The Möbius strip has provided inspiration both for sculptures and for graphical art. M. C. Escher is one of the artists who was especially fond of it and based several of his lithographs on this mathematical object. One famous one, Möbius Strip II, features ants crawling around the surface of a Möbius strip. It is also a recurrent feature in science fiction stories, such as Arthur C. Clarke's The Wall of Darkness. Science fiction stories sometimes suggest that our universe might be some kind of generalised Möbius str ...

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Möbius strip, Möbius strip - Properties, Möbius strip - Geometry and topology, Möbius strip - Möbius strip with a circular boundary, Möbius strip - Related objects, Möbius strip - Art and technology

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Ad infinitum is a Latin phrase meaning "to infinity." In context, it usually means "continue forever," and thus can be used to describe a non-terminating process, a non-terminating repeating process, or a set of instructions to be repeated "forever," among other uses. It may also be used in a manner similar to the Latin phrase "et cetera" to denote written words or a concept that continues for a lengthy period beyond what is shown. Examples include: "The series 2, 4, 6, 8, 10 ... ...

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Belts are used to mechanically link two or more rotating items. They may be used as a source of motion, to transmit power between two points, or to track relative movement. As a source of motion, a conveyor belt is one application where the belt is adapted to continually carry a load between two points. A belt may also be looped (or crossed) between two points so that the direction of rotation is reversed at the other point. Power transmission is achieved by specially designed belts and pulleys. The demands on a belt drive transmission system are ...

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In geometry, a figure is chiral (and said to have chirality) if it is not identical to its mirror image, or more particularly if it cannot be mapped to its mirror image by rotations and translations alone. Such objects come in two forms, called enantiomorphs. The word chirality is derived from the Greek χειρ (cheir), the hand, the most familiar chiral object; the word enantiomorph stems from the Greek εναντιος (enantios) 'opposite' and μορφη (morphe) 'form'. A non-chiral figure is called Including:

• Chirality mathematics - Chirality and symmetry group
• Chirality mathematics - Chirality in three dimensions
• Chirality mathematics - Chirality in two dimensions
• Chirality mathematics - Knot theory

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August Ferdinand Möbius (November 17, 1790, Schulpforta, Saxony, Germany - September 26, 1868, Leipzig) was a German mathematician and theoretical astronomer. He is best known for his discovery of the Möbius strip, a non-orientable two-dimensional surface with only one side when embedded in three-dimensional Euclidean space. It was independently discovered by Johann Benedict Listing around the same time. Möbius was the first to introduce homogeneous coordinates into projective geometry. Möbius transformations, important in ...

Including:

• August Ferdinand Möbius - References:

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Infinity is a term with very distinct, separate meanings which arise in theology, philosophy, mathematics and everyday life. Popular or colloquial usage of the term often does not accord with its more technical meanings. The word infinity comes from Latin : "Infinito", unending. In theology, for example in the work of theologians such as Duns Scotus, the infinite nature of God invokes a sense of being without constraint, rather than a sense of being unlimited in quantity. In philosophy, infinity can be attrib ...

Including:

• Infinity - History
• Infinity - Ancient view of infinity
• Infinity - Views from the Renaissance to modern times
• Infinity - Modern philosophical views
• Infinity - Infinity symbol
• Infinity - Mathematical infinity
• Infinity - Infinity in real analysis
• Infinity - Infinity in complex analysis
• Infinity - Arithmetic properties of infinity
• Infinity - Infinity in set theory
• Infinity - Mathematics without infinity
• Infinity - Use of infinity in common speech
• Infinity - Physical infinity
• Infinity - Infinity in cosmology
• Infinity - Three types of infinities
• Infinity - Infinity in science fiction
• Infinity - Note

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In mathematics, a cross-cap is a two-dimensional surface that is topologically equivalent to a Möbius strip. The term 'cross-cap', however, often implies that the surface has been deformed so that its boundary is an ordinary circle. A cross-cap that has been closed up by gluing a disc to its boundary is called a real projective plane. Two cross-caps glued together at their boundaries form a Klein bottle. An important theorem of topology, the classification theorem for surfaces, states that all two-dimensional nonorientable manifolds are spheres with some ...

