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Braid
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A braid (also referred to as a plait; /plæt/) is a complex structure or pattern formed by interlacing three or more strands of flexible material such as textile yarns, wire, or hair.[1] The simplest and most common version is a flat, solid, three-stranded structure. More complex patterns can be constructed from an arbitrary number of strands to create a wider range of structures (such as a fishtail braid, a five-stranded braid, rope braid, a French braid and a waterfall braid). The structure is usually long and narrow with each component strand functionally equivalent in zigzagging forward through the overlapping mass of the others. It can be compared with the process of weaving, which usually involves two separate perpendicular groups of strands (warp and weft).
Historically, the materials used have depended on the indigenous plants and animals available in the local area. During the Industrial Revolution, mechanized braiding equipment was invented to increase production. The braiding technique was used to make ropes[2] with both natural and synthetic fibers as well as coaxial cables for radios using copper wire.[3] In more recent times it has been used to create a covering for fuel pipes in jet aircraft and ships (first using glass fibre, then stainless steel and Kevlar). Hoses for domestic plumbing are often covered with stainless steel braid.
Hair braiding
[edit]
The oldest known reproduction of hair braiding may go back about 30,000 years: the Venus of Willendorf, a female figurine estimated to have been made between about 28,000 and 25,000 BC in modern-day Austria.[4] The Venus of Brassempouy from the southwest of France is estimated to be about 25,000 years old and shows a braided hairstyle.
Although many cultures want to take sole credit for the braid, they cannot be traced to a single origin. Like how different versions of Cinderella are traceable to nearly every culture, braids, too, are polygenetic. One early example of hair braiding takes place in 1279-1213 BCE as recorded in the story of Isis: "when some of the queen's maidens came to the well, she greeted them kindly and began to braid their hair."[5]
During the Bronze Age and Iron Age, many peoples in the Near East, Asia Minor, Caucasus, East Mediterranean and North Africa are depicted in art with braided or plaited hair and beards.[6][7] Similarly, the practice is recorded in Europe, Africa, India, China, Japan, Australasia and Central Asia.
Braiding is traditionally a social art.[citation needed] Because of the time it takes to braid hair, people have often taken time to socialize while braiding and having their hair braided. It begins with the elders making simple knots and braids for younger children. Older children watch and learn from them, start practicing on younger children, and eventually learn the traditional designs. This carries on a tradition of bonding between elders and the new generation.
Industrial history and use
[edit]Early braids had many uses, such as costume decoration, animal regalia (like camel girths), sword decoration, bowls and hats (from palm leaves), locks (such as those made in Japan to secure precious tea supplies through the use of elaborate knots), and weapons (e.g. slings).
Materials that are used in braids can vary depending on local materials. For instance, South Americans used the very fine fibers from the wool of alpaca and llama, while North American people made use of bison fibers. Throughout the world, vegetable fibers such as grass, nettle, and hemp have been used to create braids. In China, Korea, and Japan silk still remains the main material used. In the Americas, the braiding of leather is also common. Plaiting with kangaroo leather has been a widely practiced tradition in rural Australia since pioneering times. It is used in the production of fine leather belts, hatbands, bridles, dog leads, bullwhips, stockwhips, etc. Other leathers are used for the plaiting of heavier products suitable for everyday use.[8]
For nomadic peoples, braiding was a practical means of producing useful and decorative textiles. In other areas, such as the Pacific islands (where leaves and grasses are braided), and for many hill tribes, braids are made using minimal equipment. It was only when braiding became a popular occupation in the home or school, as it is in China and Japan, and when the Industrial Revolution came about, that specific tools were developed to increase production and make it easier to produce more complicated patterns of braids.
Braids are also very good for making rope and decorative objects.[9] Complex braids have been used to create hanging fibre artworks.
Gold braids and silver braids are components or trims of many kinds of formal dress, including military uniform (in epaulettes, aiguillettes, on headgear).
Ropes and cables
[edit]
Braiding creates a composite rope that is thicker than the non-interlaced strands of yarns. Braided ropes are preferred by arborists, rock climbers, and in sport sailing because they do not twist under load, as does an ordinary twisted-strand rope. These ropes consist of one or more concentric tubular braided jackets surrounding either several small twisted fibre cords, or a single untwisted yarn of straight fibres, and are known as Kernmantle ropes.
