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Polygons
A Wisdom Archive on Polygons

Polygons A selection of articles related to Polygons

We recommend this article: Polygons - 1, and also this: Polygons - 2.

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ARTICLES RELATED TO Polygons

Polygons: Encyclopedia II - Gunpowder warfare - Cannons

Gunpowder weapons had been used in China centuries before cannon appeared in Europe. They appeared in Europe in the late Middle Ages; however, for a long time European gunpowder weapons were large, unwieldy and difficult to deploy. As a result they were mainly used for attacking castles and other defences, a task that was equally well suited to undermining or non-explosive weapons. The development of siege cannon did have an important effect: it quickly made medieval castles obsolete. For several decades warfare greatly favoured the attacker ...

See also:

Gunpowder warfare, Gunpowder warfare - Cannons, Gunpowder warfare - Beginning of polygonal fortifications, Gunpowder warfare - Firearms, Gunpowder warfare - Nature of war, Gunpowder warfare - Outside of Europe, Gunpowder warfare - Ottoman Empire, Gunpowder warfare - Japan, Gunpowder warfare - Naval warfare

Read more here: » Gunpowder warfare: Encyclopedia II - Gunpowder warfare - Cannons

Polygons: Encyclopedia II - Gunpowder warfare - Outside of Europe

The gunpowder era of warfare is largely confined to Europe. This was also the time of the beginning of European exploration and colonial expansion and the lack of any significant intermediary period of gunpowder warfare proved decisive. Peoples in The Americas, Asia, and Africa fighting with medieval or even ancient warfare techniques were at a great disadvantage even if they were only a few years behind developments in Europe. Thus much of the world was annexed to European empir ...

See also:

Gunpowder warfare, Gunpowder warfare - Cannons, Gunpowder warfare - Beginning of polygonal fortifications, Gunpowder warfare - Firearms, Gunpowder warfare - Nature of war, Gunpowder warfare - Outside of Europe, Gunpowder warfare - Ottoman Empire, Gunpowder warfare - Japan, Gunpowder warfare - Naval warfare

Read more here: » Gunpowder warfare: Encyclopedia II - Gunpowder warfare - Outside of Europe

Polygons: Encyclopedia II - Ruler-and-compass construction - Ruler and compass

The "ruler" and "compass" of ruler-and-compass constructions is an idealization of rulers and compasses in the real world: The ruler is infinitely long, but it has no markings on it and has only one edge (thus making it a straightedge instead of what we usually think of as a ruler). The only thing you can use it for is to draw a line segment between two points, or to extend an existing line. The compass can be opened arbitrarily wide, but (unlike most real compasses) it also has no markings on it. ...

See also:

Ruler-and-compass construction, Ruler-and-compass construction - Ruler and compass, Ruler-and-compass construction - Constructible points and lengths, Ruler-and-compass construction - Impossible constructions, Ruler-and-compass construction - Squaring the circle, Ruler-and-compass construction - Doubling the cube, Ruler-and-compass construction - Angle trisection, Ruler-and-compass construction - Constructing regular polygons, Ruler-and-compass construction - Constructing with only ruler or only compass, Ruler-and-compass construction - Recent research, Ruler-and-compass construction - Reference

Read more here: » Ruler-and-compass construction: Encyclopedia II - Ruler-and-compass construction - Ruler and compass

Polygons: Encyclopedia II - Ruler-and-compass construction - Constructible points and lengths

There are many different ways to prove something is impossible. In this particular problem we carefully demarcate the limit of the possible, and show that to solve these problems you must transgress that limit. Using a ruler and compass, you can impose coordinates on the plane. Draw two points, and draw the line through them. Call that the x-axis, and define the length between the two points to be one. One construction that you can do is draw perpendiculars, so draw a perpendicular to your x-axis, and ...

