Möbius strip - Article Index

Encyclopedia - Möbius Strip: Encyclopedia Ii - Möbius Strip - Art And Technology
The Möbius strip has provided inspiration both for sculptures and for graphical art. M. C. Escher is one of the artists who was especial...   » Read the article

Encyclopedia - Ad Infinitum: Encyclopedia - Ad Infinitum
Ad infinitum is a Latin phrase meaning "to infinity." In context, it usually means "continue forever," and thus can be used to describe a...   » Read the article

Encyclopedia - Belt Mechanical: Encyclopedia - Belt Mechanical
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 poi...   » Read the article

Encyclopedia - Chirality Mathematics: Encyclopedia - Chirality Mathematics
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 canno...   » Read the article

Encyclopedia - August Ferdinand Möbius: Encyclopedia - August Ferdinand Möbius
August Ferdinand Möbius (November 17, 1790, Schulpforta, Saxony, Germany - September 26, 1868, Leipzig) was a German mathematician and t...   » Read the article

Encyclopedia - Infinity: Encyclopedia - Infinity
Infinity is a term with very distinct, separate meanings which arise in theology, philosophy, mathematics and everyday life. Popular or c...   » Read the article

Encyclopedia - Cross-cap: Encyclopedia - Cross-cap
In mathematics, a cross-cap is a two-dimensional surface that is topologically equivalent to a Möbius strip. The term 'cross-cap', howev...   » Read the article

Encyclopedia - Joke: Encyclopedia - Joke
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 list...   » Read the article

Encyclopedia - Manifold: Encyclopedia - Manifold
A manifold is a mathematical space which is constructed, like a patchwork, by gluing and bending together copies of simple spaces. For ex...   » Read the article

Encyclopedia - Surface: Encyclopedia - Surface
In mathematics (topology), a surface is a two-dimensional manifold. Examples arise in three-dimensional space as the boundaries of three-...   » Read the article

Encyclopedia - Manifold: Encyclopedia Ii - Manifold - Construction
A single manifold can be constructed in different ways, each stressing a different aspect of the manifold, thereby leading to a slightly ...   » Read the article

Encyclopedia - Orientability: Encyclopedia Ii - Orientability - Orientation By A Triangulation
Orientability, for surfaces, is easily defined, regardless of whether the surface is embedded in an ambient space or not. Any surface has...   » Read the article

Encyclopedia - M.c. Escher: Encyclopedia Ii - M.c. Escher - Works
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 li...   » Read the article

Encyclopedia - Real Projective Plane: Encyclopedia Ii - Real Projective Plane - Formal Construction
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...   » Read the article

Encyclopedia - Lost Highway: Encyclopedia Ii - Lost Highway - Soundtrack
The soundtrack features a number of contributions from Angelo Badalamenti, a consistent Lynch collaborator, as well as Barry Adamson, and...   » Read the article

Encyclopedia - Infinity: Encyclopedia Ii - Infinity - History
Infinity - Ancient view of infinity. The earliest known documented knowledge of infinity is presented in the Hindu Yajur Veda (ca. 1800...   » Read the article

Encyclopedia - Riemann Surface: Encyclopedia Ii - Riemann Surface - Properties And Further Definitions
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 c...   » Read the article

Encyclopedia - Promethea: Encyclopedia Ii - Promethea - Characters
Promethea - Promethea. Promethea is a young girl whose father is killed by a Christian mob in Alexandria in 411 AD. She is taken in han...   » Read the article

Encyclopedia - Euler Class: Encyclopedia Ii - Euler Class - Properties
The Euler class satisfies these useful properties: Functoriality: If is another oriented, real vector bundle and is continuous and cov...   » Read the article

Encyclopedia - Joke: Encyclopedia Ii - Joke - Types Of Jokes
Jokes often depend for humour on the unexpected, the mildly taboo (which can include the distasteful or socially improper), or the playin...   » Read the article

