153 relations: Abraham Robinson, Affine geometry, Albert Einstein, Alessandro Padoa, Alexandria, Alfred Tarski, Algebra, Analytic geometry, Andrey Kolmogorov, Angle, Angle bisector theorem, Angle trisection, Apollonius of Perga, Archimedean property, Archimedes, Architecture, Area, August Ferdinand Möbius, Axiom, Axiomatic system, Banach–Tarski paradox, Bertrand Russell, Birkhoff's axioms, Bounded function, Butterfly theorem, Cartesian coordinate system, Ceva's theorem, Circle, Classical logic, Compass-and-straightedge construction, Computer-aided design, Computer-aided manufacturing, Congruence (geometry), Consistency, Constructive proof, Coordinate system, Corresponding sides and corresponding angles, Curvature, Decidability (logic), Deductive reasoning, Degree (angle), Doubling the cube, Eleatics, Elliptic geometry, Error detection and correction, Euclid, Euclid's Elements, Euclidean distance, Euclidean relation, Existence theorem, ..., First-order logic, Formal system, Formalism (philosophy of mathematics), Foundationalism, Gödel's incompleteness theorems, General relativity, Geometric series, Geometric transformation, Geometrical optics, Geometry, Girard Desargues, Giuseppe Peano, Giuseppe Veronese, Global Positioning System, Gottfried Wilhelm Leibniz, Gravity, Greek mathematics, Gunter's chain, Heron's formula, Hilbert's axioms, Homogeneity and heterogeneity, Howard Eves, Incidence geometry, Infinitesimal, Infinity, Intuitionistic type theory, Irrational number, Isaac Newton, Isotropy, János Bolyai, Lazare Carnot, Lebesgue measure, Leonhard Euler, Level (instrument), Line (geometry), Line segment, List of interactive geometry software, Logical equivalence, Mathematical induction, Mathematical proof, Mathematics of paper folding, Menelaus's theorem, Method of exhaustion, Metric space, Moritz Pasch, Nikolai Lobachevsky, Nine-point circle, Non-Euclidean geometry, Number theory, Orange (fruit), Ordered geometry, Origami, Otto Stolz, Packing problems, Parallel postulate, Paul du Bois-Reymond, Peano axioms, Perspective (graphical), Philip Ehrlich, Pierre Wantzel, Plane (geometry), Platonic solid, Playfair's axiom, Point (geometry), Pons asinorum, Prime number, Primitive notion, Proclus, Projective geometry, Proof by contradiction, Proposition, Pythagorean theorem, Radian, Rational number, Real closed field, Real number, Rectangle, René Descartes, Right angle, Scientific modelling, Secondary school, Set theory, Similarity (geometry), Solid geometry, Sphere packing, Squaring the circle, Surveying, Synthetic geometry, Tarski's axioms, Thales of Miletus, Thales's theorem, Theodolite, Theorem, Theory of relativity, Thomas Little Heath, Three-dimensional space, Topology, Torus, Transitive relation, Type theory, University of New South Wales, Volume, Zeno's paradoxes. Expand index (103 more) » « Shrink index
Abraham Robinson (born Robinsohn; October 6, 1918 – April 11, 1974) was a mathematician who is most widely known for development of non-standard analysis, a mathematically rigorous system whereby infinitesimal and infinite numbers were reincorporated into modern mathematics.
In mathematics, affine geometry is what remains of Euclidean geometry when not using (mathematicians often say "when forgetting") the metric notions of distance and angle.
Albert Einstein (14 March 1879 – 18 April 1955) was a German-born theoretical physicist who developed the theory of relativity, one of the two pillars of modern physics (alongside quantum mechanics).
Alessandro Padoa (14 October 1868 – 25 November 1937) was an Italian mathematician and logician, a contributor to the school of Giuseppe Peano.