In electrical and electronic cables, braid is a tubular sheath made of braided strands of metal placed around a central cable for shielding against electromagnetic interference. The braid is grounded while the central conductor(s) carries the signal. The braid may be used in addition to a foil jacket to increase shielding and durability. Litz wire uses braids of thin insulated wires to carry high frequency signals with much lower losses from skin effect or to minimise proximity effect in transformers. Flat braids made of many copper wires can also be used for flexible electrical connections between large components. The numerous smaller wires comprising the braid are much more resistant to breaking under repeated motion and vibration than is a cable of larger wires.
Similar braiding is used on pressurized hoses, such as in plumbing and hydraulic brake systems in automobiles. Braiding is also used for fibres for composite reinforcements.
A property of the basic braid is that removing one strand unlinks the other two, as they are not twisted around each other. Mathematically, a braid with that property is called a Brunnian braid.
Onion and garlic
[edit]Onion and garlic stalks are often braided for storage after they are partially dried.[10][11][12]
Metaphors
[edit]Braids are often used figuratively to represent interweaving or combination, such as in, "He braided many different ideas into a new whole."
In some river and stream systems, small streams join and redivide in many places. Such stream systems are said to be braided.[13] These are often found in alluvial fans at the outlet of canyons. This is a result of heavy sediment deposition at high flows followed by re-erosion at low flows.
Gallery
[edit]-
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A gold braid on a police uniform -
A close up of a red braided USB cable.
See also
[edit]References
[edit]- ^ Kyosev, Yordan (2014). Braiding technology for textiles. Woodhead Publishing. ISBN 9780857091352.
- ^ Michael, M.; Kern, C.; Heinze, T. (2016). "Braiding processes for braided ropes". Advances in Braiding Technology. pp. 225–243. doi:10.1016/b978-0-08-100407-4.00009-0. ISBN 9780081009260.
- ^ Kyosev, Y.; Müller, B. (2016). "Lever arm braiding". Advances in Braiding Technology. pp. 209–222. doi:10.1016/b978-0-08-100407-4.00008-9. ISBN 9780081009260.
- ^ "Nude woman (Venus of Willendorf)". khanacademy.org. Archived from the original on 13 November 2014. Retrieved 1 May 2018.
- ^ Baring, Anne (1993). The myth of the goddess : evolution of an image. Jules Cashford. London: Arkana Publishing. ISBN 0-14-019292-1. OCLC 28359877.
- ^ "The history of hair, hair styles through the ages". Archived from the original on 2007-04-26. Retrieved 2007-04-29.
- ^ "BRAIDS HAIRSTYLES 2018". Archived from the original on 26 September 2018. Retrieved 26 September 2018.
- ^ Grant, Bruce, Encyclopedia of Rawhide and Leather Braiding, Cornell Maritime Press, Cambridge, Maryland, 1972. ISBN 0-87033-161-2
- ^ "Braid Hairstyles Guide - DIY". Iknowhair.com. 19 October 2010. Archived from the original on 2013-11-12. Retrieved 2013-11-22.
- ^ Accetta-Scott, Ann (2019-05-01). The Farm Girl's Guide to Preserving the Harvest: How to Can, Freeze, Dehydrate, and Ferment Your Garden's Goodness. Rowman & Littlefield. pp. 211–212. ISBN 978-1-4930-3665-3.
- ^ Winger, Jill (2016-09-28). "How to Braid Garlic • The Prairie Homestead". The Prairie Homestead. Retrieved 2023-08-12.
- ^ Damrosch, Barbara (September 8, 2017). "The stylish way to keep homegrown onions and garlic on hand". Washington Post.
- ^ Collier, Ann (2011-11-15). Using Textile Arts and Handcrafts in Therapy with Women. Jessica Kingsley Publishers. ISBN 9780857003379. Archived from the original on 5 October 2016. Retrieved 1 March 2016.