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Ruler-and-compass construction, Ruler-and-compass construction - Ruler and compass, Ruler-and-compass construction - Constructible points and lengths, Ruler-and-compass construction - Impossible constructions, Ruler-and-compass construction - Squaring the circle, Ruler-and-compass construction - Doubling the cube, Ruler-and-compass construction - Angle trisection, Ruler-and-compass construction - Constructing regular polygons, Ruler-and-compass construction - Constructing with only ruler or only compass, Ruler-and-compass construction - Recent research, Ruler-and-compass construction - Reference

Read more here: » Ruler-and-compass construction: Encyclopedia II - Ruler-and-compass construction - Constructible points and lengths

Polygons: Encyclopedia II - Ruler-and-compass construction - Constructible points and lengths

How do you prove something impossible? There are many different ways, but this particular problem we carefully demarcate the limit of the possible, and show that to solve these problems you must transgress that limit. Using a ruler and compass, you can impose coordinates on the plane. Draw two points, and draw the line through them. Call that the x-axis, and define the length between the two points to be one. One construction that you can do is draw perpendiculars, so draw a perpendicular to your x-axis, and ...

See also:

Ruler-and-compass construction, Ruler-and-compass construction - Ruler and compass, Ruler-and-compass construction - Constructible points and lengths, Ruler-and-compass construction - Impossible constructions, Ruler-and-compass construction - Squaring the circle, Ruler-and-compass construction - Doubling the cube, Ruler-and-compass construction - Angle trisection, Ruler-and-compass construction - Constructing regular polygons, Ruler-and-compass construction - Constructing with only ruler or only compass, Ruler-and-compass construction - Recent research, Ruler-and-compass construction - Reference

Read more here: » Ruler-and-compass construction: Encyclopedia II - Ruler-and-compass construction - Constructible points and lengths

Polygons: Encyclopedia II - Ruler-and-compass construction - Impossible constructions

Ruler-and-compass construction - Squaring the circle. The most famous of these problems, "squaring the circle", involves constructing a square with the same area as a given circle using only ruler and compass. Squaring the circle has been proved impossible, as it involves generating a transcendental ratio, namely . Only algebraic ratios can be constructed with ruler and compass alone. The phrase "squarin ...

See also:

Ruler-and-compass construction, Ruler-and-compass construction - Ruler and compass, Ruler-and-compass construction - Constructible points and lengths, Ruler-and-compass construction - Impossible constructions, Ruler-and-compass construction - Squaring the circle, Ruler-and-compass construction - Doubling the cube, Ruler-and-compass construction - Angle trisection, Ruler-and-compass construction - Constructing regular polygons, Ruler-and-compass construction - Constructing with only ruler or only compass, Ruler-and-compass construction - Recent research, Ruler-and-compass construction - Reference

Read more here: » Ruler-and-compass construction: Encyclopedia II - Ruler-and-compass construction - Impossible constructions

Polygons: Encyclopedia II - Gunpowder warfare - Firearms

The power of aristocracies fell throughout Western Europe during this period. Their ancestral castles were no longer useful defences. Their role in war was also eroded as the Medieval cavalry lost its central role in warfare. The cavalry made up of the elite had been fading in importance in the late Middle Ages. The English longbow and the Swiss pike had both proven their ability to devastate larger armed forces. However the proper use of the longbow required a lifetime of training making it impossible to amass very large forces while the pr ...

See also:

Gunpowder warfare, Gunpowder warfare - Cannons, Gunpowder warfare - Beginning of polygonal fortifications, Gunpowder warfare - Firearms, Gunpowder warfare - Nature of war, Gunpowder warfare - Outside of Europe, Gunpowder warfare - Ottoman Empire, Gunpowder warfare - Japan, Gunpowder warfare - Naval warfare

Read more here: » Gunpowder warfare: Encyclopedia II - Gunpowder warfare - Firearms

Polygons: Encyclopedia II - Fermat number - Fermat's little theorem and pseudoprimes

Fermat's little theorem ...Using Fermat numbers to generate infinitely many pseudoprimes... Fermat number - Other theorems about Fermat's primes. If n is a positive integer, proof = an − bn If 2n + 1 is prime, then n< ...