Alexandria (or; Arabic: الإسكندرية; Egyptian Arabic: إسكندرية; Ⲁⲗⲉⲝⲁⲛⲇⲣⲓⲁ; Ⲣⲁⲕⲟⲧⲉ) is the second-largest city in Egypt and a major economic centre, extending about along the coast of the Mediterranean Sea in the north central part of the country.
Alfred Tarski (January 14, 1901 – October 26, 1983), born Alfred Teitelbaum,School of Mathematics and Statistics, University of St Andrews,, School of Mathematics and Statistics, University of St Andrews.
Algebra (from Arabic "al-jabr", literally meaning "reunion of broken parts") is one of the broad parts of mathematics, together with number theory, geometry and analysis.
In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system.
Andrey Nikolaevich Kolmogorov (a, 25 April 1903 – 20 October 1987) was a 20th-century Soviet mathematician who made significant contributions to the mathematics of probability theory, topology, intuitionistic logic, turbulence, classical mechanics, algorithmic information theory and computational complexity.
In plane geometry, an angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle.
In geometry, the angle bisector theorem is concerned with the relative lengths of the two segments that a triangle's side is divided into by a line that bisects the opposite angle.
Angle trisection is a classical problem of compass and straightedge constructions of ancient Greek mathematics.
Apollonius of Perga (Ἀπολλώνιος ὁ Περγαῖος; Apollonius Pergaeus; late 3rdearly 2nd centuries BC) was a Greek geometer and astronomer known for his theories on the topic of conic sections.
In abstract algebra and analysis, the Archimedean property, named after the ancient Greek mathematician Archimedes of Syracuse, is a property held by some algebraic structures, such as ordered or normed groups, and fields.
Archimedes of Syracuse (Ἀρχιμήδης) was a Greek mathematician, physicist, engineer, inventor, and astronomer.
Architecture is both the process and the product of planning, designing, and constructing buildings or any other structures.
Area is the quantity that expresses the extent of a two-dimensional figure or shape, or planar lamina, in the plane.
August Ferdinand Möbius (17 November 1790 – 26 September 1868) was a German mathematician and theoretical astronomer.
An axiom or postulate is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments.
In mathematics, an axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems.
The Banach–Tarski paradox is a theorem in set-theoretic geometry, which states the following: Given a solid ball in 3‑dimensional space, there exists a decomposition of the ball into a finite number of disjoint subsets, which can then be put back together in a different way to yield two identical copies of the original ball.
Bertrand Arthur William Russell, 3rd Earl Russell, (18 May 1872 – 2 February 1970) was a British philosopher, logician, mathematician, historian, writer, social critic, political activist, and Nobel laureate.
In 1932, G. D. Birkhoff created a set of four postulates of Euclidean geometry in the plane, sometimes referred to as Birkhoff's axioms.
In mathematics, a function f defined on some set X with real or complex values is called bounded, if the set of its values is bounded.
The butterfly theorem is a classical result in Euclidean geometry, which can be stated as follows:Johnson, Roger A., Advanced Euclidean Geometry, Dover Publ., 2007 (orig. 1929).
A Cartesian coordinate system is a coordinate system that specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular directed lines, measured in the same unit of length.
Ceva's theorem is a theorem about triangles in Euclidean plane geometry.
A circle is a simple closed shape.
Classical logic (or standard logic) is an intensively studied and widely used class of formal logics.
Compass-and-straightedge construction, also known as ruler-and-compass construction or classical construction, is the construction of lengths, angles, and other geometric figures using only an idealized ruler and compass.
Computer-aided design (CAD) is the use of computer systems to aid in the creation, modification, analysis, or optimization of a design.
Computer-aided manufacturing (CAM) is the use of software to control machine tools and related ones in the manufacturing of workpieces.
In geometry, two figures or objects are congruent if they have the same shape and size, or if one has the same shape and size as the mirror image of the other.
In classical deductive logic, a consistent theory is one that does not contain a contradiction.
