Dot
A dot is a small, round mark or spot used in writing, printing, and notation, most commonly as the punctuation symbol (.) known as the full stop, period, or point, which indicates the end of a sentence or clause in many languages.[1] It also serves to separate components in abbreviations (e.g., U.S.A.), decimal fractions (e.g., 3.14), domain names in internet addresses (e.g., example.com), and as a diacritic above or below letters in various scripts to modify pronunciation (e.g., ė or ạ).[2] In addition to its orthographic roles, the dot functions as the short signal in Morse code, contrasting with the dash, and appears in mathematical notations like the dot product (⋅) for vector multiplication or enumeration lists.[1] While its precise form varies slightly across typefaces and historical scripts—such as the interpunct (·) in ancient Greek or Latin for word separation—the modern baseline dot remains a fundamental glyph in typography, essential for clarity in textual structure and numerical precision.[3]Language and orthography
Punctuation marks
The dot, alternatively termed the period in American English or the full stop in British English, functions primarily as a punctuation mark to denote the termination of a declarative sentence, thereby signaling a complete thought.[4] It also concludes most abbreviations, such as "Dr." for Doctor or "etc." for et cetera, though conventions vary by style guide and context—formal writing typically retains it, while some modern digital communication omits it for brevity.[4] In English typography, the dot is placed at the baseline without preceding space, followed by one space before the next sentence, contrasting with practices in languages like French that insert a thin space before certain marks.[4] Its historical development stems from ancient Greek innovations in textual division. Aristophanes of Byzantium, a scholar active around 257–180 BCE in Alexandria, devised an early punctuation system employing dots at three heights: a high dot (thései) for a short pause, a middle dot (stigma) for a longer one, and a low dot (pístis) to mark the end of a complete clause or sentence, addressing the challenges of scriptio continua in classical manuscripts.[5] [6] This low dot gradually evolved into the modern period during the medieval period, influenced by Isidore of Seville's 7th-century descriptions of the punctus as a sentence-ending point, and was further refined in the 15th century by Venetian printer Aldus Manutius, whose italic typefaces helped standardize its form in printed books.[5] [7] The terminology "period" derives from the Greek periodos, denoting a full circuit or complete sentence, a usage attested in English by the 16th century; "full stop" emerged in British parlance around the same era to emphasize its role in halting the flow of speech or text.[7] American preference for "period" solidified in the early 20th century, possibly reflecting printing traditions or linguistic divergence from British norms.[5] Beyond the single dot, multiples form the ellipsis (three dots, ...), which signifies omission of words or trailing thought, with spacing conventions requiring no spaces between dots but spaces around the set in running text.[8] These applications underscore the dot's foundational role in clarifying syntactic boundaries, though overuse in abbreviations or informal texting has prompted debates on evolving standards in digital eras.[4]Diacritics and typographic uses
The dot functions as a diacritic known as the tittle in the Latin script, appearing as a superscript mark over lowercase i and j to distinguish their ascenders from adjacent strokes in medieval handwriting. This usage traces to 11th-century Latin manuscripts, where scribes added the dot to prevent confusion with letters like m, n, or u in minim-heavy texts.[9][10] In Turkish orthography, adopted via the 1928 alphabet reform, the dot differentiates the front close vowel /i/ (dotted i and uppercase İ) from the back close vowel /ɯ/ (dotless ı and I), reflecting distinct phonemes absent in many Indo-European languages.[11] Similar dotted/dotless distinctions appear in other Turkic languages like Azerbaijani and Kazakh, where uppercase forms preserve the feature to maintain vowel harmony.[11] The underdot (◌̣) modifies pronunciation in select orthographies; in Vietnamese, it denotes the nặng tone—a low, abrupt, glottalized falling contour on vowels, as in ạ (from underlying /a˧˨ʔ/), one of six lexical tones essential for lexical differentiation.[12] In Yoruba, an underdot on consonants like ṣ signals a postalveolar approximant or fricative, approximating English "sh" as in ṣùpù.[13] Typographically, the interpunct (·), or middle dot, originated in classical Latin as an interword separator before scriptio continua yielded to spaced writing around the 7th–9th centuries CE.[14] In modern dictionaries, it marks syllable boundaries, as in "ra·di·o", aiding pronunciation guidance without implying pauses. It also separates elements in horizontal lists or multilingual lines, providing a neutral, non-breaking delimiter lighter than bullets or commas.[15][16]Mathematics
Operators and notation
