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Polygons | A Wisdom Archive on Polygons | | Polygons A selection of articles related to Polygons | |
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| ARTICLES RELATED TO Polygons | | | |  Polygons: Encyclopedia II - Tessellation - Regular and irregular tessellationsA regular tessellation is a highly symmetric tessellation made up of congruent regular polygons. Only three regular tessellations exist: those made up of equilateral triangles, squares, or hexagons. Other types of tessellations exist, depending on types of figures and types of pattern. There are regular versus irregular, periodic versus aperiodic, symmetric versus asymmetric, and fractal tesselations, as well as other classifications.
Penrose tilings using two different polygons are the most famous example of tessellations that ...
See also:Tessellation, Tessellation - Wallpaper groups, Tessellation - Tessellations and color, Tessellation - Tessellations with quadrilaterals, Tessellation - Regular and irregular tessellations, Tessellation - Tessellations and computer graphics, Tessellation - Number of sides of a polygon versus number of sides at a vertex, Tessellation - History Read more here: » Tessellation: Encyclopedia II - Tessellation - Regular and irregular tessellations |
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| | | | | Polygons: Encyclopedia II - Schläfli symbol - Regular polychora 4-space The Schläfli symbol of a regular polychoron is of the form {p,q,r}. It has {p} regular polygonal faces, {p,q} cells, {q,r} regular polyhedral vertex figures, and {r} regular polygonal edge figures.
There are 6 convex regular and 10 nonconvex polychora.
The smallest convex polychora is {3,3,3}, the pentachoron, and the largest is 600-cell is {3,3,5}.
All 10 nonconvex regular polychora a ...
See also:Schläfli symbol, Schläfli symbol - Regular polygons plane, Schläfli symbol - Regular polyhedra 3-space, Schläfli symbol - Regular polychora 4-space, Schläfli symbol - Higher dimensions, Schläfli symbol - Dual polytopes Read more here: » Schläfli symbol: Encyclopedia II - Schläfli symbol - Regular polychora 4-space |
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| | | | | | Polygons: Encyclopedia II - Regular polytope - Constructions
Regular polytope - Polygons.
The traditional way to "construct" a regular polygon, or indeed any other figure on the plane, is by use of a ruler (more properly, a straightedge) and a compass. Constructing some regular polygons is very easy (the easiest is perhaps the equilateral triangle), some harder, or "impossible". The simplest few regular polygons that are "impossible" to construct using just a ruler and a compass are the n-sided polygons with n equal to 7, 9, 11, 13, 14, 18, 19, 21, and so forth. (See Rule ...
See also:Regular polytope, Regular polytope - History of discovery, Regular polytope - Prehistory, Regular polytope - Greeks, Regular polytope - Star polyhedra, Regular polytope - Higher-dimensional polytopes, Regular polytope - Abstract regular polytopes, Regular polytope - Constructions, Regular polytope - Polygons, Regular polytope - Polyhedra, Regular polytope - Higher dimensions, Regular polytope - Polytopes in nature, Regular polytope - Polygons, Regular polytope - Polyhedra Read more here: » Regular polytope: Encyclopedia II - Regular polytope - Constructions |
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| | | | | Polygons: Encyclopedia II - Tessellation - Tessellations and color When discussing a tiling that is displayed in colors, to avoid ambiguity one needs to specify whether the colors are part of the tiling or just part of its illustration. See also color in symmetry.
The four color theorem states that for every tessellation of the plane, with a set of four available colors, each tile can be colored in one color such that no tiles of equal color meet at a curve of positive length. Note that the coloring guaranteed by the four-color theorem will not in general respect the symmetries of the tessellation. To produce a col ...
See also:Tessellation, Tessellation - Wallpaper groups, Tessellation - Tessellations and color, Tessellation - Tessellations with quadrilaterals, Tessellation - Regular and irregular tessellations, Tessellation - Tessellations and computer graphics, Tessellation - Number of sides of a polygon versus number of sides at a vertex, Tessellation - History Read more here: » Tessellation: Encyclopedia II - Tessellation - Tessellations and color |
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| | | | | Polygons: Encyclopedia II - Exact trigonometric constants - Table of constants Values outside 0° ... 45° angle range are trivially extracted from circle axis reflection symmetry from these values. (See Trigonometric identity)
Exact trigonometric constants - 0° Fundamental.