Including:

• Cross-cap - Cross-cap model of the real projective plane

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A joke is a short story or short series of words spoken or communicated with the intent of being laughed at or found humorous by the listener or reader. A practical joke differs in that the humour is not verbal, but mainly visual (e.g. putting a custard pie in somebody's face). Most jokes contain two components: joke setup (for example, "A man walks into a bar...") and a punchline, which, when juxtaposed with the setup, provides the necessary irony to elicit laughter from the audience. Joke - Psychology of jokesIncluding:

• Joke - Psychology of jokes
• Joke - Types of jokes
• Joke - Yo' mama jokes
• Joke - Political jokes
• Joke - Question–answer
• Joke - Dirty jokes
• Joke - Sick jokes
• Joke - Little Johnny jokes
• Joke - Ethnic jokes
• Joke - Sexist jokes
• Joke - Less offensive versions
• Joke - Blonde jokes
• Joke - Jokes about animals
• Joke - Shaggy dog stories
• Joke - You have two cows
• Joke - Duck jokes
• Joke - Religion in jokes
• Joke - Other classes of jokes

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A manifold is a mathematical space which is constructed, like a patchwork, by gluing and bending together copies of simple spaces. For example, a circle can be constructed by bending two line segments into arcs which overlap at their ends and gluing them together where they overlap. The motivation for working with manifolds is that you begin with a relatively simple space which is well understood, and build up a manifold, which may be very complicated, from copies of that simple space. By choosing different spaces as base material, di ...

Including:

• Manifold - Introduction
• Manifold - Motivational example: the circle
• Manifold - Charts atlases and transition maps
• Manifold - Construction
• Manifold - Charts
• Manifold - Patchwork
• Manifold - Zeros of a function
• Manifold - Identifying points of a manifold
• Manifold - Cartesian products
• Manifold - Gluing along boundaries
• Manifold - Topological manifolds
• Manifold - Differentiable manifolds
• Manifold - Orientability
• Manifold - Möbius strip
• Manifold - Klein bottle
• Manifold - Real projective plane
• Manifold - Other types and generalizations of manifolds
• Manifold - History

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In mathematics (topology), a surface is a two-dimensional manifold. Examples arise in three-dimensional space as the boundaries of three-dimensional solid objects. The surface of a fluid object, such as a rain drop or soap bubble, is an idealisation. To speak of the surface of a snowflake, which has a great deal of fine structure, is to go beyond the simple mathematical definition. For the nature of real surfaces see surface tension, ...

Including:

• Surface - Examples
• Surface - Definition
• Surface - Classification of closed surfaces
• Surface - Compact surfaces
• Surface - Embeddings in R³
• Surface - Differential geometry
• Surface - Some models
• Surface - Fundamental polygon
• Surface - Connected sum of surfaces
• Surface - Algebraic surface

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A single manifold can be constructed in different ways, each stressing a different aspect of the manifold, thereby leading to a slightly different viewpoint. Manifold - Charts. Perhaps the simplest way to construct a manifold is the one used in the example above of the circle. First, a subset of R2 is identified, and then an atlas covering this subset is constructed. The concept of manifold grew historically from constructions like this. Here is another example, applying this method ...

See also:

Manifold, Manifold - Introduction, Manifold - Motivational example: the circle, Manifold - Charts atlases and transition maps, Manifold - Construction, Manifold - Charts, Manifold - Patchwork, Manifold - Zeros of a function, Manifold - Identifying points of a manifold, Manifold - Cartesian products, Manifold - Manifold with boundary, Manifold - Gluing along boundaries, Manifold - Topological manifolds, Manifold - Differentiable manifolds, Manifold - Orientability, Manifold - Möbius strip, Manifold - Klein bottle, Manifold - Real projective plane, Manifold - Other types and generalizations of manifolds, Manifold - History

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Orientability, for surfaces, is easily defined, regardless of whether the surface is embedded in an ambient space or not. Any surface has a triangulation: a decomposition into triangles such that each edge on a triangle is glued to at most one other edge. We can orient each triangle, by choosing a direction for each edge (think of this as drawing an arrow on each edge) so that the arrows go from head to tail as we go around the boundary of the triangle. If we can do this so that in addition triangles sharing an edge have arrows on that edge ...

See also:

Orientability, Orientability - Examples in low dimensions, Orientability - Orientation by a triangulation, Orientability - Orientation by top-dimensional forms, Orientability - Orientation and vector bundles

Read more here: » Orientability: Encyclopedia II - Orientability - Orientation by a triangulation

Well known examples of his work include Drawing Hands, a work in which two hands are shown drawing each other, Sky and Water, in which light plays on shadow to morph fish in water into birds in the sky, and Ascending and Descending, in which lines of people ascend and descend stairs in an infinite loop, on a construction which is impossible to build and possible to draw only by taking advantage of ...

See also:

M.C. Escher, M.C. Escher - Youth, M.C. Escher - Marriage and later life, M.C. Escher - Works, M.C. Escher - Selected list of works, M.C. Escher - References in popular culture, M.C. Escher - Bibliography

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Consider a sphere, and let the great circles of the sphere be "lines", and let pairs of antipodal points be "points". It is easy to check that it obeys the axioms required of a projective plane: any pair of distinct great circles meet at a pair of antipodal points; and any two distinct pairs of antipodal points lie on a single great circle. This is the real projective plane. If we identify each point on the sphere with its antipodal point, then we get a representation of the real projective plane in ...

