Point
In geometry, a point is a fundamental primitive element defined as a location or position in space that possesses no dimension, extent, or parts.[1] This concept, originating in ancient Greek mathematics, forms the indivisible building block from which all other geometric entities—such as lines, surfaces, and solids—are constructed.[2] The classical definition of a point appears in Euclid's Elements, where it is described as "that which has no part," emphasizing its role as a zero-dimensional entity without breadth, length, or thickness.[1] In Euclidean geometry, points determine the endpoints of lines and the intersections of figures, enabling the precise description of spatial relationships.[3] For instance, two distinct points define a unique straight line, while the absence of common points between lines indicates parallelism.[4] In modern analytic geometry, points are represented using coordinate systems, where a point in the plane is denoted by an ordered pair of real numbers (x, y), corresponding to its distances along perpendicular axes from an origin.[5] This Cartesian approach, developed by René Descartes in the 17th century, bridges algebra and geometry by allowing points to be manipulated as numerical tuples, facilitating calculations in fields like physics and computer science.[6] Extensions to higher dimensions include points as n-tuples in \mathbb{R}^n, essential for vector spaces and multivariable calculus.[7] Beyond pure mathematics, the notion of a point extends to idealized models in physics, such as point particles with negligible size but defined position and momentum, though real-world applications often approximate this ideal due to quantum effects.[8] In computational geometry, points serve as vertices in algorithms for shape analysis, rendering, and data visualization, underscoring their versatility across disciplines.[9]Mathematics
Geometry and topology
In Euclidean geometry, a point is defined as a primitive entity that has position but no size, length, width, or depth, often described as "that which has no part."[10] This conceptualization serves as the foundational building block for constructing lines, planes, and other geometric figures, emphasizing its indivisibility and lack of extension.[10] Points function as primitives in axiomatic systems, where they are undefined basic terms from which all other geometric concepts are derived through axioms and theorems. In David Hilbert's Foundations of Geometry, points form one of three primitive notions alongside lines and planes, with axioms governing incidence, order, congruence, parallelism, and continuity to ensure a rigorous structure free of intuitive assumptions.[11] For instance, Hilbert's first axiom of incidence states that for every two distinct points, there exists a unique line containing them, establishing points as the atoms of geometric space.[11] In topology, a point is simply an element of a topological space X, where the space is equipped with a collection of open sets satisfying union, intersection, and containment properties.[12] Continuity at a point p \in X for a function f: X \to Y between topological spaces is defined such that for every neighborhood U of f(p) in Y, there exists a neighborhood V of p in X with f(V) \subseteq U, generalizing the geometric notion without relying on metrics.[12] Neighborhoods of a point are open sets containing it, providing the framework for local properties like connectedness and compactness.[12] A common way to specify a point in Euclidean space is through Cartesian coordinates, where a point in the plane is denoted as (x, y) relative to perpendicular axes intersecting at the origin, and in three dimensions as (x, y, z).[13] The distance between two points (x_1, y_1, z_1) and (x_2, y_2, z_2) is given by the Euclidean metric: d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}, which derives from the Pythagorean theorem and quantifies the straight-line separation in the space.[14] Fixed points in geometry include notable locations such as the center of a circle, which is the unique point equidistant from all points on the circumference and remains invariant under rotations about itself.[15] This property underscores the point's role as a symmetry pivot in circular configurations.[15]Analysis and other branches
In set theory, points function as the basic elements of sets, where a set is defined as a collection of distinct objects, and membership is denoted by the relation \in, such that p \in S indicates that point p belongs to set S.[16] A singleton set \{p\} is a specific construction containing precisely one such element p, guaranteed by the axiom of pairing, which asserts that for any objects x and y, there exists a set z = \{x, y\}, allowing singletons when x = y.[16] In real analysis, points constitute the elements within the domain of a function, serving as inputs where continuity and convergence are analyzed through neighborhoods or sequences. A limit point of a set E \subseteq \mathbb{R} is a point x such that every neighborhood of x contains at least one point of E distinct from x, equivalently, there exists a sequence in E \setminus \{x\} converging to x.[17] An accumulation point refines this by requiring that every neighborhood of x contains infinitely many points of E, ensuring the set clusters indefinitely around x; for instance, in the set E = \{1 - 1/n : n \in \mathbb{N}\}, the point 1 is an accumulation point.[17] Within metric spaces, points are classified as isolated or limit points based on their relation to the surrounding set S. An isolated point p \in S is one that is not a limit point, meaning there exists a neighborhood of p containing no other elements of S besides p itself.