In mathematics, a constructive proof is a method of proof that demonstrates the existence of a mathematical object by creating or providing a method for creating the object.
In geometry, a coordinate system is a system which uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space.
In geometry, the tests for congruence and similarity involve comparing corresponding sides and corresponding angles of polygons.
In mathematics, curvature is any of a number of loosely related concepts in different areas of geometry.
In logic, the term decidable refers to the decision problem, the question of the existence of an effective method for determining membership in a set of formulas, or, more precisely, an algorithm that can and will return a boolean true or false value that is correct (instead of looping indefinitely, crashing, returning "don't know" or returning a wrong answer).
Deductive reasoning, also deductive logic, logical deduction is the process of reasoning from one or more statements (premises) to reach a logically certain conclusion.
A degree (in full, a degree of arc, arc degree, or arcdegree), usually denoted by ° (the degree symbol), is a measurement of a plane angle, defined so that a full rotation is 360 degrees.
Doubling the cube, also known as the Delian problem, is an ancient geometric problem.
The Eleatics were a pre-Socratic school of philosophy founded by Parmenides in the early fifth century BC in the ancient town of Elea.
Elliptic geometry is a geometry in which Euclid's parallel postulate does not hold.
In information theory and coding theory with applications in computer science and telecommunication, error detection and correction or error control are techniques that enable reliable delivery of digital data over unreliable communication channels.
Euclid (Εὐκλείδης Eukleidēs; fl. 300 BC), sometimes given the name Euclid of Alexandria to distinguish him from Euclides of Megara, was a Greek mathematician, often referred to as the "founder of geometry" or the "father of geometry".
The Elements (Στοιχεῖα Stoicheia) is a mathematical treatise consisting of 13 books attributed to the ancient Greek mathematician Euclid in Alexandria, Ptolemaic Egypt c. 300 BC.
In mathematics, the Euclidean distance or Euclidean metric is the "ordinary" straight-line distance between two points in Euclidean space.
In mathematics, Euclidean relations are a class of binary relations that formalizes "Axiom 1" in Euclid's Elements: "Magnitudes which are equal to the same are equal to each other.".
In mathematics, an existence theorem is a theorem with a statement beginning 'there exist(s)..', or more generally 'for all,,...
First-order logic—also known as first-order predicate calculus and predicate logic—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science.
A formal system is the name of a logic system usually defined in the mathematical way.
In foundations of mathematics, philosophy of mathematics, and philosophy of logic, formalism is a theory that holds that statements of mathematics and logic can be considered to be statements about the consequences of certain string manipulation rules.
Foundationalism concerns philosophical theories of knowledge resting upon justified belief, or some secure foundation of certainty such as a conclusion inferred from a basis of sound premises.
Gödel's incompleteness theorems are two theorems of mathematical logic that demonstrate the inherent limitations of every formal axiomatic system containing basic arithmetic.
General relativity (GR, also known as the general theory of relativity or GTR) is the geometric theory of gravitation published by Albert Einstein in 1915 and the current description of gravitation in modern physics.
In mathematics, a geometric series is a series with a constant ratio between successive terms.
A geometric transformation is any bijection of a set having some geometric structure to itself or another such set.
Geometrical optics, or ray optics, describes light propagation in terms of rays.
Geometry (from the γεωμετρία; geo- "earth", -metron "measurement") is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space.
Girard Desargues (21 February 1591 – September 1661) was a French mathematician and engineer, who is considered one of the founders of projective geometry.
Giuseppe Peano (27 August 1858 – 20 April 1932) was an Italian mathematician and glottologist.
Giuseppe Veronese (7 May 1854 – 17 July 1917) was an Italian mathematician.
The Global Positioning System (GPS), originally Navstar GPS, is a satellite-based radionavigation system owned by the United States government and operated by the United States Air Force.