External links
[edit]
Media related to Braids at Wikimedia Commons
The dictionary definition of braid at Wiktionary- Minnesota Statute Sec.155A.28 https://www.revisor.mn.gov/statutes/cite/155A.28
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Braid
View on Grokipediafrom Grokipedia
A braid is a length of cord, ribbon, or hair formed by interlacing three or more strands of flexible material in a regular diagonal pattern.[1] This structure is created by repeatedly crossing the strands over and under one another in a specific sequence, resulting in a woven appearance that can be flat, round, or three-dimensional depending on the technique.[2]
Braids serve both practical and decorative purposes across various fields, including hairstyling, textiles, crafts, and culinary arts.[3] In hairstyling, they are a versatile technique used to organize and adorn hair, with origins tracing back potentially to the Paleolithic era, as suggested by interpretations of the Venus of Willendorf figurine from around 25,000 BCE, where the head may depict braided or plaited hair.[4] Culturally, braids have signified social status, identity, and rites in diverse societies, from ancient African communities where they encoded messages or denoted rank, to European and Asian traditions incorporating them into daily and ceremonial wear.[5] Beyond personal adornment, braids appear in rope-making for strength and durability, as well as in decorative trims like gold braid on military uniforms to indicate rank or branch of service.[6]
In mathematics and topology, a braid is conceptualized as a geometric configuration of strands fixed at two parallel boundaries that interweave without reversing direction, forming the basis for braid theory.[7] Braid groups, which capture the algebraic relations among these configurations, have significant applications in knot theory, quantum physics, and computer science, including models for particle interactions and DNA recombination.[8][9] This abstract framework, developed in the early 20th century by mathematicians like Emil Artin, underscores the braid's role as a fundamental object in modern scientific inquiry.[10]
Human and Cultural Uses
Hair Braiding
Hair braiding, the practice of intertwining strands of human hair to create structured patterns, dates back to ancient civilizations. In Egypt around 3000 BCE, skilled hairdressers crafted wigs by braiding human hair into numerous small plaits, as evidenced by artifacts and tomb reliefs depicting these styles on both men and women.[11] Similarly, archaeological findings from Hierakonpolis reveal hair extensions used for braiding as early as 3400 BCE, indicating early techniques for enhancing and styling natural hair.[12] In Africa, braiding traditions extend to at least 3500 BCE, with intricate styles such as those resembling modern box braids appearing in ancient societies for practical and symbolic purposes, as suggested by rock art and artifacts.[13] These early practices, seen in mummies and rock art across the continent, highlight braiding's role in daily life and adornment long before written records. Various techniques define hair braiding, each involving systematic weaving to achieve distinct looks. The three-strand braid, a foundational method, begins by dividing a section of hair into three equal parts; the stylist then crosses the right strand over the middle one, followed by the left strand over the new middle, repeating until the end while maintaining even tension.[14] Cornrows, a close-to-the-scalp style common in African traditions, start with parting the hair into rows using a comb; hair is gathered in small sections and woven underhand by passing strands alternately beneath the central piece, creating flat, raised tracks that lie against the head.[15] French braiding builds on the three-strand approach by starting at the crown and progressively incorporating additional hair from the sides into the outer strands with each cross-over, resulting in a voluminous, integrated plait that gathers more hair as it descends.[16] Box braids involve sectioning the hair into square or rectangular parts, adding synthetic extensions to the natural hair, and twisting or weaving three strands tightly from the roots to form uniform, cylindrical ropes that extend several inches beyond the ends.[17] In many African tribes, braids function as status symbols, conveying information about social position, age, and role within the community. For instance, among groups like the Yoruba and Igbo, specific patterns in cornrows or warrior braids denoted rank, with elaborate designs reserved for leaders or fighters to signify strength and heritage.[18] These hairstyles, often taking hours to complete, reflected tribal identity and were used in rituals or battles to inspire morale.[19] In Indigenous communities, such as those of the Diné (Navajo) and other Native American groups, braiding serves as a means of social bonding, where communal grooming sessions strengthen familial and cultural ties through shared storytelling and care.[20][21] This practice fosters emotional connections, preserving traditions passed down through generations.