See also:

Fermat number, Fermat number - Basic properties, Fermat number - Primality of Fermat numbers, Fermat number - Factorisation of Fermat numbers, Fermat number - Fermat's little theorem and pseudoprimes, Fermat number - Other theorems about Fermat's primes, Fermat number - Relationship to constructible polygons, Fermat number - Applications of Fermat numbers, Fermat number - Other interesting facts, Fermat number - Generalised Fermat numbers

Read more here: » Fermat number: Encyclopedia II - Fermat number - Fermat's little theorem and pseudoprimes

Polygons: Encyclopedia II - 3D Studio Max - Overview

3ds Max is one of the most widely-used 3D animation software. Its has strong editing capabilities, an ubiquitous plugin architecture and a long heritage on the Microsoft Windows platform. 3ds Max is mostly used by video game developers but can also be used for pre-rendered productions such as movies, special effects and architectural presentations. In addition to its modeling and animation tools, the latest versions of 3ds Max also features advanced shaders (such as ambient occlusion and subsurface scattering), dy ...

See also:

3D Studio Max, 3D Studio Max - Overview, 3D Studio Max - Modeling Methods, 3D Studio Max - Modeling with primitives, 3D Studio Max - NURMS, 3D Studio Max - Surface tool, 3D Studio Max - NURBS, 3D Studio Max - Polygon modelling, 3D Studio Max - Particle systems, 3D Studio Max - Features, 3D Studio Max - MAXScript, 3D Studio Max - Character Studio, 3D Studio Max - Mental Ray

Read more here: » 3D Studio Max: Encyclopedia II - 3D Studio Max - Overview

Polygons: Encyclopedia II - Fermat number - Basic properties

The Fermat numbers satisfy the following recurrence relations Fn = (Fn − 1 − 1)2 + 1 for n ≥ 2. Each of these relations can be proved by mathematical induction. From the last equation, we can deduce Goldbach's theorem: no two Fermat numbers share a common factor. To see this, suppose that 0 ≤ i < j and Fi and Fj have a common factor a > ...

See also:

Fermat number, Fermat number - Basic properties, Fermat number - Primality of Fermat numbers, Fermat number - Factorisation of Fermat numbers, Fermat number - Fermat's little theorem and pseudoprimes, Fermat number - Other theorems about Fermat's primes, Fermat number - Relationship to constructible polygons, Fermat number - Applications of Fermat numbers, Fermat number - Other interesting facts, Fermat number - Generalised Fermat numbers

Read more here: » Fermat number: Encyclopedia II - Fermat number - Basic properties

Polygons: Encyclopedia II - 3D Studio Max - Overview

3ds Max is one of the most widely-used 3D animation software. It has strong editing capabilities, a ubiquitous plugin architecture and a long heritage on the Microsoft Windows platform. 3ds Max is mostly used by video game developers but can also be used for pre-rendered productions such as movies, special effects and architectural presentations. In addition to its modeling and animation tools, the latest version of 3ds Max also features advanced shaders (such as ambient occlusion and subsurface scattering), dynam ...

See also:

3D Studio Max, 3D Studio Max - Overview, 3D Studio Max - Modeling Methods, 3D Studio Max - Modeling with primitives, 3D Studio Max - NURMS, 3D Studio Max - Surface tool, 3D Studio Max - NURBS, 3D Studio Max - Polygon modeling, 3D Studio Max - Particle systems, 3D Studio Max - Features, 3D Studio Max - MAXScript, 3D Studio Max - Character Studio, 3D Studio Max - Mental Ray

Read more here: » 3D Studio Max: Encyclopedia II - 3D Studio Max - Overview

Polygons: Encyclopedia II - Fermat number - Primality of Fermat numbers

Fermat numbers and Fermat primes were first studied by Pierre de Fermat, who conjectured that all Fermat numbers are prime. Indeed, the first five Fermat numbers F0,...,F4 are easily shown to be prime. However, this conjecture was refuted by Leonhard Euler in 1732 when he showed that It is interesting to note how Euler found this factorization. Euler had proved that every factor of Fn must have the form k2n+1 + 1. For n = ...