Gottfried Wilhelm (von) Leibniz (or; Leibnitz; – 14 November 1716) was a German polymath and philosopher who occupies a prominent place in the history of mathematics and the history of philosophy.
Gravity, or gravitation, is a natural phenomenon by which all things with mass or energy—including planets, stars, galaxies, and even light—are brought toward (or gravitate toward) one another.
Greek mathematics refers to mathematics texts and advances written in Greek, developed from the 7th century BC to the 4th century AD around the shores of the Eastern Mediterranean.
Gunter's chain or the surveyor's chain (also known as Gunter’s measurement or surveyor’s measurement) is a distance measuring device used for land survey.
In geometry, Heron's formula (sometimes called Hero's formula), named after Hero of Alexandria, gives the area of a triangle by requiring no arbitrary choice of side as base or vertex as origin, contrary to other formulae for the area of a triangle, such as half the base times the height or half the norm of a cross product of two sides.
Hilbert's axioms are a set of 20 assumptions proposed by David Hilbert in 1899 in his book Grundlagen der Geometrie (tr. The Foundations of Geometry) as the foundation for a modern treatment of Euclidean geometry.
Homogeneity and heterogeneity are concepts often used in the sciences and statistics relating to the uniformity in a substance or organism.
Howard Whitley Eves (10 January 1911, New Jersey – 6 June 2004) was an American mathematician, known for his work in geometry and the history of mathematics.
In mathematics, incidence geometry is the study of incidence structures.
In mathematics, infinitesimals are things so small that there is no way to measure them.
Infinity (symbol) is a concept describing something without any bound or larger than any natural number.
Intuitionistic type theory (also known as constructive type theory, or Martin-Löf type theory) is a type theory and an alternative foundation of mathematics.
In mathematics, the irrational numbers are all the real numbers which are not rational numbers, the latter being the numbers constructed from ratios (or fractions) of integers.
Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, astronomer, theologian, author and physicist (described in his own day as a "natural philosopher") who is widely recognised as one of the most influential scientists of all time, and a key figure in the scientific revolution.
Isotropy is uniformity in all orientations; it is derived from the Greek isos (ἴσος, "equal") and tropos (τρόπος, "way").
János Bolyai (15 December 1802 – 27 January 1860) or Johann Bolyai, was a Hungarian mathematician, one of the founders of non-Euclidean geometry — a geometry that differs from Euclidean geometry in its definition of parallel lines.
Lazare Nicolas Marguerite, Count Carnot (13 May 1753 – 2 August 1823) was a French mathematician, physicist and politician.
In measure theory, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of n-dimensional Euclidean space.
Leonhard Euler (Swiss Standard German:; German Standard German:; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, logician and engineer, who made important and influential discoveries in many branches of mathematics, such as infinitesimal calculus and graph theory, while also making pioneering contributions to several branches such as topology and analytic number theory.
A level is a surveying optical instrument used to establish or verify points in the same horizontal plane.
The notion of line or straight line was introduced by ancient mathematicians to represent straight objects (i.e., having no curvature) with negligible width and depth.
In geometry, a line segment is a part of a line that is bounded by two distinct end points, and contains every point on the line between its endpoints.
Interactive geometry software (IGS) or dynamic geometry environments (DGEs) are computer programs which allow one to create and then manipulate geometric constructions, primarily in plane geometry.
In logic, statements p and q are logically equivalent if they have the same logical content.
Mathematical induction is a mathematical proof technique.
In mathematics, a proof is an inferential argument for a mathematical statement.
The art of origami or paper folding has received a considerable amount of mathematical study.
Menelaus's theorem, named for Menelaus of Alexandria, is a proposition about triangles in plane geometry.
The method of exhaustion (methodus exhaustionibus, or méthode des anciens) is a method of finding the area of a shape by inscribing inside it a sequence of polygons whose areas converge to the area of the containing shape.
In mathematics, a metric space is a set for which distances between all members of the set are defined.
