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Lozenge

A lozenge is a geometric figure consisting of a rhombus with four equal sides and two acute angles opposite two obtuse angles, often visualized as a diamond shape standing on one of its points.^[1] The term also denotes a small, solid, sweetened, and flavored preparation containing one or more medicaments, designed to dissolve or disintegrate slowly in the mouth for localized or systemic effects, such as relieving sore throats or delivering nicotine for smoking cessation.^[1]^[2] In heraldry, a lozenge refers to a diamond-shaped charge on a shield, historically used to display the arms of unmarried women or widows, distinguishing it from the more common escutcheon shape for men.^[3]^[4] Originating from Middle English "losenge," derived from Anglo-French, the word first appeared in the 14th century, initially describing the geometric form before extending to shaped objects and confections.^[1] In geometry and design, lozenges appear in patterns like lozengy, where a field is divided into a crisscross of such diamonds, often alternating tinctures for visual contrast.^[3] Heraldic lozenges, sometimes confused with the similar mascle (a voided lozenge), served practical purposes in medieval armory, symbolizing honesty and constancy while accommodating gender-specific conventions in escutcheons.^[4]^[5] Pharmaceutical lozenges, also known as troches or pastilles, come in types such as hard (made from heated sugars molded for slow erosion over 5–10 minutes), soft (using polyethylene glycol or gelatin for easier dissolution), and chewable (gummy forms for pediatric use), allowing targeted delivery of ingredients like antiseptics, analgesics, or decongestants directly to the oral mucosa.^[2] Their preparation avoids high heat for sensitive drugs, ensuring stability, and they offer advantages in patient compliance for those who struggle with swallowing tablets.^[2]

Etymology and Terminology

Historical Origins

The term "lozenge" entered the English language in the early 14th century as "losange," referring to a rhombus or diamond-shaped figure with four equal sides, borrowed directly from Old French "losenge" (modern French "losange").^[6]^[7] The Old French word itself, attested in medieval texts for geometric forms like panes of glass or flat quadrilaterals, has an uncertain origin but is widely believed to stem from a pre-Roman Celtic substrate, possibly Iberian or Gaulish *lausa meaning "flat stone," evoking the shape's resemblance to a slab or tile.^[6]^[8] This etymon aligns with Vulgar Latin *lausa ("flat stone"), yielding cognates such as Provençal "lausa," Spanish "losa" (slab), and Catalan "llosa" (tile), all denoting flat, stone-like surfaces that parallel the lozenge's angular profile.^[9] The earliest recorded use of "lozenge" in English appears before 1366 in the poem The Romaunt of the Rose, where it describes a geometric diamond shape, reflecting its initial application to architectural or ornamental forms rather than comestibles or symbols.^[7] By the late 14th century, the term had expanded to denote diamond-shaped confections or wafers, a semantic shift likely influenced by the shape's prevalence in baking and confectionery.^[6] This evolution continued into the 16th century, when "lozenge" began appearing in heraldic treatises to specify the rhomboid charge—a diamond-shaped emblem on shields, often associated with women's arms—and in medical contexts for small, diamond-formulated tablets of sugar and remedies held in the mouth, as documented from the 1520s onward.^[6]^[10] Cross-cultural parallels underscore the term's roots in describing diamond-like forms across Romance languages, with Italian "losanga" borrowed from Old French "losenge" in the medieval period to mean a rhombus or lozenge shape, used similarly in geometry and heraldry.^[11] In Germanic languages, a comparable concept emerged independently as German "Raute," derived from Middle High German "rūte" (possibly linked to the cross-shaped flower of the rue plant, Ruta graveolens, or denoting a quadrangular pane), denoting a diamond or rhombus in heraldic and geometric contexts since the 13th century.^[12]^[13] These linguistic variants highlight a shared European tradition of naming oblique, stone-evoking shapes, predating specialized applications in later centuries.^[6]