See also:

Fermat number, Fermat number - Basic properties, Fermat number - Primality of Fermat numbers, Fermat number - Factorisation of Fermat numbers, Fermat number - Fermat's little theorem and pseudoprimes, Fermat number - Other theorems about Fermat's primes, Fermat number - Relationship to constructible polygons, Fermat number - Applications of Fermat numbers, Fermat number - Other interesting facts, Fermat number - Generalised Fermat numbers

Read more here: » Fermat number: Encyclopedia II - Fermat number - Primality of Fermat numbers

Polygons: Encyclopedia II - Fuchsian model - A more precise definition

To be more precise, every Riemann surface has a universal covering map that is either the Riemann sphere, the complex plane or the upper half-plane. Given a covering map , where H is the upper half-plane. The Fuchsian model of R is the quotient space . R. Note that Rh is a complete 2D hyperbolic manifold. ...

See also:

Fuchsian model, Fuchsian model - A more precise definition, Fuchsian model - Nielsen isomorphism theorem

Read more here: » Fuchsian model: Encyclopedia II - Fuchsian model - A more precise definition

Polygons: Encyclopedia II - Watershed - Analyzing watersheds

Rain gauge data is used to measure total precipitation over a watersheds, and there are different ways to interpret that data. If the gauges are many and evenly distributed over an area of uniform precipitation, using the arithmetic mean method will give good results. In the Thiessen polygon method, the watershed is divided into polygons with the rain gauge in the middle of each polygon assumed to be representative for the rainfall on the area of land included in its polygon. These polygons are made by drawing lines between gauges, then making perpend ...

See also:

Watershed, Watershed - Watersheds in ecology, Watershed - Watersheds in politics, Watershed - Analyzing watersheds, Watershed - Ocean watersheds, Watershed - Footnote

Read more here: » Watershed: Encyclopedia II - Watershed - Analyzing watersheds

Polygons: Encyclopedia II - Fuchsian model - Nielsen isomorphism theorem

The Nielsen isomorphism theorem basically states that the algebraic topology of a closed Riemann surface is the same as its geometry. More precisely, let R be a closed hyperbolic surface. Let G be the Fuchsian group of R and let be a faithful representation of G, and let ρ(G) be discrete. Then define the set A(G) = {ρ:ρ defined as above } and add to this set a topology of pointwise con ...

See also:

Fuchsian model, Fuchsian model - A more precise definition, Fuchsian model - Nielsen isomorphism theorem

Read more here: » Fuchsian model: Encyclopedia II - Fuchsian model - Nielsen isomorphism theorem

Polygons: Encyclopedia II - Schwarz-Christoffel mapping - Definition

Consider a polygon in the complex plane. The Riemann mapping theorem implies that there is a bijective holomorphic mapping f from the upper half-plane to the interior of the polygon. The function f maps the real axis to the edges of the polygon. If the polygon has interior angles α,β,γ,..., then this mapping is given by where K is a constant, and a < b < c & ...

See also:

Schwarz-Christoffel mapping, Schwarz-Christoffel mapping - Definition, Schwarz-Christoffel mapping - Example, Schwarz-Christoffel mapping - Other simple mappings, Schwarz-Christoffel mapping - Triangle, Schwarz-Christoffel mapping - Square, Schwarz-Christoffel mapping - General triangle

Read more here: » Schwarz-Christoffel mapping: Encyclopedia II - Schwarz-Christoffel mapping - Definition

Polygons: Encyclopedia II - Ramification - In complex analysis

In complex analysis, the basic model can be taken as the z → zn mapping in the complex plane, near z = 0. This is the standard local picture in Riemann surface theory, of ramification of order n. It occurs for example in the Riemann-Hurwitz formula for the effect of mappings on the genus. See also branch point. ...