In mathematical notation, the dot operator, symbolized as ⋅ (Unicode U+22C5), primarily denotes multiplication between quantities, especially in algebraic contexts where other symbols like × might confuse with variables or cross products.[17] This usage avoids ambiguity in expressions involving letters resembling multiplication signs, such as x \cdot y rather than x \times y. Gottfried Wilhelm Leibniz first proposed the dot for multiplication in a letter to Johann Bernoulli on July 29, 1698, stating a preference over the × symbol due to its potential overlap with unknowns.[18] In modern typesetting, such as LaTeX, it is rendered via the \cdot command, distinguishing it from interpuncts or decimal points in non-mathematical text.[19] A key application of the dot operator occurs in linear algebra as the notation for the dot product (also termed scalar product or inner product) of two vectors of equal dimension. For vectors \vec{u} = (u_1, u_2, \dots, u_n) and \vec{v} = (v_1, v_2, \dots, v_n) in \mathbb{R}^n, the dot product is defined as \vec{u} \cdot \vec{v} = \sum_{i=1}^n u_i v_i, producing a scalar value equal to |\vec{u}| |\vec{v}| \cos \theta, where \theta is the angle between them.[17] [20] This bilinear form underpins projections, orthogonality checks (where \vec{u} \cdot \vec{v} = 0 indicates perpendicularity), and norms (|\vec{u}| = \sqrt{\vec{u} \cdot \vec{u}}). The notation gained prominence through Josiah Willard Gibbs's vector analysis in notes published around 1881–1884, where it complemented the cross product for three-dimensional operations.[21] In computational mathematics and tensor algebra, the dot extends to generalized products, such as matrix multiplication via associative chaining (e.g., A \cdot B \cdot C for compatible matrices A, B, C), equivalent to standard row-column contractions but explicitly marked for clarity in software like Wolfram Language.[22] However, conventions vary: juxtaposition often suffices for scalar multiplication in higher algebra, reserving ⋅ for vector-specific or emphatic cases, while element-wise multiplication typically employs the Hadamard product symbol ⊙ instead.[20] These distinctions ensure precision in derivations, with the dot's geometric interpretation—projecting one vector onto another—rooted in Euclidean space properties verified empirically through coordinate computations.[17]Geometric and analytical uses
In Euclidean geometry, the dot product of two vectors \mathbf{u} and \mathbf{v} yields a scalar value given by \mathbf{u} \cdot \mathbf{v} = \|\mathbf{u}\| \|\mathbf{v}\| \cos \theta, where \theta is the angle between the vectors and \|\cdot\| denotes the Euclidean norm.[23] This formulation links the algebraic operation to geometric properties, such as directionality and magnitude scaling.[24] Algebraically, for vectors in \mathbb{R}^n, it computes as \mathbf{u} \cdot \mathbf{v} = \sum_{i=1}^n u_i v_i, enabling coordinate-based analysis in analytic geometry.[25] A primary geometric application is determining the angle between two vectors or lines: \theta = \cos^{-1} \left( \frac{\mathbf{u} \cdot \mathbf{v}}{\|\mathbf{u}\| \|\mathbf{v}\|} \right), which ranges from 0° (parallel, same direction) to 180° (opposite directions)./12%3A_Vectors_in_Space/12.03%3A_The_Dot_Product) Vectors are orthogonal if \mathbf{u} \cdot \mathbf{v} = 0, as \cos 90^\circ = 0, providing a test for perpendicularity without explicit angle computation; this holds for coordinate axes in Cartesian systems, where standard basis vectors satisfy \mathbf{e}_i \cdot \mathbf{e}_j = 0 for i \neq j.[25] The dot product also defines vector length via \|\mathbf{v}\| = \sqrt{\mathbf{v} \cdot \mathbf{v}}, foundational for distance metrics in geometric spaces.[23] In analytic geometry, the dot product facilitates projections: the scalar projection of \mathbf{u} onto \mathbf{v} is \frac{\mathbf{u} \cdot \mathbf{v}}{\|\mathbf{v}\|}, measuring the component of \mathbf{u} along \mathbf{v}'s direction, while the vector projection is \left( \frac{\mathbf{u} \cdot \mathbf{v}}{\|\mathbf{v}\|^2} \right) \mathbf{v}.[24] These decompose vectors into parallel and perpendicular components relative to a line or plane, aiding derivations of equations for lines (e.g., parametric forms via direction vectors) and distances (e.g., point-to-line distance d = \frac{\|\mathbf{PQ} \times \mathbf{d}\|}{\|\mathbf{d}\|}, where cross product magnitude relates inversely to dot-based orthogonality checks)./12%3A_Vectors_in_Space/12.03%3A_The_Dot_Product) For planes, the normal vector \mathbf{n} satisfies orthogonality to the plane via \mathbf{n} \cdot \mathbf{v} = 0 for any vector \mathbf{v} in the plane, enabling plane equations like ax + by + cz = d.[25] Further analytical uses include verifying geometric identities, such as the Cauchy-Schwarz inequality |\mathbf{u} \cdot \mathbf{v}| \leq \|\mathbf{u}\| \|\mathbf{v}\|, which bounds the dot product and underpins convergence in inner product spaces.[23] In coordinate transformations, dot products compute direction cosines between axes, essential for orthogonal projections in crystallography or engineering drawings as of standard linear algebra texts from the early 20th century onward.[24] These operations remain computationally efficient, with O(n) complexity for n-dimensional vectors, supporting applications in computer graphics for lighting models via normal-vector dots as of implementations in OpenGL specifications circa 1992./12%3A_Vectors_in_Space/12.03%3A_The_Dot_Product)Science and technology
Physical and chemical phenomena