Modern Usage and Variants

In contemporary English, "lozenge" primarily denotes either a geometric figure resembling a rhombus with two acute and two obtuse angles, or a small, medicated tablet designed to dissolve in the mouth for throat relief.^[14] The Oxford Advanced Learner's Dictionary specifies the geometric sense as "a four-sided figure shaped like a diamond, with two opposite angles more than 90° and two less than 90°," while the medicinal sense describes it as "a small sweet, often lozenge-shaped, containing medicine" such as for coughs or sore throats.^[14] Similarly, Merriam-Webster defines it as "a figure with four equal sides and two acute and two obtuse angles: diamond," alongside the pharmaceutical usage.^[1] Spelling variants include the archaic form "losange," derived from Old French, which appeared in Middle English texts around the 14th century to describe diamond-shaped objects like cakes or panes.^[6] A regional variant, "lozenger," persists in American English and northern England, particularly for the medicated tablet, often pronounced with an added "-er" sound, though "lozenge" remains the standard spelling across dialects.^[15] Regional preferences also influence usage: in British English, "lozenge" is the preferred term for the throat tablet, whereas American English more commonly employs "cough drop" for the same item, reflecting subtle lexical divergences in everyday language.^[16] Globalization has extended "lozenge" terminology into technical and international contexts, with the geometric sense adopted in Romance languages through cognates like French "losange" (rhombus) and Portuguese "losango" (lozenge shape), often in mathematical or heraldic discussions.^[6] In non-European languages, English-derived forms appear in scientific literature; for instance, Russian uses "lozenge" transliterations in geometric contexts, while the medicinal term translates to "пастилка" but retains "lozenge" in global pharmaceutical branding.^[17] This adoption highlights the term's utility in cross-cultural technical communication, particularly in fields like geometry and medicine.^[18]

Geometry

Definition and Basic Shape

A lozenge is a quadrilateral with four equal sides, making it an equilateral figure, though its interior angles are not necessarily equal or right angles, which distinguishes it from a square.^[19] In geometric terms, it is a type of parallelogram where opposite sides are parallel and all sides share the same length.^[20] Visually, a lozenge is often represented as a diamond shape, akin to a square rotated by 45 degrees, emphasizing its elongated or tilted appearance.^[19] A simple ASCII diagram illustrates this basic form:

  /\
 /  \
\  /
 \/ 
  /\
 /  \
\  /
 \/

In modern geometry, "lozenge" is synonymous with "rhombus," encompassing any such equilateral quadrilateral regardless of angle restrictions, though some definitions specify acute angles of 45 degrees for precision.^[19]

Mathematical Properties

The lozenge, as a geometric shape synonymous with a rhombus, exhibits several key mathematical properties derived from its equilateral parallelogram structure, where all four sides are of equal length a. The perimeter of a lozenge is simply $4a, reflecting the equality of its sides.^[21]^[22] A fundamental attribute is the configuration of its diagonals, denoted as d_1 and d_2, which bisect each other at right angles and also bisect the vertex angles. These diagonals are perpendicular bisectors, satisfying the relation d_1^2 + d_2^2 = 4a^2. The area of a lozenge is given by the formula \frac{1}{2} d_1 d_2, which arises from the perpendicular division into four right triangles.^[21]^[22] Regarding its angles, opposite angles in a lozenge are equal, and adjacent angles are supplementary, summing to $180^\circ. This property holds as a consequence of the parallelogram nature of the shape.^[21]^[22] Special cases include the square, which occurs when all angles are $90^\circ, making the diagonals equal in length. The lozenge is both a special parallelogram (with equal sides) and a special kite (with two pairs of adjacent equal sides).^[21]^[22]

Heraldry

Symbolic Representation

In heraldry, the lozenge is defined as a diamond-shaped charge, forming a rhombus with equal sides and two acute angles opposite two obtuse angles, typically oriented with one point to chief and the opposite to base.^[10] This shape serves as a standard ordinary, often depicted with its longer axis palewise, though proportions may vary at the artist's discretion to fit the shield's composition.^[23] Geometrically, it represents a rhombus rotated 45 degrees from a square.^[10] Blazoning standardizes the lozenge's depiction, specifying its tincture, position, and modifications such as being charged with other elements or voided. For instance, a solid lozenge is simply blazoned by its color and placement, as in the arms of Sire Gerard de Braybrok: "Argent, seven lozenges gules," where multiple lozenges are arranged without voids.^[24] The term "lozenge throughout" indicates the charge extends to touch all four edges of the shield with its points, emphasizing its expansive form, while a plain lozenge may be smaller and centered.^[25] Voided lozenges, pierced through the center, are conventionally blazoned as mascles in modern heraldry to denote the opening, which is typically one-third the diameter of the outer shape.^[23] An example appears in the arms of Monsire Henry de Ferrers: "Gules, six lozenges percés d’or," though contemporary usage would render this as mascles.^[24] The lozenge is distinguished from related charges by its proportions and structure: unlike the fusil, which is narrower and taller—often derived from segments of an indented ordinary like a bend—the lozenge maintains a more equilateral diamond form, with width about three-quarters of its height.^[25]^[23] Early heraldry sometimes blurred this line, using the terms interchangeably, as in the Percy arms: "Azure, five fusils conjoined in fess or," which could resemble lozenges in depiction.^[25] The mascle, by contrast, is inherently a voided lozenge, not a separate shape, and fields covered in such patterns are termed masculy, differing from lozengy (solid lozenges) or fusily (elongated fusils).^[10] These distinctions ensure precise graphical representation in armorial bearings, such as Richard Mountague's: "Argent, three lozenges conjoined in fesse gules, between three ogresses, in chief a mullet."^[24]

