|
Absolute Geometry
Absolute geometry is a geometry based on an axiom system for Euclidean geometry without the parallel postulate or any of its alternatives. Traditionally, this has meant using only the first four of Euclid's postulates. The term was introduced by János Bolyai in 1832. It is sometimes referred to as neutral geometry, as it is neutral with respect to the parallel postulate. The first four of Euclid's postulates are now considered insufficient as a basis of Euclidean geometry, so other systems (such as Hilbert's axioms without the parallel axiom) are used instead. Properties In Euclid's ''Elements'', the first 28 Propositions and Proposition 31 avoid using the parallel postulate, and therefore are valid in absolute geometry. One can also prove in absolute geometry the exterior angle theorem (an exterior angle of a triangle is larger than either of the remote angles), as well as the Saccheri–Legendre theorem, which states that the sum of the measures of the angles in a triang ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometry
Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician who works in the field of geometry is called a ''List of geometers, geometer''. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point (geometry), point, line (geometry), line, plane (geometry), plane, distance, angle, surface (mathematics), surface, and curve, as fundamental concepts. Originally developed to model the physical world, geometry has applications in almost all sciences, and also in art, architecture, and other activities that are related to graphics. Geometry also has applications in areas of mathematics that are apparently unrelated. For example, methods of algebraic geometry are fundamental in Wiles's proof of Fermat's Last Theorem, Wiles's proof of Fermat's ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Circle
A circle is a shape consisting of all point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre. The distance between any point of the circle and the centre is called the radius. The length of a line segment connecting two points on the circle and passing through the centre is called the diameter. A circle bounds a region of the plane called a Disk (mathematics), disc. The circle has been known since before the beginning of recorded history. Natural circles are common, such as the full moon or a slice of round fruit. The circle is the basis for the wheel, which, with related inventions such as gears, makes much of modern machinery possible. In mathematics, the study of the circle has helped inspire the development of geometry, astronomy and calculus. Terminology * Annulus (mathematics), Annulus: a ring-shaped object, the region bounded by two concentric circles. * Circular arc, Arc: any Connected ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Completeness (logic)
In mathematical logic and metalogic, a formal system is called complete with respect to a particular property if every formula having the property can be derived using that system, i.e. is one of its theorems; otherwise the system is said to be incomplete. The term "complete" is also used without qualification, with differing meanings depending on the context, mostly referring to the property of semantical validity. Intuitively, a system is called complete in this particular sense, if it can derive every formula that is true. Other properties related to completeness The property converse to completeness is called soundness: a system is sound with respect to a property (mostly semantical validity) if each of its theorems has that property. Forms of completeness Expressive completeness A formal language is ''expressively complete'' if it can express the subject matter for which it is intended. Functional completeness A set of logical connectives associated with a formal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Congruence (geometry)
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. More formally, two sets of points are called congruent if, and only if, one can be transformed into the other by an isometry, i.e., a combination of rigid motions, namely a translation, a rotation, and a reflection. This means that either object can be repositioned and reflected (but not resized) so as to coincide precisely with the other object. Therefore, two distinct plane figures on a piece of paper are congruent if they can be cut out and then matched up completely. Turning the paper over is permitted. In elementary geometry the word ''congruent'' is often used as follows. The word ''equal'' is often used in place of ''congruent'' for these objects. *Two line segments are congruent if they have the same length. *Two angles are congruent if they have the same measure. *Two circles are congruent if they have ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ordered Geometry
Ordered geometry is a form of geometry featuring the concept of intermediacy (or "betweenness") but, like projective geometry, omitting the basic notion of measurement. Ordered geometry is a fundamental geometry forming a common framework for affine, Euclidean, absolute, and hyperbolic geometry (but not for projective geometry). History Moritz Pasch first defined a geometry without reference to measurement in 1882. His axioms were improved upon by Peano (1889), Hilbert (1899), and Veblen (1904). Euclid anticipated Pasch's approach in definition 4 of ''The Elements'': "a straight line is a line which lies evenly with the points on itself". Primitive concepts The only primitive notions in ordered geometry are points ''A'', ''B'', ''C'', ... and the ternary relation of intermediacy 'ABC''which can be read as "''B'' is between ''A'' and ''C''". Definitions The ''segment'' ''AB'' is the set of points ''P'' such that 'APB'' The ''interval'' ''AB'' is the segment ''AB'' and it ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Incidence (geometry)
In geometry, an incidence relation is a heterogeneous relation that captures the idea being expressed when phrases such as "a point ''lies on'' a line" or "a line is ''contained in'' a plane" are used. The most basic incidence relation is that between a point, , and a line, , sometimes denoted . If ''P'' and ''l'' are incident, , the pair is called a ''flag''. There are many expressions used in common language to describe incidence (for example, a line ''passes through'' a point, a point ''lies in'' a plane, etc.) but the term "incidence" is preferred because it does not have the additional connotations that these other terms have, and it can be used in a symmetric manner. Statements such as "line intersects line " are also statements about incidence relations, but in this case, it is because this is a shorthand way of saying that "there exists a point that is incident with both line and line ". When one type of object can be thought of as a set of the other type of object (''vi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Frames Of Reference