Encyclopedia - Infinity: Encyclopedia Ii - Infinity - History
Infinity - Ancient view of infinity. The earliest known documented knowledge of infinity is presented in the Veda- Yajur Veda which sta...   » Read the article

Encyclopedia - Manifold: Encyclopedia Ii - Manifold - Construction
A single manifold can be constructed in different ways, each stressing a different aspect of the manifold, thereby leading to a slightly ...   » Read the article

Encyclopedia - Octopus Card: Encyclopedia Ii - Octopus Card - Obtaining And Using An Octopus Card
An on-loan Octopus card can be purchased at Mass Transit Railway (MTR) and Kowloon Canton Railway (KCR) stations. No identification is re...   » Read the article

Encyclopedia - Klein Bottle: Encyclopedia Ii - Klein Bottle - Properties
Topologically, the Klein bottle can be defined as the square [0,1] × [0,1] with sides identified by the relations (0,y) ~ (1,y) for 0
...   » Read the article

Encyclopedia - Surface Integral: Encyclopedia Ii - Surface Integral - Advanced Issues
Let us notice that we defined the surface integral by using a parametrization of the surface S. We know that a given surface might have s...   » Read the article

Encyclopedia - Chirality Mathematics: Encyclopedia Ii - Chirality Mathematics - Chirality In Three Dimensions
In three dimensions, every figure which possesses a plane of symmetry or a center of symmetry is achiral. (A plane of symmetry of a figur...   » Read the article

Encyclopedia - Joke: Encyclopedia Ii - Joke - Types Of Jokes
Jokes often depend for humour on the unexpected, the mildly taboo (which can include the distasteful or socially improper), or the playin...   » Read the article

Encyclopedia - Surface: Encyclopedia Ii - Surface - Definition
In what follows, all surfaces are considered to be second-countable 2-dimensional manifolds. More precisely: a topological surface (with ...   » Read the article

Encyclopedia - Fiber Bundle: Encyclopedia Ii - Fiber Bundle - Formal Definition
A fiber bundle consists of the data (E, B, π, F), where E, B, and F are topological spaces and π : E → B is a continuous surject...   » Read the article

Encyclopedia - Infinity: Encyclopedia Ii - History
Infinity - Ancient view of infinity. The earliest known documented knowledge of infinity is presented in the Hindu Yajur Veda (ca. 1800...   » Read the article

Encyclopedia - Infinity: Encyclopedia Ii - Use Of Infinity In Common Speech
In common parlance, infinity is often used in a hyperbolic sense. For example, "The movie was infinitely boring, but we had to wait forev...   » Read the article

Encyclopedia - Infinity: Encyclopedia Ii - Physical Infinity
In physics, approximations of real numbers are used for continuous measurements and natural numbers are used for discrete measurements (i...   » Read the article

Encyclopedia - Infinity: Encyclopedia Ii - Mathematical Infinity
Infinity - Infinity in real analysis. In real analysis, the symbol , called "infinity", denotes an unbounded limit. means that x grows...   » Read the article

Encyclopedia - Manifold: Encyclopedia Ii - Manifold - Charts, Atlases And Transition Maps
Charts A coordinate map, a coordinate chart, or simply a chart of a manifold is an invertible map between a subset of the manifold and a ...   » Read the article

Encyclopedia - Infinity: Encyclopedia Ii - Infinity In Science Fiction
The Hitchhiker's Guide to the Galaxy contains the following definition of infinity: "Bigger than the biggest thing ever and then some, m...   » Read the article

Encyclopedia - Joke: Encyclopedia Ii - Joke - Psychology Of Jokes
Why we laugh has been the subject of serious academic study, examples being: Sigmund Freud's "Jokes and Their Relationship to the Uncons...   » Read the article

Encyclopedia - Riemann Surface: Encyclopedia Ii - Riemann Surface - Formal Definition
Let X be a Hausdorff space. A homeomorphism from an open subset U⊂X to a subset of C is called a chart. Two charts f and g whose domain...   » Read the article

