kayaking)

    All sorts of climbers carry Prusiks as standard equipment “just in case”. Prusiks are unlikely to be needed on short climbs where the climber can be readily lowered to the ground; conversely, they may prove useful where the climber cannot be lowered, for instance from a high cliff or due to a hazard underneath the climber.

    Prusiks can be tied using other climbing equipment, such as slings already carried by the climber. Three loops allow the climber to pass a knot in the rope, a difficult task without a third loop.

    Kayaking:
    For kayaking, a Prusik can be used in a similar way as in climbing; for rescuing people and equipment from a river, a Prusik or two with a set of pulleys to create a Z-drag is preferred.

    Camping, hiking and bushcraft:
    The Prusik Knot can be used to attach a tarpaulin or shelter to a ridge line, as they can easily be slid along the line to the required position without damaging the line.[10]

    See also

    References

    1. ↑ Climbing School. Barrons Educational Series Inc. 1989. pp. 78–79. ISBN 0812059697.
    2. ↑ “The Prusik Knot or Triple Sliding Hitch”. Animated Knots. Archived from the original on 14 July 2022. Retrieved 6 March 2016.
    3. ↑ Learning to Rock Climb. Sierra Club Books. 1981. pp. 116–118. ISBN 0871562812.
    4. ↑ Gaines, Bob; Martin, Jason D. (2014-05-20). Rock Climbing: The AMGA Single Pitch Manual. Rowman & Littlefield. ISBN 9781493009626. Double Fisherman’s Knot (aka the Barrel or Grapevine) … is the preferred knot to use for joining nylon cord into a loop to make a cordelette or prusik loop.
    5. ↑ “Becket”. 4 July 2023.
    6. ↑ Steven M. Cox and Kris Fulsaal, Mountaineering: the Freedom of the Hills -7th ed. (Seattle: The Mountaineers Books, 2003), chapter 8.
    7. ↑ Ashley, Clifford W. (1944). The Ashley Book of Knots, p.300. Doubleday. ISBN 0-385-04025-3.
    8. ↑ “French prusik – OZultimate.com canyoning”. OZUltimate.com. Archived from the original on 17 October 2021.
    9. ↑ “Geared Nema17 Extruder – RepRapWiki”. RepRap.org. Archived from the original on 27 January 2010. Retrieved 10 February 2010.
    10. ↑ “11 Useful Knots for Tarp Shelters”. Tactical.com. 29 April 2021. Archived from the original on 17 September 2021.

    Further reading

    External links

    This article is adapted from “Prusik” on Wikipedia, written by its contributors, and used under CC BY-SA 4.0. Text on this page is available under the same licence. Source snapshot: Wikipedia via Kiwix, 2026-07-20.

  • Blake’s hitch

    Blake’s Hitch
    Blake's hitch
    Category Hitch
    Origin Heinz Prohaska
    Related Prusik knot, Sailor’s hitch, Bachmann knot, Klemheist knot
    Releasing Not Jamming
    Typical use Climbing
    ABoK NA

    The Blake’s hitch is a friction hitch commonly used by arborists and tree climbers as an ascending knot. Unlike other common climbing hitches, which often use a loop of cord, the Blake’s hitch is formed using the end of a rope. Although it is a stable knot, it is often backed up with a stopper knot, such as a figure-of-eight knot, for safety. It is used for both ascending and descending, and is preferred by many arborists over other hitches, such as the taut-line hitch, as it is less prone to binding.

    History

    The first known presentation of this knot was made by Heinz Prohaska in an Austrian guides periodical in 1981; in 1990, he presented it in a caving journal, Nylon Highway. Separately, Jason Blake discovered the knot for himself and presented it to the arborist community in a letter to Arbor Age in 1994, after which it was enthusiastically adopted by arborists. It has since become well known under the name “Blake’s hitch.”

    Usage

    This hitch has two conventional forms – the 4/2 and the 5/3 – although other variations are possible.

    The 4/2 version has four total turns, with the tail passing up through the bottom two.

    The 5/3 version has five total turns, with the tail passing up through the bottom three.

    The hitch is dressed and set tight enough to provide enough grip for the applied load without being tighter than necessary.

    This hitch is most commonly used with 12-13mm (1/2 inch) static climbing ropes.

    To prevent failure in slippery rope Heinz advises adding a round turn to the 4/2 knot, creating a 5/2.

    To prevent failure due to rope stiffness, both add a round turn and tuck the tail one full turn higher, resulting in the 5/3.

    Tying

    After passing the tail round the standing end, the tail then must pass back behind the standing line and up through the desired number of turns of the coil.

    In practice it helps to insert the thumb under the lower turns to facilitate threading the tail.

    A stopper knot is added to the tail after tying to prevent failure.

    Incorrect tying – by not passing the tail behind the standing line after looping – can lead to failure.

    Blake's hitch

    See also

    References

    External links

    This article is adapted from “Blake's hitch” on Wikipedia, written by its contributors, and used under CC BY-SA 4.0. Text on this page is available under the same licence. Source snapshot: Wikipedia via Kiwix, 2026-07-20.

  • Prime knot

    Prime knot
    Simplest prime link

    In knot theory, a prime knot or prime link is a knot that is, in a certain sense, indecomposable. Specifically, it is a non-trivial knot which cannot be written as the knot sum of two non-trivial knots. Knots that are not prime are said to be composite knots or composite links. It can be a nontrivial problem to determine whether a given knot is prime or not.

    A family of examples of prime knots are the torus knots. These are formed by wrapping a circle around a torus p times in one direction and q times in the other, where p and q are coprime integers.

    Knots are characterized by their crossing numbers. The simplest prime knot is the trefoil with three crossings. The trefoil is actually a (2, 3)-torus knot. The figure-eight knot, with four crossings, is the simplest non-torus knot. For any positive integer n, there are a finite number of prime knots with n crossings. The first few values for exclusively prime knots (sequence A002863 in the OEIS) and for prime or composite knots (sequence A086825 in the OEIS) are given in the following table. As of June 2025, prime knots up to 20 crossings have been fully tabulated. [1]

    n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
    Number of prime knots
    with n crossings    		 0 0 1 1 2 3 7 21 49 165 552 2176 9988 46972 253293 1388705 8053393 48266466 294130458 1847319428
    Composite knots 0 0 0 0 0 2 1 5 … … … … … …
    Total 0 0 1 1 2 5 8 26 … … … … … …

    Enantiomorphs are counted only once in this table and the following chart (i.e. a knot and its mirror image are considered equivalent).

    Prime knot
    A chart of all prime knots with seven or fewer crossings, not including mirror-images, plus the unknot (which is not considered prime).

    Schubert’s theorem

    A theorem due to Horst Schubert (1919–2001) states that every knot can be uniquely expressed as a connected sum of prime knots.[2]

    See also

    References

    1. ↑ Thistlethwaite, M. “The enumeration and classification of prime 20–crossing knots” University of Tennessee, 2025. https://web.math.utk.edu/~morwen/k20v3.pdf
    2. ↑ Schubert, H. “Die eindeutige Zerlegbarkeit eines Knotens in Primknoten”. S.-B Heidelberger Akad. Wiss. Math.-Nat. Kl. 1949 (1949), 57–104.

    External links

    This article is adapted from “Prime knot” on Wikipedia, written by its contributors, and used under CC BY-SA 4.0. Text on this page is available under the same licence. Source snapshot: Wikipedia via Kiwix, 2026-07-20.