[21] Modern hair braiding relies on specialized tools and materials to facilitate precise and durable styles. Rat-tail combs, with their long, thin handles, are essential for clean partings and sectioning, allowing stylists to create even rows or patterns on the scalp.[22] Beads made of plastic, wood, or metal add decorative elements, threaded onto the ends of braids to enhance aesthetics and cultural expression, while hair extensions—typically synthetic fibers or human hair—extend length and volume for styles like box braids.[23] These materials, combined with holding sprays or oils, ensure longevity without compromising hair integrity.[22] When performed tension-free, hair braiding offers health benefits, including scalp protection from environmental damage and promotion of hair growth. Protective styles like cornrows or box braids shield strands from friction, UV exposure, and manipulation, helping retain moisture and reduce breakage to support length retention over time.[24] Gentle braiding minimizes traction on follicles, preventing conditions like alopecia while allowing the scalp to breathe and encouraging nutrient delivery for healthier growth.[25][26] However, excessive tightness can lead to scalp irritation, underscoring the importance of professional application.[27]Decorative and Textile Braids
Decorative and textile braids encompass a diverse array of interlaced structures created from flexible materials, primarily for aesthetic enhancement in clothing, accessories, and crafts. These braids differ from functional ropes by emphasizing intricate patterns and ornamental qualities, often involving multi-strand techniques that produce symmetrical or repeating motifs. Historically rooted in various cultures, they serve both artistic and utilitarian roles, such as trimming garments or forming jewelry components. One prominent type is kumihimo, a traditional Japanese braiding method originating in the Heian period (794–1185 CE), where multiple strands—typically silk cords—are interlaced on a loom-like device called a marudai or takadai to form flat or round braids.[28] This technique, known as karakumi in its early flat form, combines elements of braiding and weaving using paired strands to achieve symmetrical patterns, often finished with tassels for decorative appeal.[28] Similarly, Celtic knot braids draw from ancient interlacing designs that emerged in the 3rd–4th centuries CE, influenced by Roman interlace patterns and later adapted into Insular art for textiles and metalwork.[29] These knots feature continuous, looping strands without clear beginnings or ends, symbolizing eternity, and were incorporated into embroidered fabrics and woven accessories during the medieval era.[29] In medieval Europe, braided laces, or "points," were essential for securing clothing elements like hose to doublets, crafted from twisted or interlaced cords of linen or wool to provide both durability and subtle decoration.[30] These evolved into more ornate uses in jewelry and garment trims by the 14th century, where braiders employed finger-loop methods to create narrow, patterned cords for lacing bodices or adorning belts.[31] This tradition persisted into modern crafts, notably in friendship bracelets, which trace their origins to indigenous Central American knotting practices and gained popularity in the United States during the 1970s as symbols of camaraderie.[32] Common materials for decorative braids include yarn for soft, colorful textiles; leather for sturdy accessories like belts or watchbands; and fine wire for metallic jewelry, each allowing distinct textures and durabilities.[32] Techniques vary by material but often involve over-under interlacing: for instance, chevron patterns emerge from mirroring forward and backward knots in even-strand setups, creating V-shaped zigzags ideal for bracelets, while spiral designs result from uneven tension in four-strand braids, producing a helical twist suited to leather thongs.[32] Contemporary applications blend these traditions with macramé-inspired braids, where knotted cords of cotton or synthetic yarns form bohemian fashion items like handbags, necklaces, and hair accessories, reviving 1970s aesthetics in sustainable, handmade designs.[33] These modern pieces emphasize eco-friendly materials and customizable patterns, appearing in high-street jewelry and artisanal markets as versatile accents for everyday wear.[33]Industrial and Practical Applications
Ropes and Cables
Braided ropes and cables have been essential for load-bearing applications since ancient times, with evidence of hemp-based constructions dating back to around 8000 BCE in Mesopotamia, where fibers were twisted and woven into durable strands for hauling and binding.[34] These early ropes, often made from natural materials like hemp, flax, and sisal, provided the tensile strength needed for tasks such as construction and agriculture, evolving through trial and error to incorporate braiding techniques that enhanced durability under tension.[35] Modern rope construction methods build on these foundations, employing techniques like laid, plaited, and kernmantle designs to optimize tensile strength. Laid ropes consist of three or more strands twisted together in opposing directions, creating a balanced structure that resists unraveling while distributing load across the fibers.