1.
Tonne
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The SI symbol for the tonne is t, adopted at the same time as the unit itself in 1879. Its use is also official, for the metric ton, within the United States, having been adopted by the US National Institute of Standards and it is a symbol, not an abbreviation, and should not be followed by a period. Informal and non-approved symbols or abbreviations include T, mT, MT, in French and all English-speaking countries that are predominantly metric, tonne is the correct spelling. Before metrication in the UK the unit used for most purposes was the Imperial ton of 2,240 pounds avoirdupois, equivalent to 1,016 kg, differing by just 1. 6% from the tonne. Ton and tonne are both derived from a Germanic word in use in the North Sea area since the Middle Ages to designate a large cask. A full tun, standing about a high, could easily weigh a tonne. An English tun of wine weighs roughly a tonne,954 kg if full of water, in the United States, the unit was originally referred to using the French words millier or tonneau, but these terms are now obsolete. The Imperial and US customary units comparable to the tonne are both spelled ton in English, though they differ in mass, one tonne is equivalent to, Metric/SI,1 megagram. Equal to 1000000 grams or 1000 kilograms, megagram, Mg, is the official SI unit. Mg is distinct from mg, milligram, pounds, Exactly 1000/0. 453 592 37 lb, or approximately 2204.622622 lb. US/Short tons, Exactly 1/0. 907 184 74 short tons, or approximately 1.102311311 ST. One short ton is exactly 0.90718474 t, imperial/Long tons, Exactly 1/1. 016 046 9088 long tons, or approximately 0.9842065276 LT. One long ton is exactly 1.0160469088 t, for multiples of the tonne, it is more usual to speak of thousands or millions of tonnes. Kilotonne, megatonne, and gigatonne are more used for the energy of nuclear explosions and other events. When used in context, there is little need to distinguish between metric and other tons, and the unit is spelt either as ton or tonne with the relevant prefix attached. *The equivalent units columns use the short scale large-number naming system used in most English-language countries. †Values in the equivalent short and long tons columns are rounded to five significant figures, ǂThough non-standard, the symbol kt is also sometimes used for knot, a unit of speed for sea-going vessels, and should not be confused with kilotonne. A metric ton unit can mean 10 kilograms within metal trading and it traditionally referred to a metric ton of ore containing 1% of metal. In the case of uranium, the acronym MTU is sometimes considered to be metric ton of uranium, in the petroleum industry the tonne of oil equivalent is a unit of energy, the amount of energy released by burning one tonne of crude oil, approximately 42 GJ
2.
Megatron
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Megatron is a character from the Transformers franchise, created by Hasbro in 1984, based on a toy design by Takara. The original Megatron was the Decepticon Leader, which could transform into three different types of guns, a Walther P38 handgun, a beam cannon. He is a sentient robotic lifeform from the planet Cybertron and the leader of the villainous Decepticons as well as the archenemy of the Autobot leader Optimus Prime, Megatron is usually depicted as having risen up from the lowly worker to become a champion in gladiatorial combat. As a gladiator, he took the legendary name Megatronus as his own, Prime would later use his teachings against him when he became corrupt. He has the ability to transform between his robot shape and various weapons or vehicles, but these alternate-modes, his origins and even personality and this origin is considered the most consistent between the various incarnations. Further differences are listed in the sections below. The original Megatronus, better known as the Fallen, was one of the original Thirteen Primes created by Primus, Megatronus betrayed his creator by siding with Primus dark twin, the malevolent planet-eater Unicron. In the final battle between Primus and Unicron, Megatronus fell victim to the fate as his master, sucked through a black hole into another dimension. However, while Unicron emerged in another universe, the Fallen was not so fortunate, another being addressed simply as Megatron is an apocalyptic figure said to bring about death and destruction through alteration of the time stream. Possible interpretations in the book of him are Alpha and Omega, the Fallen made his first appearance in the second volume of Transformers, The War Within. The character appears as the main antagonist in the live action film, Transformers, Revenge of the Fallen. Director Michael Bay described him as apocalyptic and he has the ability to teleport and generate a shockwave upon reappearance. He wields a spear as his weapon and has the ability of telekinesis, Megatronus is also mentioned as one of Thirteen in Transformers, Exodus and later appeared as the main antagonist in the first-season finale of Transformers, Robots in Disguise, voiced by Gil Gerard. Megatronuss history would be retconned in Transformers, The Covenant of Primus, Megatronus was one of the original thirteen Primes. Megatronus commissioned Solus Prime to create the Requiem Blaster, following the War of the Primes, he exiled himself from Cybertron after unintentionally murdering Solus Prime. Megatron is the founder of the Decepticon uprising and their most feared leader, bob Budiansky, the writer for the Marvel Comics series, stated that originally Hasbro took issue with the name, saying it sounded too frightening. Budiansky responded that as the villain, that was the point. Hasbro later agreed with his reasoning, and approved the name Megatron, the name itself probably comes from the Greek word megas, although in one of his interviews Budiansky claimed that it is in fact a portmanteau of electronic and megaton
3.