See also:

Real projective plane, Real projective plane - Formal construction

Read more here: » Real projective plane: Encyclopedia II - Real projective plane - Formal construction

The soundtrack features a number of contributions from Angelo Badalamenti, a consistent Lynch collaborator, as well as Barry Adamson, and Trent Reznor. Also appearing are tracks from David Bowie, Nine Inch Nails, The Smashing Pumpkins, Lou Reed, Marilyn Manson, and Rammstein. Some tracks were recorded at a sound studio in Prague. ...

See also:

Lost Highway, Lost Highway - Structure, Lost Highway - Responses, Lost Highway - Comparisons, Lost Highway - Principal cast, Lost Highway - Soundtrack, Lost Highway - Screenplay, Lost Highway - Opera, Lost Highway - Other uses

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Infinity - Ancient view of infinity. The earliest known documented knowledge of infinity is presented in the Hindu Yajur Veda (ca. 1800 BC - 800 BC) which states that "if you remove a part from infinity or add a part to infinity, still what remains is infinity". The Indian Jaina mathematical text Surya Prajinapti (ca. 400 BC) classifies all numbers into three sets: enumerable, innumerable and infinite. It recognises five different types of infinity: infinite in one and tw ...

See also:

Infinity, Infinity - History, Infinity - Ancient view of infinity, Infinity - Views from the Renaissance to modern times, Infinity - Modern philosophical views, Infinity - Infinity symbol, Infinity - Mathematical infinity, Infinity - Infinity in real analysis, Infinity - Infinity in complex analysis, Infinity - Arithmetic properties of infinity, Infinity - Infinity in set theory, Infinity - Mathematics without infinity, Infinity - Use of infinity in common speech, Infinity - Physical infinity, Infinity - Infinity in cosmology, Infinity - Three types of infinities, Infinity - Infinity in science fiction, Infinity - Note

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A function f : M → N between two Riemann surfaces M and N is called holomorphic if for every chart g in the atlas of M and every chart h in the atlas of N, the map h o f o g-1 is holomorphic (as a function from C to C) wherever it is defined. The composition of two holomorphic maps is holomorphic. The two Riemann surfaces M and N are called conformally equivalent if there exists a bijective holomorph ...

See also:

Riemann surface, Riemann surface - Formal definition, Riemann surface - Examples, Riemann surface - Properties and further definitions, Riemann surface - History, Riemann surface - In art and literature

Read more here: » Riemann surface: Encyclopedia II - Riemann surface - Properties and further definitions

Promethea - Promethea. Promethea is a young girl whose father is killed by a Christian mob in Alexandria in 411 AD. She is taken in hand by the twin gods Thoth and Hermes who tell her that if she goes with them into the Immateria - a plane of existence home to the imagination - she will no longer be just a little girl but a story living eternally. "Promethea" then is manifested in a series of avatars over the 19th and 20th centuries, culminating in the involvem ...

See also:

Promethea, Promethea - Plot summary, Promethea - Common themes, Promethea - Weeping Gorilla Comix, Promethea - Experimental media, Promethea - Characters, Promethea - Promethea, Promethea - Published Collections, Promethea - Awards & Recognition

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The Euler class satisfies these useful properties: Functoriality: If is another oriented, real vector bundle and is continuous and covered by an orientation-preserving map , then e(F) = f * e(E). In particular, e(f * E) = f * e(E). Orientation: If is E with the opposite orientation, then . ...

See also:

Euler class, Euler class - Formal definition, Euler class - Properties, Euler class - Relations to other invariants, Euler class - Example: Line bundles over the circle

Read more here: » Euler class: Encyclopedia II - Euler class - Properties

Jokes often depend for humour on the unexpected, the mildly taboo (which can include the distasteful or socially improper), or the playing on stereotypes and other cultural myths. Many jokes fit into more than one category. Joke - Mathematical jokes. Main article: Mathematical joke There are numerous jokes related to mathematics. Many of them are in-jokes, bu ...

See also:

Joke, Joke - Psychology of jokes, Joke - Types of jokes, Joke - Mathematical jokes, Joke - Yo' mama jokes, Joke - Political jokes, Joke - Self-deprecating humor, Joke - Question–answer, Joke - Dirty jokes, Joke - Sick jokes, Joke - Little Johnny jokes, Joke - Ethnic jokes, Joke - Sexist jokes, Joke - Less offensive versions, Joke - Blonde jokes, Joke - Jokes about animals, Joke - Shaggy dog stories, Joke - You have two cows, Joke - Duck jokes, Joke - Religion in jokes, Joke - Other classes of jokes, Joke - External links

Read more here: » Joke: Encyclopedia II - Joke - Types of jokes