[36] Plaited ropes, a form of braiding, interweave multiple strands—often eight or more—in a crisscross pattern, which allows for greater flexibility and even stress distribution compared to simple twisting.[37] Kernmantle ropes feature a braided outer sheath (mantle) encasing a core of parallel or twisted yarns (kern), where the core bears most of the load for high tensile strength, and the sheath protects against abrasion.[38] The shift from natural to synthetic materials accelerated post-World War II, with nylon emerging as a groundbreaking fiber in the late 1930s and gaining widespread use in ropes by the 1940s due to its superior strength-to-weight ratio and elasticity. Subsequent developments included polyester in the 1950s for low-stretch mooring lines, polypropylene in the 1950s for lightweight and buoyant options, and high-performance fibers like aramids (e.g., Kevlar) and ultra-high-molecular-weight polyethylene (UHMWPE, e.g., Dyneema) from the 1960s-1980s for exceptional strength and cut resistance.[39] This evolution enabled ropes to handle heavier loads with less bulk, as seen in applications like maritime ship rigging, where braided nylon cables secure sails and moor vessels against dynamic forces; climbing, where kernmantle designs absorb falls; and construction, for lifting materials in cranes and scaffolding.[40] For instance, a 1-inch diameter double-braided nylon rope typically achieves a breaking strength of around 10,000 kg, far surpassing traditional hemp equivalents.[41] Braided constructions offer key advantages over purely twisted ropes, including reduced kinking during handling and more uniform stress distribution, which prevents localized failures and enhances overall longevity under repeated loading.[42] This makes them ideal for industrial uses requiring reliability, though related braiding techniques also appear in electrical shielding for electromagnetic protection.[43]Electrical and Shielding Applications
Braided shielding plays a crucial role in electrical applications by providing electromagnetic interference (EMI) protection through interwoven metallic wires, typically copper or tinned copper, that form a flexible mesh around cable conductors. This construction, often referred to as a copper wire braid, creates a Faraday cage-like barrier that attenuates external electromagnetic fields and prevents signal leakage, essential for maintaining signal integrity in sensitive electronics. Coverage ratios, defined as the optical percentage of the braid that blocks light (indicating shielding density), typically range from 70% to 95%, with higher ratios like 85% offering superior EMI suppression for high-frequency applications.[44][45] The development of braided shielding traces back to the late 19th century, when it was initially employed to insulate early telegraph and electrical wires against interference as these systems entered homes and businesses. By the early 20th century, advancements in telephony led to the 1929 patent by Lloyd Espenschied and Herman Affel for coaxial cables, which incorporated braided metallic shields to enable reliable long-distance transmission over open wires, marking a shift from simple spacing techniques to structured EMI mitigation. This evolved into modern coaxial cables used in telecommunications, where braided shields balance flexibility and protection, evolving alongside foil alternatives for comprehensive shielding in broadband and RF systems.[46][47] In automotive wiring harnesses, braided shields encase bundles of wires to safeguard against EMI from ignition systems and motors, while their inherent flexibility accommodates the repeated bending and vibration in vehicle environments, enhancing durability over millions of cycles. Similarly, in audio equipment, such as microphone and instrument cables, braided copper meshes reduce noise and hum from external RF sources, preserving audio fidelity during dynamic use like stage performances, where the braid's suppleness prevents signal degradation from movement. These applications leverage the braid's mechanical strength, which adds tensile reinforcement without sacrificing bend radius, making it ideal for harnesses in confined spaces.[48][49][50] Manufacturing braided sheaths for electrical cables primarily utilizes maypole or spiral braiding machines, which interlace carriers of wire around a central core at controlled speeds to form uniform patterns. In maypole machines, carriers follow a figure-eight path on a rotating bed, producing a diamond or regular weave with adjustable braid angles typically between 30° and 60° to optimize coverage and flexibility; steeper angles increase density but require more material. Spiral braiding, conversely, employs rotating spindles for helical application, allowing precise overlap control—often 15-20° for minimal gaps—suited for high-speed production of EMI shields in coaxial assemblies. These processes ensure low-resistance paths for grounding while maintaining the sheath's expandability for easy installation.[51][52][53]Culinary Uses
Onion and Garlic Braids