Exact trigonometric constants - 3° - 60-sided polygon.
Exact trigonometric constants - 6° - 30-side ...
See also:Exact trigonometric constants, Exact trigonometric constants - Table of constants, Exact trigonometric constants - 0° Fundamental, Exact trigonometric constants - 3° - 60-sided polygon, Exact trigonometric constants - 6° - 30-sided polygon, Exact trigonometric constants - 9° - 20-sided polygon, Exact trigonometric constants - 12° - 15-sided polygon, Exact trigonometric constants - 15° - 12-sided polygon, Exact trigonometric constants - 18° - 10-sided polygon, Exact trigonometric constants - 21° - Sum 9° + 12°, Exact trigonometric constants - 22.5° - Octagon, Exact trigonometric constants - 24° - Sum 12° + 12°, Exact trigonometric constants - 27° - Sum 12° + 15°, Exact trigonometric constants - 30° - Hexagon, Exact trigonometric constants - 33° - Sum 15° + 18°, Exact trigonometric constants - 36° - Pentagon, Exact trigonometric constants - 39° - Sum 18°+ 21°, Exact trigonometric constants - 42° - Sum 21° + 21°, Exact trigonometric constants - 45° - Square, Exact trigonometric constants - Notes, Exact trigonometric constants - Uses for constants, Exact trigonometric constants - Derivation triangles, Exact trigonometric constants - How can the trig values for sin and cos be calculated?, Exact trigonometric constants - The trivial ones, Exact trigonometric constants - n π over 10, Exact trigonometric constants - n π over 20, Exact trigonometric constants - n π over 30, Exact trigonometric constants - n π over 60, Exact trigonometric constants - How can the trig values for tan and cot be calculated?, Exact trigonometric constants - Plans for simplifying, Exact trigonometric constants - Rationalize the denominator, Exact trigonometric constants - Split a fraction in two, Exact trigonometric constants - Squaring and square rooting, Exact trigonometric constants - Simplification of nested radical expressions Read more here: » Exact trigonometric constants: Encyclopedia II - Exact trigonometric constants - Table of constants |
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| | | | | Polygons: Encyclopedia II - Exact trigonometric constants - Notes
Exact trigonometric constants - Uses for constants.
As an example of the use of these constants, consider a dodecahedron with the following volume:
Using
this can be simplified to:
Exact trigonometric constants - Derivation triangles.
The derivation of sine, cosine, and tangent constants into radial forms ...
See also:Exact trigonometric constants, Exact trigonometric constants - Table of constants, Exact trigonometric constants - 0° Fundamental, Exact trigonometric constants - 3° - 60-sided polygon, Exact trigonometric constants - 6° - 30-sided polygon, Exact trigonometric constants - 9° - 20-sided polygon, Exact trigonometric constants - 12° - 15-sided polygon, Exact trigonometric constants - 15° - 12-sided polygon, Exact trigonometric constants - 18° - 10-sided polygon, Exact trigonometric constants - 21° - Sum 9° + 12°, Exact trigonometric constants - 22.5° - Octagon, Exact trigonometric constants - 24° - Sum 12° + 12°, Exact trigonometric constants - 27° - Sum 12° + 15°, Exact trigonometric constants - 30° - Hexagon, Exact trigonometric constants - 33° - Sum 15° + 18°, Exact trigonometric constants - 36° - Pentagon, Exact trigonometric constants - 39° - Sum 18°+ 21°, Exact trigonometric constants - 42° - Sum 21° + 21°, Exact trigonometric constants - 45° - Square, Exact trigonometric constants - Notes, Exact trigonometric constants - Uses for constants, Exact trigonometric constants - Derivation triangles, Exact trigonometric constants - How can the trig values for sin and cos be calculated?, Exact trigonometric constants - The trivial ones, Exact trigonometric constants - n π over 10, Exact trigonometric constants - n π over 20, Exact trigonometric constants - n π over 30, Exact trigonometric constants - n π over 60, Exact trigonometric constants - How can the trig values for tan and cot be calculated?, Exact trigonometric constants - Plans for simplifying, Exact trigonometric constants - Rationalize the denominator, Exact trigonometric constants - Split a fraction in two, Exact trigonometric constants - Squaring and square rooting, Exact trigonometric constants - Simplification of nested radical expressions Read more here: » Exact trigonometric constants: Encyclopedia II - Exact trigonometric constants - Notes |
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| | | | | Polygons: Encyclopedia II - Regular polytope - Polytopes in nature
Regular polytope - Polygons.