In chemistry, Lewis electron-dot symbols represent the valence electrons of an atom by placing dots around its elemental symbol, with a maximum of two dots per side to illustrate electron distribution and facilitate understanding of chemical bonding.[26] These symbols extend to Lewis structures, where shared electron pairs form covalent bonds depicted as lines between atoms, while unshared pairs remain as dots, enabling prediction of molecular geometry and reactivity based on the octet rule.[27] Introduced by Gilbert N. Lewis in his 1916 paper, this notation underscores phenomena such as electron pairing in stable molecules and the tendency of atoms to achieve noble gas configurations through bonding.[28] The dot symbol also appears in chemical formulas to denote non-covalent associations, such as in hydrated salts like copper(II) sulfate pentahydrate (CuSO₄·5H₂O), where it separates the anhydrous compound from waters of crystallization incorporated in the crystal lattice.[29] This convention highlights hydration phenomena, where water molecules stabilize ionic structures through hydrogen bonding and electrostatic interactions, influencing solubility and thermal decomposition behaviors, as water is released upon heating without altering the core formula.[26] In physical chemistry and materials science, quantum dots are nanoscale semiconductor particles, typically 2–10 nm in diameter, exhibiting quantum confinement effects that discretize energy levels akin to artificial atoms.[30] Their bandgap energy increases inversely with size due to electron wavefunction restriction, resulting in size-tunable photoluminescence where smaller dots emit higher-energy (blue-shifted) light; for instance, cadmium selenide dots of 2 nm emit blue, while 6 nm ones emit red.[31] This phenomenon, predicted in the 1970s and experimentally realized in colloidal solutions by Alexei Ekimov in 1983 and Louis Brus in 1984, enables applications in light-emitting devices and biological imaging by exploiting enhanced quantum yields and resistance to photobleaching.[30] The 2023 Nobel Prize in Chemistry recognized these developments for controlling matter at atomic scales.[32] Carbon dots, a class of fluorescent carbon-based nanoparticles under 10 nm, demonstrate similar photoluminescent properties arising from surface states and quantum confinement, with excitation-dependent emission spanning UV to near-infrared.[33] Synthesized via hydrothermal or pyrolysis methods from organic precursors, they exhibit biocompatibility and low toxicity compared to heavy-metal quantum dots, facilitating phenomena like energy transfer in photocatalytic reactions and cellular imaging without significant quenching.[34]Computing and digital formats
In computing, the dot (.) serves as a delimiter in filenames to separate the base name from the file extension, which indicates the file's format or type to operating systems and applications. This convention originated in early UNIX systems around 1970 and became widespread with MS-DOS in 1981, allowing software to infer handling based on the suffix following the dot.[35] For instance, a file named "document.txt" uses ".txt" to denote a plain text format, while ".exe" signals an executable on Windows. Multiple dots may appear, as in "archive.tar.gz", where ".tar.gz" represents a compressed tar archive, though only the final extension is typically parsed by default tools.[35] Files with the ".dot" extension specifically store templates in older versions of Microsoft Word, containing predefined formatting, styles, and macros for generating consistent documents; these were introduced with Word 97 in 1996 and largely superseded by .dotx in Office 2007 for XML-based storage.[36] Independently, ".dot" files also hold source code in the DOT language, a plain-text syntax for describing directed and undirected graphs, nodes, edges, and attributes, as implemented in the Graphviz toolkit developed by AT&T Bell Labs and first released in 1991.[37] DOT supports UTF-8 encoding by default and enables hierarchical layouts via subgraphs and clusters, with tools like the "dot" layout engine producing visualizations in formats such as SVG or PNG; for example, a simple graph is defined asdigraph G { A -> B; }.[37][38]
Dot notation, employing the dot operator, is a syntactic feature in object-oriented and structured programming languages for accessing object attributes, methods, or members, providing concise chaining over alternatives like bracket notation. In Python, introduced in 1991, it accesses instance variables as object.attribute, resolving via attribute lookup tables for efficiency. Similarly, in JavaScript (standardized ECMAScript since 1997), obj.property retrieves values from prototypes or scopes, with origins traceable to Smalltalk's message-passing syntax in the 1970s influencing modern implementations. Languages like Java and C# extend this for qualified names, such as ClassName.staticMethod(), enforcing encapsulation and namespace resolution at compile time.[39][40]
In networking, dot-decimal notation represents IPv4 addresses as four decimal octets separated by dots, e.g., 192.168.0.1, per Internet Engineering Task Force standards defined in RFC 791 (1981), enabling human-readable parsing of 32-bit binary addresses into ranges for subnetting and routing. Dots also delineate hierarchical components in domain names within URLs, separating subdomains, the second-level domain, and top-level domain (TLD), as in "www.example.com", per DNS specifications in RFC 1035 (1987); a trailing dot denotes the root zone for absolute resolution, preventing search domain appendage. Periods are encoded as %2E in URL paths to avoid misinterpretation as path separators, though unencoded dots remain valid in most components except leading/trailing in hosts.[41][42]