Onion and garlic braids are created by weaving the dried stems of harvested bulbs into strands, typically using a three-strand pattern on heavy-duty string or rope for easy hanging and display. After curing the bulbs in a dry, well-ventilated area for about a week to reduce moisture content, the tops are left intact while the bulbs are interlaced progressively, starting with one bulb per strand and crossing them like a French braid to secure the structure. This method works well for softneck garlic varieties and onions with flexible tops, allowing for braids of 10 to 20 bulbs or more, depending on the desired length.[54][55] The practice originated as a traditional storage technique in Mediterranean and European farming communities, where it facilitated drying and preservation post-harvest to combat rot in humid climates. In regions like Italy and Sicily, farmers have long braided garlic and onions to utilize airflow around the spaced bulbs, a method documented in agricultural customs dating back centuries for maintaining crop quality through winter. Similar interlacing techniques for alliums appear in broader European rural traditions to ensure even drying and accessibility.[56][57][58] Braiding provides key benefits for preservation, including enhanced airflow that circulates around each bulb to prevent moisture buildup and fungal growth, thereby deterring rot and common pests like mold or insects. The elevated hanging position further reduces ground contact and humidity exposure, allowing properly cured braids to store in cool, dark spaces for 6 to 12 months without significant sprouting or spoilage. Garlic's natural antimicrobial properties in the bulbs add an extra layer of pest resistance during extended storage.[54][59][60] Culturally, onion and garlic braids serve as symbols of abundance and prosperity, often displayed during harvest festivals to celebrate successful yields and communal labor. In Italian traditions, such as those in Abruzzo, intricately braided garlic strands are showcased at local markets and events like the Red Garlic Festival in Sulmona, representing agricultural heritage and warding off misfortune. Similar displays appear in European harvest celebrations, where braids embody the fruits of the harvest and are hung in homes for protection and good fortune.[57][61]Braided Baked Goods
Braided baked goods feature intricate weaving of dough strands that enhance both their visual appeal and structural integrity during baking. For challah, this technique originated in 15th-century Central and Eastern Europe, where it became common for enriched Sabbath breads among Ashkenazi Jewish communities, evolving from earlier loaves into a distinctly ceremonial form.[62][63] The plaited design symbolizes unity, with intertwined strands representing the cohesion of the Jewish people and the binding of physical life to the divine.[64][65] Preparation begins with an enriched yeast dough, typically incorporating eggs, butter or oil, and sugar for tenderness. For challah, the risen dough is divided into three to six equal portions, each rolled into uniform strands about 12-14 inches long and slightly tapered at the ends to facilitate weaving. The strands are pinched firmly together at one end to secure the braid, then crossed over one another in a repeating pattern—such as pulling the outer strand under the middle for a three-strand version—until the opposite end, where the loose ends are pinched and tucked underneath for a neat finish.[66][67] Brioche, a French enriched bread originating in Normandy in the 1400s, follows a similar braiding process, with its higher butter content making the dough softer; after dividing into strands, bakers braid tightly and pinch the ends to maintain shape, often forming a loaf or ring to accommodate the dough's richness.[68][69][70] The braided loaf then undergoes a final proofing for 45-60 minutes until puffy, followed by an egg wash for shine and optional light scoring along the strands to guide expansion and add decorative flair.[71] Beyond aesthetics, braiding provides practical advantages in baking. The interwoven structure offers stability to soft, enriched doughs prone to spreading, promoting even heat distribution for uniform cooking and a tender crumb without dense spots.[72][73] This results in consistent baking times, typically 30-40 minutes at 350°F (175°C), yielding a golden crust while retaining internal moisture.[74] Cultural variations highlight regional adaptations of this technique. Italian pane intrecciato, or braided bread, uses a simple three-strand plait from a lightly sweetened dough, proofed for about 1 hour after shaping, and baked at 350°F for 40 minutes to achieve a fluffy texture ideal for holidays.[74] These examples underscore braiding's role in elevating everyday dough into festive, structurally sound pastries across traditions. In other regions, such as Latin America, braided breads like pan de yema incorporate similar weaving for holiday meals, adding to the global diversity of the practice.[75]Mathematical and Scientific Concepts
Braid Theory