See also:

Ramification, Ramification - In complex analysis, Ramification - In algebraic topology, Ramification - In algebraic number theory, Ramification - In local fields

Read more here: » Ramification: Encyclopedia II - Ramification - In complex analysis

Polygons: Encyclopedia II - Fundamental domain - Example

The existence and description of a fundamental domain is in general something requiring painstaking work to establish. The diagram to the right shows part of the construction of the fundamental domain for the action of the modular group Γ on the upper half-plane H. This famous diagram appears in all classical books on elliptic modular functions. (It was probably well known to C. F. Gauss, who dealt with fundamental domains in the guise of the reduction theory of quadratic forms.) Here, each triangular region (bounded by the bl ...

See also:

Fundamental domain, Fundamental domain - Example

Read more here: » Fundamental domain: Encyclopedia II - Fundamental domain - Example

Polygons: Encyclopedia II - Ramification - In algebraic topology

In a covering map the Euler-Poincaré characteristic should multiply by the number of sheets; ramification can therefore be detected by some dropping from that. The z → zn mapping shows this as a local pattern: if we exclude 0, looking at 0 < |z| < 1 say, we have (from the homotopy point of view) the circle mapped to itself by the n-th power map (Euler-Poincaré characteristic 0), but with the whole disk the Euler-Poincaré characteristic is 1, n-1 being the 'lost' points ...

See also:

Ramification, Ramification - In complex analysis, Ramification - In algebraic topology, Ramification - In algebraic number theory, Ramification - In local fields

Read more here: » Ramification: Encyclopedia II - Ramification - In algebraic topology

Polygons: Encyclopedia II - Ramification - In algebraic number theory

Ramification in algebraic number theory means prime numbers factorising into some repeated prime ideal factors. Let R be the ring of integers of an algebraic number field K and P a prime ideal of R. For each extension field L of K we can consider the integral closure S of R in L and the ideal PS of S. This may or may not be prime, but assuming [L:K] is finite it is a product of prime ideals P1 ...

See also:

Ramification, Ramification - In complex analysis, Ramification - In algebraic topology, Ramification - In algebraic number theory, Ramification - In local fields

Read more here: » Ramification: Encyclopedia II - Ramification - In algebraic number theory

Polygons: Encyclopedia II - Polytope - Simplicial decomposition

Now given any convex hull in r-dimensional space (but not in any (r-1)-plane, say) we can take linearly independent subsets of the vertices, and define r-simplices with them. In fact, you can choose several simplices in this way such that their union as sets is the original hull, and the intersection of any two is either empty or an s-simplex (for some s < r). For example, in the plane a square (convex hull of its corners) is the union of the two triangles (2-simplices), defined ...

See also:

Polytope, Polytope - Convex polytopes, Polytope - Simplicial decomposition, Polytope - Uses

Read more here: » Polytope: Encyclopedia II - Polytope - Simplicial decomposition

Polygons: Encyclopedia II - Rasterisation - Acceleration Techniques

To extract the maximum performance out of any rasterization engine, a minimum number of polygons should be sent to the renderer. A number of acceleration techniques have been developed over time to cull out objects which can not be seen. Backface Culling The simplest way to cull polygons is to cull all polygons which face away from the viewer. This is known as backface culling. Since most 3d objects are fully enclosed, polygons facing away from a viewer are always blocked by polygons facing towards the viewer unless the viewer ...

See also:

Rasterisation, Rasterisation - Introduction, Rasterisation - Basic Approach, Rasterisation - Transformations, Rasterisation - Clipping, Rasterisation - Scan Conversion, Rasterisation - Acceleration Techniques, Rasterisation - Further Refinements, Rasterisation - Texture Filtering, Rasterisation - Environment Mapping, Rasterisation - Bump Mapping, Rasterisation - Level of Detail, Rasterisation - Shadows, Rasterisation - Hardware Acceleration

Read more here: » Rasterisation: Encyclopedia II - Rasterisation - Acceleration Techniques

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