Encyclopedia - Euler Class: Encyclopedia Ii - Euler Class - Formal Definition
The Euler class e(E) is an element of the integral cohomology group , constructed as follows. An orientation of E amounts to a continuo...   » Read the article

Encyclopedia - Manifold: Encyclopedia Ii - Manifold - History
The first to have conceived clearly of curves and surfaces as spaces by themselves was possibly Carl Friedrich Gauss, the founder of intr...   » Read the article

Encyclopedia - Octopus Card: Encyclopedia Ii - Octopus Card - Operator
As of 2005, Octopus Cards Limited (OCL), the operator of Octopus, is a joint-venture between six transit companies, namely MTR Corporatio...   » Read the article

Encyclopedia - Octopus Card: Encyclopedia Ii - Octopus Card - Back-end Technology And Operations
The Octopus system was designed by AES ProData (Hong Kong) Limited, now known as ERG Transit Systems, a member of the ERG Group based in ...   » Read the article

Encyclopedia - Octopus Card: Encyclopedia Ii - Octopus Card - History
The MTR network adopted a system of recirculated magnetic plastic cards when it started operations in 1979. These cards were either used ...   » Read the article

Encyclopedia - Octopus Card: Encyclopedia Ii - Octopus Card - Adoption Of The Octopus Card
The Octopus system was launched in 1997, and 3 million cards were issued within the first three months. The main reason for the quick suc...   » Read the article

Encyclopedia - Octopus Card: Encyclopedia Ii - Octopus Card - Comparison With Other Transit Card Systems
Mass transit agencies have been using stored value, pre-paid cards for electronic ticketing since the 1970s. This market started to move ...   » Read the article

Encyclopedia - Octopus Card: Encyclopedia Ii - Octopus Card - Comparison With Other Electronic Cash Systems
Mondex specifically cited the widespread popularity of Octopus as the reason for withdrawing from the Hong Kong market in 2002. This is d...   » Read the article

Encyclopedia - Octopus Card: Encyclopedia Ii - Octopus Card - Name And Logo
The Chinese name for the Octopus card literally means "eight places pass." Eight is a significant number in Chinese in that it is often u...   » Read the article

Encyclopedia - Surface Integral: Encyclopedia Ii - Surface Integral - Surface Integrals Of Differential 2-forms
Let be a differential 2-form defined on the surface S, and let be an orientation preserving parametrization of S with (s,t) in D. T...   » Read the article

Encyclopedia - Manifold: Encyclopedia Ii - Manifold - Introduction
A manifold is a space that looks, locally, like a Euclidean space of some fixed dimension. This may be one of the familiar one, two, or t...   » Read the article

Encyclopedia - Infinity: Encyclopedia Ii - Infinity - Use Of Infinity In Common Speech
In common parlance, infinity is often used in a hyperbolic sense. For example, "The movie was infinitely boring, but we had to wait forev...   » Read the article

Encyclopedia - Manifold: Encyclopedia Ii - Manifold - Motivational Example: The Circle
The circle is the simplest example of a topological manifold after Euclidean space itself. Consider, for instance, the circle of radius 1...   » Read the article

Encyclopedia - Manifold: Encyclopedia Ii - Manifold - Charts Atlases And Transition Maps
Charts A coordinate map, a coordinate chart, or simply a chart of a manifold is an invertible map between a subset of the manifold and a ...   » Read the article

Encyclopedia - Manifold: Encyclopedia Ii - Manifold - Orientability
Consider a topological manifold with charts mapping to Rn. Given an ordered basis for Rn, a chart causes its piece of the manifold to its...   » Read the article

Encyclopedia - Manifold: Encyclopedia Ii - Manifold - Topological Manifolds
The simplest kind of manifold to define is the topological manifold, which looks locally like some "ordinary" Euclidean space Rn. Formall...   » Read the article