In mathematics, braid theory is the topological study of braids, which are geometric objects consisting of a finite number of strands or curves embedded in three-dimensional space, connecting a fixed set of points on a top plane to corresponding points on a parallel bottom plane, considered up to ambient isotopy that preserves the endpoints.[76] These strands do not intersect except possibly at the endpoints, and the isotopy ensures that the braids can be continuously deformed into one another without passing through each other or moving the fixed points.[77] This framework extends classical knot theory by focusing on open-ended tangles rather than closed loops, allowing for the analysis of intertwining paths in a controlled geometric setting.[78] The foundations of braid theory were established by Emil Artin in 1925 through his seminal work "Theorie der Zopfe," where he introduced braids as a means to classify and study these structures systematically, building directly on the emerging field of knot theory to address limitations in representing open tangles.[79] Artin's approach formalized braids as equivalence classes under isotopy, providing tools to convert them into algebraic representations while preserving their topological essence.[80] This development marked a pivotal shift, enabling deeper insights into the topology of three-dimensional embeddings that knot theory alone could not fully capture.[81] Key properties of braids distinguish between Artin braids, which permit permutations of the strand endpoints (allowing strands to swap positions), and pure braids, a subgroup where each strand returns to its original endpoint without permutation, preserving the identity of individual paths.[78] A fundamental operation in braid theory is the closure of a braid, achieved by connecting the top endpoints to the bottom ones via non-intersecting arcs outside the braid's space, resulting in a link or knot; Alexander's theorem (1923) guarantees that every oriented knot or link can be represented as the closure of some braid.[82] For instance, a braid word— a sequence describing crossings—can be projected and closed to yield a standard knot diagram, facilitating the study of link invariants.[83] These properties underpin the geometric foundations, with algebraic encodings via braid groups providing a complementary representation.[84] Braid theory finds significant applications in physics, particularly in modeling the supercoiling of DNA molecules, where intertwined double helices behave as braids during replication and recombination, with closure operations simulating catenane formation.[85] In quantum mechanics, braids describe the paths of anyonic particles in two-dimensional systems, where the topological braiding of worldlines encodes non-Abelian statistics robust against local perturbations, as explored in models of topological quantum computing.[86] These applications highlight braid theory's role in bridging abstract topology with physical phenomena involving persistent entanglement.[87]Braid Groups
The braid group on strands is defined as the fundamental group of the unordered configuration space of points in the Euclidean plane, capturing the homotopy classes of motions of these points without collisions.[88] This topological interpretation, established through fibrations of configuration spaces, provides a geometric foundation for the group structure.[89] Algebraically, is presented by generators for , where each represents the positive crossing of the -th strand over the -th strand, with all other strands fixed.[76] The defining relations, known as the Artin relations, govern the interactions between these generators. The braid relation for adjacent generators is for , reflecting the equivalence of certain crossing sequences under isotopy.[76] Additionally, far commutativity holds for non-adjacent generators: ensuring that distant strands do not interfere.[76] These relations suffice to define the group, as proven by Artin, who showed that any braid can be reduced to a canonical form using them.[90] A representative example is the 3-braid group , generated by and subject to the single braid relation . This group is isomorphic to the fundamental group of the complement of the trefoil knot in , highlighting connections between braids and knot complements.[91] Regarding computability, the word problem in braid groups is solvable via algorithms that reduce words to unique normal forms, such as Garside's normal form, enabling efficient verification of equality between braid words despite the groups' infinite nature.[92] In modern applications, braid groups facilitate path planning in robotics, where trajectories of multiple tethered or untethered agents are modeled as braids to ensure collision-free motions and detect entanglements in configuration spaces.[93] Similarly, in cryptography, braid-based encryption protocols exploit the computational hardness of problems like the conjugacy search problem in , proposing public-key systems resistant to classical attacks, though vulnerabilities have been identified in some schemes.[94] These uses underscore the braid groups' utility beyond pure algebra, linking to practical computational challenges. In braid theory, these algebraic elements correspond geometrically to isotopy classes of strand diagrams in the plane.Symbolism and Metaphors
Cultural Symbolism