Regular polygon
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In Euclidean geometry, a regular polygon is a polygon that is equiangular and equilateral. Regular polygons may be convex or star, in the limit, a sequence of regular polygons with an increasing number of sides becomes a circle, if the perimeter is fixed, or a regular apeirogon, if the edge length is fixed. These properties apply to all regular polygons, whether convex or star, a regular n-sided polygon has rotational symmetry of order n. All vertices of a regular polygon lie on a common circle and that is, a regular polygon is a cyclic polygon. Together with the property of equal-length sides, this implies that every regular polygon also has a circle or incircle that is tangent to every side at the midpoint. Thus a regular polygon is a tangential polygon, a regular n-sided polygon can be constructed with compass and straightedge if and only if the odd prime factors of n are distinct Fermat primes. The symmetry group of a regular polygon is dihedral group Dn, D2, D3. It consists of the rotations in Cn, together with reflection symmetry in n axes that pass through the center, if n is even then half of these axes pass through two opposite vertices, and the other half through the midpoint of opposite sides. If n is odd then all pass through a vertex. All regular simple polygons are convex and those having the same number of sides are also similar. An n-sided convex regular polygon is denoted by its Schläfli symbol, for n <3 we have two degenerate cases, Monogon, degenerate in ordinary space. Digon, a line segment, degenerate in ordinary space. In certain contexts all the polygons considered will be regular, in such circumstances it is customary to drop the prefix regular. For instance, all the faces of uniform polyhedra must be regular, for n >2 the number of diagonals is n 2, i. e.0,2,5,9. for a triangle, square, pentagon, hexagon. The diagonals divide the polygon into 1,4,11,24, for a regular n-gon inscribed in a unit-radius circle, the product of the distances from a given vertex to all other vertices equals n. For a regular simple n-gon with circumradius R and distances di from a point in the plane to the vertices. For a regular n-gon, the sum of the distances from any interior point to the n sides is n times the apothem. This is a generalization of Vivianis theorem for the n=3 case, the sum of the perpendiculars from a regular n-gons vertices to any line tangent to the circumcircle equals n times the circumradius
4.
Edge (geometry)
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For edge in graph theory, see Edge In geometry, an edge is a particular type of line segment joining two vertices in a polygon, polyhedron, or higher-dimensional polytope. In a polygon, an edge is a segment on the boundary. In a polyhedron or more generally a polytope, an edge is a segment where two faces meet. A segment joining two vertices while passing through the interior or exterior is not an edge but instead is called a diagonal. In graph theory, an edge is an abstract object connecting two vertices, unlike polygon and polyhedron edges which have a concrete geometric representation as a line segment. However, any polyhedron can be represented by its skeleton or edge-skeleton, conversely, the graphs that are skeletons of three-dimensional polyhedra can be characterized by Steinitzs theorem as being exactly the 3-vertex-connected planar graphs. Any convex polyhedrons surface has Euler characteristic V − E + F =2, where V is the number of vertices, E is the number of edges and this equation is known as Eulers polyhedron formula. Thus the number of edges is 2 less than the sum of the numbers of vertices and faces, for example, a cube has 8 vertices and 6 faces, and hence 12 edges. In a polygon, two edges meet at each vertex, more generally, by Balinskis theorem, at least d edges meet at every vertex of a convex polytope. Similarly, in a polyhedron, exactly two faces meet at every edge, while in higher dimensional polytopes three or more two-dimensional faces meet at every edge. Thus, the edges of a polygon are its facets, the edges of a 3-dimensional convex polyhedron are its ridges, archived from the original on 4 February 2007
5.