Encyclopedia - Euler Class: Encyclopedia Ii - Euler Class - Relations To Other Invariants
In the special case when the bundle E in question is the tangent bundle of a compact, oriented, r-dimensional manifold, the Euler class i...   » Read the article

Encyclopedia - Manifold: Encyclopedia Ii - Manifold - Introduction
A manifold is a space that looks, locally, like a Euclidean space of some fixed dimension. This may be one of the familiar one, two, or t...   » Read the article

Encyclopedia - Fiber Bundle: Encyclopedia Ii - Fiber Bundle - Structure Groups And Transition Functions
Fiber bundles often come with a group of symmetries which describe the matching conditions between overlapping local trivialization chart...   » Read the article

Encyclopedia - Fiber Bundle: Encyclopedia Ii - Fiber Bundle - Sections
A section (or cross section) of a fiber bundle is a continuous map f : B → E such that π(f(x))=x for all x in B. Since bundles do...   » Read the article

Encyclopedia - Klein Bottle: Encyclopedia Ii - Klein Bottle - Parametrization
The "figure 8" immersion of the Klein bottle has a particularly simple parametrization: In this immersion, the self-intersection cir...   » Read the article

Encyclopedia - M.c. Escher: Encyclopedia Ii - M.c. Escher - Youth
Maurits Cornelis, or Mauk as he was to be nicknamed, was born in Leeuwarden (Friesland), the Netherlands. He was the youngest son of civi...   » Read the article

Encyclopedia - Surface Integral: Encyclopedia Ii - Surface Integral - Surface Integrals Of Vector Fields
Consider a vector field v on S, that is, for each x in S, v(x) is a vector. Then the integral of v over S is called the flux. Imagine tha...   » Read the article

Encyclopedia - M.c. Escher: Encyclopedia Ii - M.c. Escher - Marriage And Later Life
Escher travelled to Italy regularly in the following years. It was in Italy that he first met Jetta Umiker, the woman whom he married in ...   » Read the article

Encyclopedia - Fiber Bundle: Encyclopedia Ii - Fiber Bundle - Examples
Let E = B × F and let π : E → B be the projection onto the first factor. Then E is a fiber bundle over B. Here E is not just loc...   » Read the article

Encyclopedia - Surface: Encyclopedia Ii - Surface - Connected Sum Of Surfaces
Given two surfaces M and M', their connected sum M # M' is obtained by removing a disk in each of them and gluing them along the newly fo...   » Read the article

Encyclopedia - Chirality Mathematics: Encyclopedia Ii - Chirality Mathematics - Chirality In Two Dimensions
In two dimensions, every figure which possesses an axis of symmetry is achiral, and it can be shown that every bounded achiral figure mus...   » Read the article

Encyclopedia - Joke: Encyclopedia Ii - Joke - Psychology Of Jokes
Why we laugh has been the subject of serious academic study, examples being: Sigmund Freud's "Jokes and Their Relationship to the Uncons...   » Read the article

Encyclopedia - Surface: Encyclopedia Ii - Surface - Classification Of Closed Surfaces
There is a complete classification of closed (i.e compact without boundary) connected, surfaces up to homeomorphism. Any such surface fal...   » Read the article

Encyclopedia - Surface: Encyclopedia Ii - Surface - Some Models
To make some models of various surfaces, attach the sides of these squares (A with A, B with B) so that the directions of the arrows matc...   » Read the article

Encyclopedia - Surface: Encyclopedia Ii - Surface - Fundamental Polygon
Each closed surface can be constructed from an even sided oriented polygon, called a fundamental polygon by pairwise identification of it...   » Read the article

Encyclopedia - Orientability: Encyclopedia Ii - Orientability - Orientation By Top-dimensional Forms
Another way of thinking about orientability is thinking of it as a choice of "right handedness" vs. "left handedness" at each point in th...   » Read the article