In physics and astronomy, a frame of reference (or reference frame) is an abstract coordinate system, whose origin, orientation, and scale have been specified in physical space. It is based on a set of reference points, defined as geometric points whose position is identified both mathematically (with numerical coordinate values) and physically (signaled by conventional markers). An important special case is that of '' inertial reference frames'', a stationary or uniformly moving frame. For ''n'' dimensions, reference points are sufficient to fully define a reference frame. Using rectangular Cartesian coordinates, a reference frame may be defined with a reference point at the origin and a reference point at one unit distance along each of the ''n'' coordinate axes. In Einsteinian relativity, reference frames are used to specify the relationship between a moving observer and the phenomenon under observation. In this context, the term often becomes observational frame o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperbolic Rotation
In linear algebra, a squeeze mapping, also called a squeeze transformation, is a type of linear map that preserves Euclidean area of regions in the Cartesian plane, but is ''not'' a rotation or shear mapping. For a fixed positive real number , the mapping :(x, y) \mapsto (ax, y/a) is the ''squeeze mapping'' with parameter . Since :\ is a hyperbola, if and , then and the points of the image of the squeeze mapping are on the same hyperbola as is. For this reason it is natural to think of the squeeze mapping as a hyperbolic rotation, as did Émile Borel in 1914, by analogy with ''circular rotations'', which preserve circles. Logarithm and hyperbolic angle The squeeze mapping sets the stage for development of the concept of logarithms. The problem of finding the area bounded by a hyperbola (such as is one of quadrature. The solution, found by Grégoire de Saint-Vincent and Alphonse Antonio de Sarasa in 1647, required the natural logarithm function, a new concept. Some insight ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wayback Machine
The Wayback Machine is a digital archive of the World Wide Web founded by Internet Archive, an American nonprofit organization based in San Francisco, California. Launched for public access in 2001, the service allows users to go "back in time" to see how websites looked in the past. Founders Brewster Kahle and Bruce Gilliat developed the Wayback Machine to provide "universal access to all knowledge" by preserving archived copies of defunct web pages. The Wayback Machine's earliest archives go back at least to 1995, and by the end of 2009, more than 38.2 billion webpages had been saved. As of November 2024, the Wayback Machine has archived more than 916 billion web pages and well over 100 petabytes of data. History The Internet Archive has been archiving cached web pages since at least 1995. One of the earliest known pages was archived on May 8, 1995. Internet Archive founders Brewster Kahle and Bruce Gilliat launched the Wayback Machine in San Francisco, California ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
American Academy Of Arts And Sciences
The American Academy of Arts and Sciences (The Academy) is one of the oldest learned societies in the United States. It was founded in 1780 during the American Revolution by John Adams, John Hancock, James Bowdoin, Andrew Oliver, and other Founding Fathers of the United States. It is headquartered in Cambridge, Massachusetts. Membership in the academy is achieved through a nominating petition, review, and election process. The academy's quarterly journal, '' Dædalus'', is published by the MIT Press on behalf of the academy, and has been open-access since January 2021. The academy also conducts multidisciplinary public policy research. Laurie L. Patton has served as President of the Academy since January 2025. History The Academy was established by the Massachusetts legislature on May 4, 1780, charted in order "to cultivate every art and science which may tend to advance the interest, honor, dignity, and happiness of a free, independent, and virtuous people." The sixty-tw ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gilbert N
Gilbert may refer to: People and fictional characters *Gilbert (given name), including a list of people and fictional characters * Gilbert (surname), including a list of people Places Australia * Gilbert River (Queensland) * Gilbert River (South Australia) Kiribati * Gilbert Islands, a chain of atolls and islands in the Pacific Ocean United States * Gilbert, Arizona, a town * Gilbert, Arkansas, a town * Gilbert, Florida, the airport of Winterhaven * Gilbert, Iowa, a city * Gilbert, Louisiana, a village * Gilbert, Michigan, and unincorporated community * Gilbert, Minnesota, a city * Gilbert, Nevada, ghost town * Gilbert, Ohio, an unincorporated community * Gilbert, Pennsylvania, an unincorporated community * Gilbert, South Carolina, a town * Gilbert, West Virginia, a town * Gilbert, Wisconsin, an unincorporated community * Mount Gilbert (other), various mountains * Gilbert River (Oregon) Outer space * Gilbert (lunar crater) * Gilbert (Martian crater) Arts a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Edwin B
The name Edwin means "wealth-friend". It comes from (wealth, good fortune) and (friend). Thus the Old English form is Ēadwine, a name widely attested in early medieval England. Edwina is the feminine form of the name. Notable people and characters with the name include: Historical figures * Edwin of Northumbria (died 632 or 633), King of Northumbria and Christian saint * Edwin (son of Edward the Elder) (died 933) * Eadwine of Sussex (died 982), Ealdorman of Sussex * Eadwine of Abingdon (died 990), Abbot of Abingdon * Edwin, Earl of Mercia (died 1071), brother-in-law of Harold Godwinson (Harold II) * Edwin Sandys (bishop) (1519–1588), Archbishop of York Modern era * E. W. Abeygunasekera, Sri Lankan Sinhala politician * Edwin Abbott Abbott (1838–1926), English schoolmaster, theologian, and Anglican priest * Edwin Ariyadasa (1922–2021), Sri Lankan Sinhala journalist * Edwin Arrieta Arteaga (died 2023), Colombian murder victim * Edwin Austin Abbey (1852–191 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