Moritz Pasch (8 November 1843, Breslau, Prussia (now Wrocław, Poland) – 20 September 1930, Bad Homburg, Germany) was a German mathematician of Jewish ancestry specializing in the foundations of geometry.
Nikolai Ivanovich Lobachevsky (a; –) was a Russian mathematician and geometer, known primarily for his work on hyperbolic geometry, otherwise known as Lobachevskian geometry and also his fundamental study on Dirichlet integrals known as Lobachevsky integral formula.
In geometry, the nine-point circle is a circle that can be constructed for any given triangle.
In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those specifying Euclidean geometry.
Number theory, or in older usage arithmetic, is a branch of pure mathematics devoted primarily to the study of the integers.
The orange is the fruit of the citrus species ''Citrus'' × ''sinensis'' in the family Rutaceae.
Ordered geometry is a form of geometry featuring the concept of intermediacy (or "betweenness") but, like projective geometry, omitting the basic notion of measurement.
) is the art of paper folding, which is often associated with Japanese culture.
Otto Stolz (3 July 1842 – 23 November 1905) was an Austrian mathematician noted for his work on mathematical analysis and infinitesimals.
Packing problems are a class of optimization problems in mathematics that involve attempting to pack objects together into containers.
In geometry, the parallel postulate, also called Euclid's fifth postulate because it is the fifth postulate in Euclid's ''Elements'', is a distinctive axiom in Euclidean geometry.
Paul David Gustav du Bois-Reymond (2 December 1831 – 7 April 1889) was a German mathematician who was born in Berlin and died in Freiburg.
In mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are axioms for the natural numbers presented by the 19th century Italian mathematician Giuseppe Peano.
Perspective (from perspicere "to see through") in the graphic arts is an approximate representation, generally on a flat surface (such as paper), of an image as it is seen by the eye.
Philip Ehrlich is Professor at Department of Philosophy of Ohio University.
Pierre Laurent Wantzel (5 June 1814 in Paris – 21 May 1848 in Paris) was a French mathematician who proved that several ancient geometric problems were impossible to solve using only compass and straightedge.
In mathematics, a plane is a flat, two-dimensional surface that extends infinitely far.
In three-dimensional space, a Platonic solid is a regular, convex polyhedron.
In geometry, Playfair's axiom is an axiom that can be used instead of the fifth postulate of Euclid (the parallel postulate): In a plane, given a line and a point not on it, at most one line parallel to the given line can be drawn through the point.
In modern mathematics, a point refers usually to an element of some set called a space.
In geometry, the statement that the angles opposite the equal sides of an isosceles triangle are themselves equal is known as the pons asinorum, typically translated as "bridge of asses".
A prime number (or a prime) is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers.
In mathematics, logic, and formal systems, a primitive notion is an undefined concept.
Proclus Lycaeus (8 February 412 – 17 April 485 AD), called the Successor (Greek Πρόκλος ὁ Διάδοχος, Próklos ho Diádokhos), was a Greek Neoplatonist philosopher, one of the last major classical philosophers (see Damascius).
Projective geometry is a topic in mathematics.
In logic, proof by contradiction is a form of proof, and more specifically a form of indirect proof, that establishes the truth or validity of a proposition.
The term proposition has a broad use in contemporary analytic philosophy.
In mathematics, the Pythagorean theorem, also known as Pythagoras' theorem, is a fundamental relation in Euclidean geometry among the three sides of a right triangle.
The radian (SI symbol rad) is the SI unit for measuring angles, and is the standard unit of angular measure used in many areas of mathematics.
In mathematics, a rational number is any number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator.
In mathematics, a real closed field is a field F that has the same first-order properties as the field of real numbers.
In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line.
In Euclidean plane geometry, a rectangle is a quadrilateral with four right angles.
René Descartes (Latinized: Renatus Cartesius; adjectival form: "Cartesian"; 31 March 1596 – 11 February 1650) was a French philosopher, mathematician, and scientist.