[18] In contrast, a limit point p of S has every neighborhood containing some q \in S with q \neq p, highlighting the topological density around such points; a set is closed if it includes all its limit points.[18] In complex analysis, points lie in the complex plane \mathbb{C}, where they represent locations for evaluating holomorphic functions, but certain points manifest as singularities that disrupt analyticity. A pole is an isolated singularity at a point z_0 where the function tends to infinity, classified by order n if the Laurent series principal part terminates with the term a_{-n} (z - z_0)^{-n} and a_{-n} \neq 0; for example, f(z) = 1/z has a simple pole (order 1) at z = 0.[19] Algebraic geometry treats points as the solutions to systems of polynomial equations, forming the zero loci that define affine or projective varieties. An affine algebraic set V(S) in \mathbb{A}^n consists of points ( \xi_1, \dots, \xi_n ) \in \mathbb{C}^n satisfying f(\xi_1, \dots, \xi_n) = 0 for all polynomials f in the ideal S \subseteq \mathbb{C}[x_1, \dots, x_n].[20] Intersection points of curves arise as common solutions to multiple such equations; Bézout's theorem establishes that two projective plane curves of degrees m and n without common components intersect in exactly mn points, counted with multiplicity, as seen in the four intersection points of two conics.[20]Natural sciences
Physics
In classical mechanics, a point particle serves as an idealized model representing an object with all its mass concentrated at a single geometric point, disregarding any finite size or internal structure to simplify the analysis of motion.[21] This approximation is valid for many macroscopic bodies where the scale of forces and trajectories far exceeds the object's dimensions, allowing the trajectory to be determined solely by external forces.[22] The dynamics of such a particle are governed by Newton's second law of motion, expressed as \mathbf{F} = m \mathbf{a}, where \mathbf{F} is the net force acting on the particle, m is its mass, and \mathbf{a} is its acceleration; this relation directly connects applied forces to changes in velocity, enabling predictions of paths under influences like gravity or friction.[23] In the framework of special relativity, points extend to four-dimensional spacetime as world points, which trace the history of a particle's motion through coordinates of time and space.[24] The geometry of this spacetime is flat and described by the Minkowski metric, given by ds^2 = -c^2 dt^2 + dx^2 + dy^2 + dz^2, where c is the speed of light, t is the time coordinate, and x, y, z are spatial coordinates; this metric quantifies invariant intervals between world points, ensuring the laws of physics remain consistent across inertial frames.[24] Unlike Euclidean space, the negative sign for the time component introduces hyperbolic structure, fundamental to relativistic effects such as time dilation and length contraction observed in high-speed phenomena.[25] In optics, the focal point represents the precise location where incident light rays, parallel to the optical axis, converge after refraction through a lens, forming a critical point for image formation in instruments like cameras and microscopes./University_Physics_III_-Optics_and_Modern_Physics(OpenStax)/02%3A_Geometric_Optics_and_Image_Formation/2.05%3A_Thin_Lenses) For a thin lens, the position of this point relative to the object and image is determined by the thin lens formula, \frac{1}{f} = \frac{1}{u} + \frac{1}{v}, where f is the focal length (a property of the lens material and curvature), u is the object distance from the lens, and v is the image distance; positive values indicate real images for converging lenses, while diverging lenses produce virtual focal points.[26] This equation derives from Snell's law applied to ray tracing, providing a foundational tool for designing optical systems with predictable ray convergence.[27] Point sources model emitters of waves or fields that originate from an infinitesimal location, assuming isotropic propagation in all directions, which simplifies calculations for phenomena like sound, light, or electromagnetic radiation.[28] A prominent example occurs in electrostatics, where the interaction between two point charges q_1 and q_2 separated by distance r obeys Coulomb's law, F = k \frac{q_1 q_2}{r^2}, with F as the magnitude of the electrostatic force, k as the Coulomb constant ($8.99 \times 10^9 \, \mathrm{N \cdot m^2 / C^2}), and the force directed along the line joining the charges; this inverse-square dependence arises from the geometry of flux spreading over a spherical surface./17%3A_Electric_Charge_and_Field/17.3%3A_Coulombs_Law) Such models extend to wave propagation, where point sources generate spherical wavefronts whose intensity diminishes as $1/r^2, underpinning approximations in antenna design and acoustic modeling.[29] In general relativity, singularities manifest as point-like regions where spacetime curvature diverges to infinity, often simplifying the description of extreme gravitational collapse.[30] For black holes, the central singularity is idealized as a zero-volume point containing all the mass, surrounded by an event horizon that acts as a one-way boundary for infalling matter and light.[31] These point-like models, derived from solutions like the Schwarzschild metric, highlight the breakdown of classical predictability at the singularity, motivating quantum gravity theories to resolve such infinities while preserving the horizon's role in observable phenomena like gravitational lensing.[30]Chemistry and biology