Historical and Cultural Applications

The lozenge emerged in heraldry during the medieval period in Europe, with the practice of armorial bearings originating around the mid-12th century as a means of identifying knights in battle and tournaments. By the late 13th century, the lozenge—a diamond-shaped form—began to be associated with women's arms, as seen in early examples such as a 1262 seal depicting a woman's coat. This shape was used on escutcheons for women, including unmarried maidens who displayed their paternal arms, distinguishing their heraldry from the shield traditionally reserved for men due to its militaristic connotations. In cases involving illegitimate lines, female heirs could bear their father's arms on a lozenge, often with a mark of difference to denote cadency, allowing transmission of lineage despite non-primogeniture status.^[26]^[27]^[28] Culturally, the lozenge symbolized femininity and marital alliances within noble society, serving as a non-combative emblem that highlighted women's roles in forging dynastic ties through marriage. For instance, upon marriage, a woman's arms were impaled with her husband's on a shared shield or lozenge, visually representing the union and her integration into his lineage; this convention underscored the lozenge's role in denoting social status and inheritance rights for women. Unmarried women, or maidens, bore their father's arms alone on the lozenge, sometimes accented with a blue ribbon to signify eligibility, reinforcing its association with maidenhood and potential alliances. Such practices emphasized the lozenge's function in preserving female heraldic identity amid patriarchal structures. Examples include the lozenge-shaped arms of Eleanor of Castile, Queen Consort of Edward I of England, and Jeanne d'Auvergne, Queen Consort of France.^[29]^[28]^[27] The use of the lozenge evolved significantly, becoming more standardized by the 15th century in Britain, France, and the Low Countries. A 1561 decree by English heralds mandated the lozenge for women's arms, solidifying its exclusivity. In the 19th century, amid a broader revival of interest in medieval heraldry during the Victorian era, the lozenge saw renewed decorative application in architecture, seals, and badges, particularly for royal consorts. This resurgence reflected a romanticized return to chivalric traditions, enhancing the lozenge's presence in British and French armorial art. In modern heraldry as of 2025, the lozenge remains the standard shape for displaying women's arms in English and most European traditions, though ovals are used in Scottish heraldry.^[27]^[30]

Medicine

Composition and Formulation

Medicinal lozenges are solid oral dosage forms designed for slow dissolution in the mouth, primarily consisting of a base that provides structure and sweetness. The core composition includes a sugar-based matrix, such as sucrose or corn syrup, or artificial sweeteners like sorbitol for sugar-free variants, which form the bulk of the lozenge and aid in controlled release.^[31] Active ingredients, such as menthol for its cooling sensation or benzocaine as a local anesthetic, are blended in at specific concentrations to deliver therapeutic effects locally in the oral cavity.^[31] Binders like acacia gum or gelatin ensure cohesion, while flavorings such as eucalyptus oil or fruit extracts improve acceptability and mask any bitterness from the actives.^[32] Formulation types differ based on texture and dissolution profile, with hard lozenges produced by compressing a powdered mixture of the base, actives, fillers (e.g., lactose or microcrystalline cellulose), and lubricants (e.g., magnesium stearate) into tablet form using tableting equipment.^[31] In contrast, soft lozenges employ a gelatin or polyethylene glycol (PEG) base, often mixed with glycerin and water, which is heated, combined with ingredients, and poured into molds to solidify, resulting in a more pliable product suitable for slower, moisture-dependent dissolution.^[32] Manufacturing for hard lozenges involves dry blending followed by compression at controlled pressures to achieve uniformity, whereas soft types use a fusion and molding process, with subsequent drying to low moisture content for stability (typically around 20% for gelatin-based pastilles to maintain pliability while preventing microbial growth).^[31]^[33] Regulatory standards from agencies like the FDA and EMA emphasize safe, approved ingredients and formulations that promote effective oral dissolution without rapid disintegration. The FDA's Inactive Ingredients Database specifies excipients such as mannitol, citric acid, and acesulfame potassium that have been used in approved lozenges, ensuring compatibility and limiting potential irritants.^[34] Guidelines require dissolution testing to verify that lozenges release actives appropriately in simulated oral conditions, typically using apparatuses like USP Apparatus 2 with media mimicking saliva pH for bioavailability assessment.^[35] EMA similarly mandates excipient declarations and quality controls under good manufacturing practices to guarantee solubility and absence of undeclared allergens in the final product.^[36]