In various African cultures, such as among the Himba people of Namibia, braided hairstyles serve as important indicators of social status, including marital status and age; for instance, unmarried girls typically wear two plaits (ozondato) hanging forward over their faces, while married women or new mothers adopt an Erembe headdress crafted from animal skin to signify their transition to adulthood and family roles.[95][96][97] This practice underscores braids' role in communal identity and life stages, extending beyond aesthetics to encode personal and societal narratives. Among Native American traditions, sweetgrass braids hold profound spiritual significance in smudging ceremonies, where the plant—regarded as the sacred hair of Mother Earth—is braided, dried, and burned to purify spaces, attract positive energies, and symbolize healing, peace, and interconnectedness with the natural world across tribes like the Cree and Lakota.[98][99] These braids, often consisting of three strands representing unity, are used in rituals to honor ancestors and foster harmony, highlighting braids as conduits for sacred communication.[100] In Renaissance art, braided hair frequently conveyed layers of symbolism related to virtue, status, and sensuality; Sandro Botticelli's portraits, such as Portrait of a Young Woman, depict women with elaborate bound braids intertwined with pearls, signaling marital eligibility or wedded status while evoking erotic undertones as a "threshold to desire" in Florentine society.[101][102] This artistic motif extended to modern tattoos, where braided designs—such as rope-like patterns—represent resilience, unity, and the strength to endure challenges, often chosen to embody personal empowerment and interconnected life journeys.[103] Global indigenous traditions further illustrate braids' symbolic depth; in Māori culture, tā moko tattoos incorporate interlocking linear and spiral motifs symbolizing whakapapa (genealogy), cultural heritage, and prestige, as these chiseled designs narrate an individual's ancestry and societal role.[104] The symbolism of braids has evolved historically from associations with fertility in ancient depictions—such as the stylized braided caps on Paleolithic Venus figurines like the Venus of Willendorf, interpreted as emblems of abundance and the divine feminine—to markers of empowerment in 20th-century movements; during the 1970s Black Power and feminist eras, Black women reclaimed cornrows and box braids as acts of cultural defiance and self-affirmation against Eurocentric beauty standards, transforming them into icons of racial pride and autonomy.[4][105][106][107]Literary and Idiomatic Uses
In English literature and idiom, the term "braid" has evolved from its etymological roots to embody themes of interlacing, complexity, and transformation. Originating from Old English bregdan, meaning "to move quickly, pull, shake, or weave," the word entered Middle English as breyden around the 14th century, shifting toward connotations of plaiting or twisting strands together. This Proto-Germanic base, akin to Old Norse bregða ("to draw or brandish"), initially carried senses of sudden action or deceit, as in a "braid" denoting a trick or stratagem by circa 1200. By the 19th century, poetic applications highlighted aesthetic and symbolic depth; for instance, Alfred Lord Tennyson's Locksley Hall (1842) evokes "a swarm of fire-flies tangled in a silver braid" to depict celestial beauty and entanglement, while W.S. Gilbert's lyric "Braid the Raven Hair" (1867) uses braiding as a metaphor for enhancing feminine allure through deliberate adornment.[108][109][110][111] Idiomatic expressions extend "braid" into figurative realms, often symbolizing unity amid multiplicity or intricate pathways. The phrase "braided river," borrowed from geomorphology to describe rivers with interwoven channels, serves as a literary metaphor for meandering, resilient journeys; in Dorothy Tse's The Braided River: Migration and the Personal Essay (2019), it illustrates the nonlinear flow of migrant identities across cultures, emphasizing convergence and divergence in personal narratives. Similarly, "braided fates" evokes intertwined destinies, as in Karla M. Jay's historical novel When We Were Brave (2019), where it captures the linked lives of characters navigating World War II's upheavals, underscoring shared vulnerability and hope. These idioms draw briefly from cultural symbolism of braids as bonds but adapt them to verbal expressions of relational complexity.[112][113] In literature, braids frequently represent identity, preparation, and narrative structure. Homer's Odyssey employs "lovely braids" (eukómōn) as an epithet for goddesses like Calypso, symbolizing divine elegance and otherworldly poise, as when she is described as "the nymph with lovely braids" in Book 5, highlighting her allure amid Odysseus's trials. Modern works amplify this for cultural identity; in Cherie Dimaline's The Marrow Thieves (2017), protagonist Frenchie's long braid signifies Indigenous pride and resilience against colonial erasure, embodying a tactile link to heritage and survival. The concept of "braided narratives" further illustrates structural metaphor, where stories interweave multiple threads for depth—exemplified in Jason Fagone's essay "The Jessica Simulation" (2020), which braids personal loss with AI ethics to explore emotional entanglement in contemporary nonfiction. Such uses prioritize conceptual unity over isolation, reflecting braids' enduring role in evoking cohesion in fragmented human experiences.[114]References
- https://groupprops.subwiki.org/wiki/Braid_group:B3