In geometry and trigonometry, a right angle is an angle of exactly 90° (degrees), corresponding to a quarter turn.
Scientific modelling is a scientific activity, the aim of which is to make a particular part or feature of the world easier to understand, define, quantify, visualize, or simulate by referencing it to existing and usually commonly accepted knowledge.
A secondary school is both an organization that provides secondary education and the building where this takes place.
Set theory is a branch of mathematical logic that studies sets, which informally are collections of objects.
Two geometrical objects are called similar if they both have the same shape, or one has the same shape as the mirror image of the other.
In mathematics, solid geometry is the traditional name for the geometry of three-dimensional Euclidean space.
In geometry, a sphere packing is an arrangement of non-overlapping spheres within a containing space.
Squaring the circle is a problem proposed by ancient geometers.
Surveying or land surveying is the technique, profession, and science of determining the terrestrial or three-dimensional positions of points and the distances and angles between them.
Synthetic geometry (sometimes referred to as axiomatic or even pure geometry) is the study of geometry without the use of coordinates or formulas.
Tarski's axioms, due to Alfred Tarski, are an axiom set for the substantial fragment of Euclidean geometry, called "elementary," that is formulable in first-order logic with identity, and requiring no set theory.
Thales of Miletus (Θαλῆς (ὁ Μιλήσιος), Thalēs; 624 – c. 546 BC) was a pre-Socratic Greek philosopher, mathematician, and astronomer from Miletus in Asia Minor (present-day Milet in Turkey).
In geometry, Thales's theorem states that if A, B, and C are distinct points on a circle where the line is a diameter, then the angle ∠ABC is a right angle.
A theodolite is a precision instrument for measuring angles in the horizontal and vertical planes.
In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and generally accepted statements, such as axioms.
The theory of relativity usually encompasses two interrelated theories by Albert Einstein: special relativity and general relativity.
Sir Thomas Little Heath (5 October 1861 – 16 March 1940) was a British civil servant, mathematician, classical scholar, historian of ancient Greek mathematics, translator, and mountaineer.
Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called parameters) are required to determine the position of an element (i.e., point).
In mathematics, topology (from the Greek τόπος, place, and λόγος, study) is concerned with the properties of space that are preserved under continuous deformations, such as stretching, crumpling and bending, but not tearing or gluing.
In geometry, a torus (plural tori) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis coplanar with the circle.
In mathematics, a binary relation over a set is transitive if whenever an element is related to an element and is related to an element then is also related to.
In mathematics, logic, and computer science, a type theory is any of a class of formal systems, some of which can serve as alternatives to set theory as a foundation for all mathematics.
The University of New South Wales (UNSW; branded as UNSW Sydney) is an Australian public research university located in the Sydney suburb of Kensington.
Volume is the quantity of three-dimensional space enclosed by a closed surface, for example, the space that a substance (solid, liquid, gas, or plasma) or shape occupies or contains.
Zeno's paradoxes are a set of philosophical problems generally thought to have been devised by Greek philosopher Zeno of Elea (c. 490–430 BC) to support Parmenides' doctrine that contrary to the evidence of one's senses, the belief in plurality and change is mistaken, and in particular that motion is nothing but an illusion.
Applications of Euclidean geometry, Axioms of geometry, Classical geometry, Euclid axioms, Euclid fifth postulate, Euclid postulates, Euclid's axioms, Euclid's fourth postulate, Euclid's postulates, Euclid's second postulate, Euclid's third postulate, Euclidean Geometry, Euclidean axioms, Euclidean geometry of the plane, Euclidean plane geometry, Euclidian geometry, Fundamental concepts of geometry, Geometry Postulates, Geometry in R2, Geometry of Euclid, Noncoordinate geometry, Orthogonal geometry, Planar geometry, Plane Geometry, Plane geometry, Real plane, Two dimensional geometry.