Therapeutic Uses and History

Medicinal lozenges have ancient origins, with the oldest known recipe dating to around 1550 BC in an Egyptian papyrus, using honey and herbs for throat ailments.^[37] They also appear in traditional Chinese medicine from the Eastern Han Dynasty (25–220 AD).^[38] In Europe, by the 17th century, sugar pastilles formulated by apothecaries delivered herbal remedies for throat discomfort, leveraging sugar's preservative and soothing properties to coat and protect irritated mucous membranes.^[39] These early European formulations, often infused with botanicals like licorice or mint, served as demulcents to alleviate soreness and irritation, marking an evolution of lozenges as a convenient oral dosage form for local symptom relief.^[40] By the 19th century, lozenges gained widespread popularity with the addition of menthol, a compound isolated from peppermint oil in the 1860s, which provided a cooling sensation and enhanced cough suppression.^[41] A key milestone was the creation of Fisherman's Friend in 1865 by English pharmacist James Lofthouse, who developed a strong menthol-eucalyptus lozenge to aid fishermen suffering from respiratory issues in harsh conditions.^[42] This era saw mass production enabled by the Industrial Revolution, transforming lozenges from artisanal remedies into accessible over-the-counter treatments for common ailments.^[40] Therapeutically, lozenges primarily soothe sore throats by forming a protective film over inflamed tissues, suppress coughs through antitussive agents, and relieve dry mouth by stimulating saliva production via demulcents or mild anesthetics like benzocaine. For instance, menthol-based varieties desensitize nerve endings to reduce irritation, while specialized types incorporate antihistamines such as diphenhydramine to address allergy-induced throat symptoms like itching or swelling.^[43] Dry mouth relief is achieved through sugar-free formulations with xylitol or sorbitol, which moisturize oral tissues without promoting decay.^[44] Clinical evidence supports their efficacy for symptom management, with meta-analyses of randomized trials demonstrating significant reductions in throat pain intensity compared to placebo; for example, ambroxol lozenges yielded a mean pain score decrease of 0.11 points on a 6-point Verbal Rating Scale (0-5) (p < 0.0001), equating to moderate relief in acute pharyngitis.^[45] Other studies report substantial pain reductions for anti-inflammatory variants like flurbiprofen, with multiple-dose use over 24 hours providing up to 59% greater relief compared to placebo.^[46] These benefits are most pronounced for non-infectious conditions, with lozenges providing rapid, localized action lasting up to 2 hours post-dissolution.^[47]

Other Contexts

Architecture and Design

The lozenge, a rhombus-shaped motif, has been employed in architectural tiling for its ability to form seamless tessellations, leveraging geometric properties that allow interlocking without gaps or overlaps. In Islamic architecture, lozenge tessellations appear prominently in the intricate girih tile patterns of the Alhambra Palace in Granada, Spain, where rhombi with angles derived from multiples of 36 degrees contribute to quasi-crystalline designs dating to the 14th century.^[48] These patterns, documented in medieval scrolls like the Topkapi Scroll from the early 16th century, were constructed using compass and straightedge to create non-repeating, infinite motifs that adorn walls and floors in mosques and palaces.^[49] Similarly, in Gothic architecture, lozenge meshes adorn surfaces such as the net vaults of Wells Cathedral in England (12th–13th centuries), where they form a decorative lattice enhancing the verticality and lightness of the structure.^[50] Beyond tiling, the lozenge serves as a versatile decorative element in various historical styles. In Romanesque buildings, lozenge-shaped motifs appear in moldings and window surrounds, as seen in the geometric carvings of Durham Cathedral (11th–12th centuries), where diamond patterns alternate with chevrons to add rhythmic ornamentation to arches and piers.^[51] During the Victorian era (19th century), lozenges featured in wallpaper and flooring designs, often as repeating geometric panels in Arts and Crafts-inspired interiors, exemplified by patterns from the Aesthetic Movement that combined lozenges with floral motifs for domestic elegance.^[52] In Art Deco architecture of the 1920s–1930s, stylized lozenge patterns emerged as low-relief motifs on facades and interiors, integrating with zigzags and chevrons to evoke modernity and luxury.^[53] In contemporary architecture, lozenge grids are generated through parametric design software, enabling complex, algorithm-driven facades that adapt to structural and aesthetic needs. Tools like Grasshopper for Rhino facilitate the creation of lozenge-based lattices, drawing on historical tiling principles for innovative applications.^[54] A notable example is Zaha Hadid Architects' Pierresvives library in Montpellier, France (2012), where the facade incorporates floating lozenge geometries in green-tinted glass and perforated aluminum panels, producing dynamic wave-like effects that respond to light and movement.^[55] This approach extends to other parametric projects, such as adaptive Islamic-inspired patterns in modern Middle Eastern developments, where software simulates quasi-periodic lozenge arrangements for sustainable, visually intricate envelopes.^[56]