Numerous regular polygons may be seen in nature. In the world of minerals, crystals often have faces which are triangular, square or hexagonal. Quasicrystals can even have regular pentagons as faces. Another fascinating example of regular polygons occurring as a result of geological processes may be seen at the Giant's Causeway in Ireland, or at the Devil's Postpile in California, where the cooling of lava has for ...
See also:Regular polytope, Regular polytope - History of discovery, Regular polytope - Prehistory, Regular polytope - Greeks, Regular polytope - Star polyhedra, Regular polytope - Higher-dimensional polytopes, Regular polytope - Abstract regular polytopes, Regular polytope - Constructions, Regular polytope - Polygons, Regular polytope - Polyhedra, Regular polytope - Higher dimensions, Regular polytope - Polytopes in nature, Regular polytope - Polygons, Regular polytope - Polyhedra Read more here: » Regular polytope: Encyclopedia II - Regular polytope - Polytopes in nature |
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| | | | | Polygons: Encyclopedia II - Exact trigonometric constants - Plans for simplifying
Exact trigonometric constants - Rationalize the denominator.
If the denominator is a square root, multiply the numerator and denominator by that radical.
If the denominator is the sum or difference of two terms, multiply the numerator and denominator by the conjugate of the denominator. The conjugate is the identical, except the sign between the terms is changed.
S ...
See also:Exact trigonometric constants, Exact trigonometric constants - Table of constants, Exact trigonometric constants - 0° Fundamental, Exact trigonometric constants - 3° - 60-sided polygon, Exact trigonometric constants - 6° - 30-sided polygon, Exact trigonometric constants - 9° - 20-sided polygon, Exact trigonometric constants - 12° - 15-sided polygon, Exact trigonometric constants - 15° - 12-sided polygon, Exact trigonometric constants - 18° - 10-sided polygon, Exact trigonometric constants - 21° - Sum 9° + 12°, Exact trigonometric constants - 22.5° - Octagon, Exact trigonometric constants - 24° - Sum 12° + 12°, Exact trigonometric constants - 27° - Sum 12° + 15°, Exact trigonometric constants - 30° - Hexagon, Exact trigonometric constants - 33° - Sum 15° + 18°, Exact trigonometric constants - 36° - Pentagon, Exact trigonometric constants - 39° - Sum 18°+ 21°, Exact trigonometric constants - 42° - Sum 21° + 21°, Exact trigonometric constants - 45° - Square, Exact trigonometric constants - Notes, Exact trigonometric constants - Uses for constants, Exact trigonometric constants - Derivation triangles, Exact trigonometric constants - How can the trig values for sin and cos be calculated?, Exact trigonometric constants - The trivial ones, Exact trigonometric constants - n π over 10, Exact trigonometric constants - n π over 20, Exact trigonometric constants - n π over 30, Exact trigonometric constants - n π over 60, Exact trigonometric constants - How can the trig values for tan and cot be calculated?, Exact trigonometric constants - Plans for simplifying, Exact trigonometric constants - Rationalize the denominator, Exact trigonometric constants - Split a fraction in two, Exact trigonometric constants - Squaring and square rooting, Exact trigonometric constants - Simplification of nested radical expressions Read more here: » Exact trigonometric constants: Encyclopedia II - Exact trigonometric constants - Plans for simplifying |
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| | | | | Polygons: Encyclopedia II - Schläfli symbol - Regular polyhedra 3-space The Schläfli symbol of a polyhedron is {p,q} if its faces are p-gons, and each vertex is surrounded by q faces.