In chemistry, the boiling point of a substance is defined as the temperature at which its vapor pressure equals the surrounding atmospheric pressure, allowing the liquid to transition into a gas throughout the bulk of the material.[32] For water under standard atmospheric pressure of 1 atm, this occurs at 100°C, a value that exemplifies how boiling points are influenced by intermolecular forces such as hydrogen bonding, which require additional thermal energy to overcome for phase change.[33] Similarly, the melting point is the temperature at which a solid converts to a liquid, reflecting the energy needed to disrupt the ordered crystal lattice through overcoming intermolecular attractions like van der Waals forces or ionic bonds.[34] These phase transition points are fundamental in processes like purification and material synthesis, where variations in pressure or impurities can shift the temperatures.[33] The dew point, relevant to both chemistry and atmospheric science, represents the temperature at which air becomes fully saturated with water vapor, leading to condensation when cooled further at constant pressure.[35] This saturation point is crucial for understanding humidity and phase behavior in gaseous mixtures, as it indicates the onset of liquid formation from vapor without a defined boiling-like transition.[35] In phase diagrams, the critical point marks the endpoint of the liquid-gas coexistence curve, where the distinction between liquid and gas phases vanishes due to identical densities and properties under supercritical conditions.[36] For water, this occurs at 374°C and 218 atm, beyond which the substance exists as a supercritical fluid with unique solvating capabilities.[37] In biology, points often denote specific anatomical or structural features. Acupuncture points are discrete locations along the meridian system, a network of pathways in traditional Chinese medicine believed to channel vital energy (qi), with over 360 such points mapped on the human body for therapeutic stimulation.[38] The meridian system comprises 12 principal meridians linked to organ systems and extremities, plus eight extraordinary meridians, supported by anatomical evidence of connective tissue alignments that may facilitate signal transmission.[39] However, the existence and anatomical basis of meridians remain controversial in modern science, with many studies finding no distinct physiological structures corresponding to traditional descriptions. In zoology, antler points refer to the tines or projections on deer antlers, used in classifying trophy quality and age; for instance, points are categorized as typical (symmetric, arising from the main beam) or non-typical (irregular), influencing Boone and Crockett scoring systems for white-tailed deer.[40] Botanical growth points, such as apical meristems, are undifferentiated tissues at the tips of shoots and roots responsible for primary growth through continuous cell division, enabling elongation and organ formation.[41] These meristems, organized in a tunica-corpus structure in many angiosperms, produce new cells that differentiate into stems, leaves, and vascular tissues, with hormonal regulation like auxin promoting directional growth.[41] In deer biology, antler points also correlate with maturity, as yearling bucks typically exhibit fewer points that increase with age due to testosterone-driven development cycles.[40]Units of measurement
Typography and printing
In typography, the point serves as a fundamental unit for measuring type size and other elements in printing and design. Originating in the 18th century, the system was first formalized by French typefounder Pierre Simon Fournier in 1737, who divided the traditional cicéro type size into 12 points based on the French foot, establishing a scalable framework for punch-cutting and type composition.[42] Fournier's point measured approximately 0.349 mm, aiming to standardize measurements across varying national units.[43] This system was refined by François-Ambroise Didot in the late 18th century, who adjusted the point to exactly 1/864 of the French royal foot (or 1/72 of the royal inch), resulting in a size of about 0.376 mm or 0.0148 inches, which became the European standard known as the Didot point.[44] In the 20th century, the adoption of digital typography led to the PostScript point, defined precisely as 1/72 of an inch (approximately 0.0139 inches or 0.3528 mm), promoted by Adobe Systems to ensure consistency in desktop publishing and vector graphics.[45] This PostScript standard, slightly smaller than the Didot point, remains the global norm for font sizing in modern software.[46] Type sizes are expressed in points, referring to the height of the metal body or em square containing the glyphs; for instance, a 12-point font has a body height of 12 points, though the actual cap height or x-height varies by typeface design.[47] Related spacing units include the em, equal to the point size and historically the width of a capital M in the font, used for indents, paragraph spacing, and em dashes; and the en, half an em or the width of a capital N, applied to narrower gaps like en dashes and half-line spaces.[48] These units, derived from letter widths in hot-metal typesetting, facilitate proportional layout in both print and digital media.[49] The decimal point, or radix point, functions as a typographic symbol (typically '.' in English or ',' in many European languages) to separate the integer and fractional parts of numerals in printed matter.[50] The earliest known use of the decimal point dates to the 1440s in the work of Venetian merchant and astronomer Giovanni Bianchini; its modern usage was popularized by Scottish mathematician John Napier, who in his 1617 work Rabdologia employed both the period and comma as separators for decimal fractions, building on earlier concepts to simplify arithmetic in logarithmic tables and calculations.[51] In list formatting, bullet points—symbolized by a solid dot (•) or similar glyph—provide visual emphasis and structure to items, originating as an ancient punctuation mark in European manuscripts and evolving into a standard typographic device for unordered lists in print since the 19th century.Other measurement contexts