Games and Recreation

Lozenge shapes, recognized as rhombi composed of two equilateral triangles, play a prominent role in recreational mathematics through tiling puzzles that challenge players to cover regions without gaps or overlaps. One seminal example is the calissons puzzle, which involves tiling a large regular hexagon—subdivided into small equilateral triangles—with lozenge tiles, each covering a pair of adjacent triangles. This problem, named after a diamond-shaped French confection, was classically posed in the 18th century and later rigorously analyzed, revealing that the number of distinct tilings for a hexagon of side length n equals the n-th Catalan number, C_n = \frac{1}{n+1} \binom{2n}{n}. These puzzles emphasize geometric dissection and combinatorial counting, often extending to more complex regions like Aztec diamonds or mutilated hexagons, fostering conceptual understanding of symmetry and enumeration in recreational math. Beyond pure puzzles, lozenge tilings inspire polyform-based challenges akin to polyominoes, where players manipulate sets of connected lozenges (or polyiamonds) to form larger figures or solve dissections. Recreational math enthusiasts use physical sets or software to explore configurations, such as assembling polyiamonds into squares or other polygons, highlighting spatial reasoning and pattern recognition without requiring advanced computation.^[57] In modern board games, lozenges appear as playing pieces or grid elements in abstract strategy titles, promoting territorial control and placement tactics. For instance, Taiji (2007) employs lozenge-shaped wooden pieces on a square board, where players alternate placements to maximize connected areas of their color while inadvertently aiding opponents through shared dual-sided effects.^[58] This design encourages strategic depth, as each lozenge influences both players' territories, blending connection mechanics with area majority goals. Digital adaptations further integrate lozenge grids into accessible recreational formats. The puzzle game Lozenge (2018), available on platforms like Steam, tasks players with filling irregular boards using randomly generated shapes, scoring based on coverage efficiency and combo chains.^[59] Such apps democratize tiling challenges, often incorporating procedural generation for replayability, and echo traditional polyiamond puzzles in a touch-friendly interface suitable for casual play.

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Arabic / Islamic geometry 02 - Catnaps design
... lozenge formed by four pentagons. When accuracy in laying tiles is certain, rectangular tiles are easy to lay. Lozenge shaped tiles take up inaccuracies ...
[47]
https://pmc.ncbi.nlm.nih.gov/articles/PMC7202229/
[48]
A Guide to English Gothic Architecture - The Historic England Blog
May 5, 2022 · The following examples trace the development of Gothic architecture in England. ... Its surface is adorned with a mesh of lozenges that are more ...Missing: tessellation | Show results with:tessellation
[49]
Romanesque architecture, an introduction - Smarthistory
The decoration is often quite simple, using geometric shapes rather than floral or curvilinear patterns. Common shapes used include diapers—squares or lozenges— ...
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https://heritagecalling.com/2022/05/05/what-is-gothic-architecture/
[51]
Art Deco Architecture - Colorado Preservation, Inc.
Stylized, geometric patterns used as decorative motifs. Patterns include zigzags, chevron and lozenge, sunrise, and floral. Stylized, low-relief figure ...
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Parametric House
Parametric House is a trusted platform for Grasshopper3D & Parametric design, offering tutorials, tools, and resources for architects & designers worldwide.
[53]
Pierresvives by Zaha Hadid - ICON Magazine
Dec 12, 2012 · The facade presents a striking geometry of floating lozenges that seem suspended in waves of green-tinted glass. The big curvaceous cutaways are ...
[54]
# 18 Islamic Architectural by Using Parametric Algorithms - Scribd
The document discusses the development of principles for Islamic architectural elements using parametric algorithms. It provides background on the history ...
[55]
(PDF) The Calissons Puzzle - ResearchGate
Jul 5, 2023 · The solutions involve equivalence between calisson tilings, stepped surfaces and certain DAG cuts that avoid passing through a set of edges that ...
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Lozenge on Steam
14-day returnsTry this new gameplay with an awesome twist. The objective here is to fill the board using the given random shapes. You score with every shape you put there.