The Schläfli symbols of the Platonic solids are:
for the tetrahedron : {3,3}
for the cube : {4,3}
for the octahedron : {3,4}
for the dodecahedron : {5,3}
for the icosahedron : {3,5}
Schläfli symbols may also be defined for regular tessellations of Euclidean or hyperbolic space in a similar way.
In addition to the 5 convex regular polyhedra, th ...
See also:Schläfli symbol, Schläfli symbol - Regular polygons plane, Schläfli symbol - Regular polyhedra 3-space, Schläfli symbol - Regular polychora 4-space, Schläfli symbol - Higher dimensions, Schläfli symbol - Dual polytopes Read more here: » Schläfli symbol: Encyclopedia II - Schläfli symbol - Regular polyhedra 3-space |
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| | | | | Polygons: Encyclopedia II - Computer graphics - Texturing Polygon surfaces (the sequence of faces) can contain data corresponding to not only a color, but in more advanced software, can be a virtual canvas for a picture, or other rasterized image. Such an image is placed onto a face, or series of faces and is called a Texture.
Textures add a new degree of customization as to how a faces & polygons will ultimately look after being shaded, depending on the shading method, a ...
See also:Computer graphics, Computer graphics - Computer graphics 2D, Computer graphics - Computer graphics 3D, Computer graphics - Shading, Computer graphics - Texturing, Computer graphics - Toolkits and APIs Read more here: » Computer graphics: Encyclopedia II - Computer graphics - Texturing |
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| | | | | Polygons: Encyclopedia II - Computer graphics - Computer graphics 3D With the birth of the workstation computers (like LISP machines, paintbox computers and Silicon Graphics workstations) came the 3D computer graphics, based on vector graphics. Instead of the computer storing information about points, lines and curves on a 2-Dimensional plane, the computer stores the location of points, lines and typically faces (to construct a polygon) in 3-Dimensional Space.
3-Dimensional polygons are the life blood of virtually all 3D computer graphics. As a result, most 3D graphics engines are based around storing ...
See also:Computer graphics, Computer graphics - Computer graphics 2D, Computer graphics - Computer graphics 3D, Computer graphics - Shading, Computer graphics - Texturing, Computer graphics - Toolkits and APIs Read more here: » Computer graphics: Encyclopedia II - Computer graphics - Computer graphics 3D |
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| | | | | Polygons: Encyclopedia II - Computer graphics - Computer graphics, 3D With the birth of the workstation computers (like LISP machines, paintbox computers and Silicon Graphics workstations) came the 3D computer graphics, based on vector graphics. Instead of the computer storing information about points, lines and curves on a 2-Dimensional plane, the computer stores the location of points, lines and typically faces (to construct a polygon) in 3-Dimensional Space.
3-Dimensional polygons are the life blood of virtually all 3D computer graphics. As a result, most 3D graphics engines are based around storing ...
See also:Computer graphics, Computer graphics - Computer graphics, 2D, Computer graphics - Computer graphics, 3D, Computer graphics - Shading, Computer graphics - Texturing, Computer graphics - Toolkits and APIs Read more here: » Computer graphics: Encyclopedia II - Computer graphics - Computer graphics, 3D |
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| | | | | Polygons: 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 match:
sphere
real projective plane
Klein bottle
torus
...
See also:Surface, 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 Read more here: » Surface: Encyclopedia II - Surface - Some models |
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| | | | | Polygons: Encyclopedia II - Computer graphics - Computer graphics 3D With the birth of the workstation computers (like LISP machines, paintbox computers and Silicon Graphics workstations) came the 3D computer graphics, based on vector graphics. Instead of the computer storing information about points, lines and curves on a 2-Dimensional plane, the computer stores the location of points, lines and typically faces (to construct a polygon) in 3-Dimensional Space.
3-Dimensional polygons are the life blood of virtually all 3D computer graphics. As a result, most 3D graphics engines are based around storing ...