Vertex (geometry)
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In geometry, a vertex is a point where two or more curves, lines, or edges meet. As a consequence of this definition, the point where two lines meet to form an angle and the corners of polygons and polyhedra are vertices. A vertex is a point of a polygon, polyhedron, or other higher-dimensional polytope. However, in theory, vertices may have fewer than two incident edges, which is usually not allowed for geometric vertices. However, a smooth approximation to a polygon will also have additional vertices. A polygon vertex xi of a simple polygon P is a principal polygon vertex if the diagonal intersects the boundary of P only at x and x, there are two types of principal vertices, ears and mouths. A principal vertex xi of a simple polygon P is called an ear if the diagonal that bridges xi lies entirely in P, according to the two ears theorem, every simple polygon has at least two ears. A principal vertex xi of a simple polygon P is called a mouth if the diagonal lies outside the boundary of P. Any convex polyhedrons surface has Euler characteristic V − E + F =2, where V is the number of vertices, E is the number of edges and this equation is known as Eulers polyhedron formula. Thus the number of vertices is 2 more than the excess of the number of edges over the number of faces, for example, a cube has 12 edges and 6 faces, and hence 8 vertices
6.
Dihedral group
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In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections. Dihedral groups are among the simplest examples of groups, and they play an important role in group theory, geometry. The notation for the group of order n differs in geometry. In geometry, Dn or Dihn refers to the symmetries of the n-gon, in abstract algebra, Dn refers to the dihedral group of order n. The geometric convention is used in this article, a regular polygon with n sides has 2 n different symmetries, n rotational symmetries and n reflection symmetries. Usually, we take n ≥3 here. The associated rotations and reflections make up the dihedral group D n, if n is odd, each axis of symmetry connects the midpoint of one side to the opposite vertex. If n is even, there are n/2 axes of symmetry connecting the midpoints of opposite sides, in either case, there are n axes of symmetry and 2 n elements in the symmetry group. Reflecting in one axis of symmetry followed by reflecting in another axis of symmetry produces a rotation through twice the angle between the axes, as with any geometric object, the composition of two symmetries of a regular polygon is again a symmetry of this object. With composition of symmetries to produce another as the binary operation, the following Cayley table shows the effect of composition in the group D3. R0 denotes the identity, r1 and r2 denote counterclockwise rotations by 120° and 240° respectively, for example, s2s1 = r1, because the reflection s1 followed by the reflection s2 results in a rotation of 120°. The order of elements denoting the composition is right to left, the composition operation is not commutative. In all cases, addition and subtraction of subscripts are to be performed using modular arithmetic with modulus n, if we center the regular polygon at the origin, then elements of the dihedral group act as linear transformations of the plane. This lets us represent elements of Dn as matrices, with composition being matrix multiplication and this is an example of a group representation. For example, the elements of the group D4 can be represented by the eight matrices. In general, the matrices for elements of Dn have the following form, rk is a rotation matrix, expressing a counterclockwise rotation through an angle of 2πk/n. Sk is a reflection across a line makes an angle of πk/n with the x-axis. Further equivalent definitions of Dn are, D1 is isomorphic to Z2, D2 is isomorphic to K4, the Klein four-group. D1 and D2 are exceptional in that, D1 and D2 are the only abelian dihedral groups, Dn is a subgroup of the symmetric group Sn for n ≥3
7.
Internal and external angles
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In geometry, an angle of a polygon is formed by two sides of the polygon that share an endpoint. For a simple polygon, regardless of whether it is convex or non-convex, a polygon has exactly one internal angle per vertex. If every internal angle of a polygon is less than 180°. In contrast, an angle is an angle formed by one side of a simple polygon. The sum of the angle and the external angle on the same vertex is 180°. The sum of all the angles of a simple polygon is 180° where n is the number of sides. The formula can be proved using induction and starting with a triangle for which the angle sum is 180°. The sum of the angles of any simple convex or non-convex polygon is 360°. The interior angle concept can be extended in a consistent way to crossed polygons such as star polygons by using the concept of directed angles, in other words, 360k° represents the sum of all the exterior angles. For example, for convex and concave polygons k =1, since the exterior angle sum is 360°