In statistics and economics, a percentage point serves as a unit to denote the arithmetic difference between two percentages, emphasizing absolute rather than relative change. For instance, if a rate rises from 20% to 25%, this constitutes a 5 percentage point increase, whereas the relative change would be 25% of the original value.[52] This distinction avoids confusion in reporting variations, such as in polling data or economic indicators where proportional growth might otherwise be misinterpreted.[53] In finance, the basis point (often abbreviated as bp or bps) represents one hundredth of a single percentage point, or 0.01%, facilitating precise quantification of minor adjustments in rates or yields. Commonly applied to interest rates, bond pricing, and credit spreads, a shift from 2.75% to 3.25% equates to a 50 basis point rise, enabling clear communication in trading and policy discussions.[54] This unit originated in the fixed-income markets to express small differentials without cumbersome decimals.[55] Educational systems employ the point as a numerical score corresponding to letter grades in computing the grade point average (GPA), a cumulative measure of academic performance. Under the standard 4.0 scale prevalent in many U.S. institutions, an A earns 4.0 points, a B 3.0 points, a C 2.0 points, a D 1.0 point, and an F 0 points; the GPA is then derived by summing (grade points × course credits) and dividing by total credits attempted.[56] Variations exist, such as plus/minus adjustments (e.g., A- at 3.7 points), but the core system prioritizes weighted averaging for holistic evaluation.[57]Language and notation
Punctuation and symbols
In writing systems, the point most commonly refers to the full stop or period (.), a punctuation mark used to indicate the end of a declarative sentence or an abbreviation. This symbol evolved from the ancient Greek stigme teleia, a dot placed high in the line to denote a long pause or sentence end, as introduced by the scholar Aristophanes of Byzantium in the 3rd century BCE for marking rhythmic breaks in poetry. By the 7th century CE, Isidore of Seville adapted it into a grammatical tool, distinguishing it from other dots for pauses of varying lengths, and it solidified as the modern period during the Renaissance with the rise of printing presses.[58] Variants of the point include the exclamation mark (!) and question mark (?), which emerged in medieval Europe to convey emotional emphasis and interrogative tone, respectively. The exclamation mark originated in the 15th century, likely derived from Latin notations like "io" (an expression of joy) overlaid with a period, evolving from musical notation in Gregorian chants to signal rising inflection and excitement. Similarly, the question mark traces to the medieval punctus interrogativus, a curved symbol indicating queries, which by the late 8th century had stylized into its current form to denote rising intonation in spoken language.[58][59] In historical typography, the colon (:) and semicolon (;) were considered related "points" as stacked or hybrid dot forms for intermediate pauses. The colon, consisting of two vertically aligned dots, evolved from medieval forms such as the punctus elevatus (a raised point from chant notation, resembling an inverted semicolon), gaining its modern role in the Renaissance for introducing lists or explanations. The semicolon, invented in 1494 by Italian printer Aldus Manutius in Venice, combined a comma with a period to signify a pause longer than a comma but shorter than a full stop, facilitating clearer separation in complex sentences during the early print era.[58][60] In programming languages, the point often denotes the dot operator (.), used for accessing members of structures or objects, as in C wherestruct_point.x retrieves the x field of a struct_point variable. This operator, introduced in the original C language specification in 1978, enables direct reference to components within composite data types, distinguishing it from the arrow operator (->) for pointers.[61]
Braille, a tactile writing system for the visually impaired, employs raised points—small embossed dots arranged in a 2x3 cell grid—to represent letters, numbers, and symbols. Invented by Louis Braille in 1824, each character uses up to six points numbered 1 through 6 (left column top to bottom: 1-3; right: 4-6), with configurations like dot 1 alone for "a" or dots 1-2-4 for "b," allowing blind readers to interpret text by touch through standardized patterns.[62]
The decimal point, a dot separating the integer and fractional parts of a number (e.g., 3.14), briefly references numerical notation but is distinct from sentence-ending uses, with early appearances in 1440s Italian merchant records predating widespread adoption.[63]