See also:Computer graphics, Computer graphics - Computer graphics 2D, Computer graphics - Computer graphics 3D, Computer graphics - Shading, Computer graphics - Image based rendering, Computer graphics - Texturing, Computer graphics - Toolkits and APIs Read more here: » Computer graphics: Encyclopedia II - Computer graphics - Computer graphics 3D |
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| | | | | Polygons: Encyclopedia II - 3D Studio Max - Modeling Methods There are 5 basic modeling methods:
Modeling with primitives
NURMS (subdivision surfaces)
Surface tool/Editable patch object
NURBS
Polygon modelling
3D Studio Max - Modeling with primitives.
This is a basic method, in which one models something using only boxes, spheres, cones, cylinders and other predefined objects. One may also apply boolean operations, including subtract, cut and connect. For example, one can make two spheres which will work as blobs that will connect with each other. This is called "blob-mesh ...
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 - Modeling Methods |
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| | | | | Polygons: Encyclopedia II - 3D Studio Max - Modeling Methods There are 5 basic modeling methods:
Modeling with primitives
NURMS (subdivision surfaces)
Surface tool/Editable patch object
NURBS
Polygon modeling
3D Studio Max - Modeling with primitives.
This is a basic method, in which one models something using only boxes, spheres, cones, cylinders and other predefined objects. One may also apply boolean operations, including subtract, cut and connect. For example, one can make two spheres which will work as blobs that will connect with each other. This is called "blob-mesh ...
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 - Modeling Methods |
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| | | | | Polygons: Encyclopedia II - Computer graphics - Texturing Polygon surfaces (the sequence of faces) can contain data corresponding to not only a color, but in more advanced software, can be a virtual canvas for a picture, or other rasterized image. Such an image is placed onto a face, or series of faces and is called a Texture.
Textures add a new degree of customization as to how a faces & polygons will ultimately look after being shaded, depending on the shading method, a ...
See also:Computer graphics, Computer graphics - Computer graphics 2D, Computer graphics - Computer graphics 3D, Computer graphics - Shading, Computer graphics - Image based rendering, Computer graphics - Texturing, Computer graphics - Toolkits and APIs Read more here: » Computer graphics: Encyclopedia II - Computer graphics - Texturing |
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| | | | | Polygons: Encyclopedia II - Surface - Definition In what follows, all surfaces are considered to be second-countable 2-dimensional manifolds.
More precisely: a topological surface (with boundary) is a Hausdorff space in which every point has an open neighbourhood homeomorphic to either an open subset of E2 (Euclidean 2-space) or an open subset of the closed half of E2. The set of points which have an open neighbourhood homeomorphic to En is called the interior of the manifold; it is always non-empty. The complement of the interior, is called the boundary; it is a (1 ...
See also:Surface, 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 Read more here: » Surface: Encyclopedia II - Surface - Definition |
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| | | | | Polygons: 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 falls into one of two infinite collections:
Spheres with g handles attached (called g-fold tori). These are orientable surfaces with Euler characteristic 2-2g, also called surfaces of genus g.
Spheres with k projective planes attached. These are non-orientable surfaces with Euler characteristic 2-k.
Therefore Euler characteristic and orientability describe a compact surfaces up to homeomor ...
See also:Surface, 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 Read more here: » Surface: Encyclopedia II - Surface - Classification of closed surfaces |
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| | | | | Polygons: 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 formed boundary components.
We use the following notation.
sphere: S
torus: T
Klein bottle: K
Projective plane: P
Facts:
S # S = S
S # M = M
P # P = K
P # K = P # T
We use a shorthand ...
See also:Surface, 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 Read more here: » Surface: Encyclopedia II - Surface - Connected sum of surfaces |
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| | | | | Polygons: Encyclopedia II - Gunpowder warfare - Nature of war For the most part the wars were not particularly deadly by later standards. Armies were slow moving in an era before good roads and canals. Battles of maneuver were common with armies circling one another, often for months, with no direct conflict. By far the most common battles were sieges, hugely time-consuming and expensive affairs, but ones with only limited casualties. The indecisive nature of conflict meant wars were long and endemic. Conflicts stretched on for decades and many sta ...
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 - Nature of war |
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