Encyclopedia - Orientability: Encyclopedia Ii - Orientability - Examples In Low Dimensions
Surfaces we normally encounter in every day life are orientable. For example, sphere, plane, torus. Example of non-orientable surfaces ar...   » Read the article

Encyclopedia - Manifold: Encyclopedia Ii - Manifold - Differentiable Manifolds
For more details on this topic, see differentiable manifold. It is easy to define topological manifolds, but it is very hard to work w...   » Read the article

Encyclopedia - Manifold: Encyclopedia Ii - Manifold - Topological Manifolds
For more details on this topic, see topological manifold. The simplest kind of manifold to define is the topological manifold, which l...   » Read the article

Encyclopedia - Manifold: Encyclopedia Ii - Manifold - Orientability
Consider a topological manifold with charts mapping to Rn. Given an ordered basis for Rn, a chart causes its piece of the manifold to its...   » Read the article

Encyclopedia - Manifold: Encyclopedia Ii - Manifold - History
The first to have conceived clearly of curves and surfaces as spaces by themselves was possibly Carl Friedrich Gauss, the founder of intr...   » Read the article

Encyclopedia - Surface Integral: Encyclopedia Ii - Surface Integral - Surface Integrals Of Scalar Fields
Consider a surface S on which a scalar field f is defined. If we think of S as made of some material, and for each x in S the number f(x)...   » Read the article

Encyclopedia - Infinity: Encyclopedia Ii - Infinity - Mathematical Infinity
Infinity - Infinity in real analysis. In real analysis, the symbol , called "infinity", denotes an unbounded limit. means that x grows...   » Read the article

Encyclopedia - Manifold: Encyclopedia Ii - Manifold - Charts Atlases And Transition Maps
Charts A coordinate map, a coordinate chart, or simply a chart of a manifold is an invertible map between a subset of the manifold and a ...   » Read the article

Encyclopedia - Manifold: Encyclopedia Ii - Manifold - Motivational Example: The Circle
The circle is the simplest example of a topological manifold after Euclidean space itself. Consider, for instance, the circle of radius 1...   » Read the article

Encyclopedia - Infinity: Encyclopedia Ii - Infinity - Use Of Infinity In Common Speech
In common parlance, infinity is often used in a hyperbolic sense. For example, "The movie was infinitely boring, but we had to wait forev...   » Read the article

Encyclopedia - Infinity: Encyclopedia Ii - Infinity - Infinity In Science Fiction
The Hitchhiker's Guide to the Galaxy contains the following definition of infinity: "Bigger than the biggest thing ever and then some, m...   » Read the article

Encyclopedia - Infinity: Encyclopedia Ii - Infinity - Physical Infinity
In physics, approximations of real numbers are used for continuous measurements and natural numbers are used for discrete measurements (i...   » Read the article

Encyclopedia - Promethea: Encyclopedia Ii - Promethea - Common Themes
The series has been both criticized for acting as a mouthpiece for Moore's religious beliefs and praised for the beauty of its artwork an...   » Read the article

Encyclopedia - Promethea: Encyclopedia Ii - Promethea - Published Collections
Starting with Book 2, the trade paperbacks for Promethea were first released in hardcover, a rare occurrence for collections of regularly...   » Read the article

Encyclopedia - Möbius Strip: Encyclopedia Ii - Möbius Strip - Related Objects
A closely related "strange" geometrical object is the Klein bottle. A Klein bottle can be produced by gluing two Möbius strips together ...   » Read the article

Encyclopedia - Möbius Strip: Encyclopedia Ii - Möbius Strip - Möbius Strip With A Circular Boundary
Topologically, the boundary of a Möbius strip is a circle. Under the usual embeddings of the strip in Euclidean space, as above, this bo...   » Read the article

Encyclopedia - Möbius Strip: Encyclopedia Ii - Möbius Strip - Geometry And Topology
One way to represent the Möbius strip as a subset of R3 is using the parametrization: where and . This creates a Möbius strip...   » Read the article