1.
Quadrilateral
–
In Euclidean plane geometry, a quadrilateral is a polygon with four edges and four vertices or corners. Sometimes, the quadrangle is used, by analogy with triangle. The origin of the quadrilateral is the two Latin words quadri, a variant of four, and latus, meaning side. Quadrilaterals are simple or complex, also called crossed, simple quadrilaterals are either convex or concave. The interior angles of a simple quadrilateral ABCD add up to 360 degrees of arc and this is a special case of the n-gon interior angle sum formula × 180°. All non-self-crossing quadrilaterals tile the plane by repeated rotation around the midpoints of their edges, any quadrilateral that is not self-intersecting is a simple quadrilateral. In a convex quadrilateral, all angles are less than 180°. Irregular quadrilateral or trapezium, no sides are parallel, trapezium or trapezoid, at least one pair of opposite sides are parallel. Isosceles trapezium or isosceles trapezoid, one pair of sides are parallel. Alternative definitions are a quadrilateral with an axis of symmetry bisecting one pair of opposite sides, parallelogram, a quadrilateral with two pairs of parallel sides. Equivalent conditions are that opposite sides are of length, that opposite angles are equal. In other words, parallelograms include all rhombi and all rhomboids, rhombus or rhomb, all four sides are of equal length. An equivalent condition is that the diagonals bisect each other. Rhomboid, a parallelogram in which adjacent sides are of unequal lengths, not all references agree, some define a rhomboid as a parallelogram which is not a rhombus. Rectangle, all four angles are right angles, an equivalent condition is that the diagonals bisect each other and are equal in length. Square, all four sides are of length, and all four angles are right angles. An equivalent condition is that opposite sides are parallel, that the diagonals bisect each other. A quadrilateral is a if and only if it is both a rhombus and a rectangle
2.
Parallelogram
–
In Euclidean geometry, a parallelogram is a simple quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length, by comparison, a quadrilateral with just one pair of parallel sides is a trapezoid in American English or a trapezium in British English. The three-dimensional counterpart of a parallelogram is a parallelepiped, rhomboid – A quadrilateral whose opposite sides are parallel and adjacent sides are unequal, and whose angles are not right angles Rectangle – A parallelogram with four angles of equal size. Rhombus – A parallelogram with four sides of equal length, square – A parallelogram with four sides of equal length and angles of equal size. A simple quadrilateral is a if and only if any one of the following statements is true. Two pairs of opposite angles are equal in measure, one pair of opposite sides are parallel and equal in length. Each diagonal divides the quadrilateral into two congruent triangles, the sum of the squares of the sides equals the sum of the squares of the diagonals. It has rotational symmetry of order 2, the sum of the distances from any interior point to the sides is independent of the location of the point. Thus all parallelograms have all the properties listed above, and conversely, if just one of statements is true in a simple quadrilateral. Opposite sides of a parallelogram are parallel and so will never intersect, the area of a parallelogram is twice the area of a triangle created by one of its diagonals. The area of a parallelogram is also equal to the magnitude of the cross product of two adjacent sides. Any line through the midpoint of a parallelogram bisects the area, any non-degenerate affine transformation takes a parallelogram to another parallelogram. A parallelogram has rotational symmetry of order 2, if it also has exactly two lines of reflectional symmetry then it must be a rhombus or an oblong. If it has four lines of symmetry, it is a square. The perimeter of a parallelogram is 2 where a and b are the lengths of adjacent sides, unlike any other convex polygon, a parallelogram cannot be inscribed in any triangle with less than twice its area. The centers of four squares all constructed either internally or externally on the sides of a parallelogram are the vertices of a square. If two lines parallel to sides of a parallelogram are constructed concurrent to a diagonal, then the parallelograms formed on opposite sides of that diagonal are equal in area, the diagonals of a parallelogram divide it into four triangles of equal area. All of the formulas for general convex quadrilaterals apply to parallelograms
3.
Kite (geometry)
–
In Euclidean geometry, a kite is a quadrilateral whose four sides can be grouped into two pairs of equal-length sides that are adjacent to each other. In contrast, a parallelogram also has two pairs of sides, but they are opposite to each other rather than adjacent. Kite quadrilaterals are named for the wind-blown, flying kites, which often have this shape, kites are also known as deltoids, but the word deltoid may also refer to a deltoid curve, an unrelated geometric object. A kite, as defined above, may be convex or concave. A concave kite is called a dart or arrowhead, and is a type of pseudotriangle. If all four sides of a kite have the same length, if a kite is equiangular, meaning that all four of its angles are equal, then it must also be equilateral and thus a square. A kite with three equal 108° angles and one 36° angle forms the hull of the lute of Pythagoras. The kites that are cyclic quadrilaterals are exactly the ones formed from two congruent right triangles. That is, for these kites the two angles on opposite sides of the symmetry axis are each 90 degrees. These shapes are called right kites and they are in fact bicentric quadrilaterals, among all the bicentric quadrilaterals with a given two circle radii, the one with maximum area is a right kite. The tiling that it produces by its reflections is the deltoidal trihexagonal tiling, among all quadrilaterals, the shape that has the greatest ratio of its perimeter to its diameter is an equidiagonal kite with angles π/3, 5π/12, 5π/6, 5π/12. Its four vertices lie at the three corners and one of the midpoints of the Reuleaux triangle. In non-Euclidean geometry, a Lambert quadrilateral is a kite with three right angles. A quadrilateral is a if and only if any one of the following conditions is true. One diagonal is the bisector of the other diagonal. One diagonal is a line of symmetry, one diagonal bisects a pair of opposite angles. The kites are the quadrilaterals that have an axis of symmetry along one of their diagonals, if crossings are allowed, the list of quadrilaterals with axes of symmetry must be expanded to also include the antiparallelograms. Every kite is orthodiagonal, meaning that its two diagonals are at angles to each other
4.
Edge (geometry)
–
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)
–
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
–
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.
Area
–
Area is the quantity that expresses the extent of a two-dimensional figure or shape, or planar lamina, in the plane. Surface area is its analog on the surface of a three-dimensional object. It is the analog of the length of a curve or the volume of a solid. The area of a shape can be measured by comparing the shape to squares of a fixed size, in the International System of Units, the standard unit of area is the square metre, which is the area of a square whose sides are one metre long. A shape with an area of three square metres would have the area as three such squares. In mathematics, the square is defined to have area one. There are several formulas for the areas of simple shapes such as triangles, rectangles. Using these formulas, the area of any polygon can be found by dividing the polygon into triangles, for shapes with curved boundary, calculus is usually required to compute the area. Indeed, the problem of determining the area of plane figures was a motivation for the historical development of calculus. For a solid such as a sphere, cone, or cylinder. Formulas for the areas of simple shapes were computed by the ancient Greeks. Area plays an important role in modern mathematics, in addition to its obvious importance in geometry and calculus, area is related to the definition of determinants in linear algebra, and is a basic property of surfaces in differential geometry. In analysis, the area of a subset of the plane is defined using Lebesgue measure, in general, area in higher mathematics is seen as a special case of volume for two-dimensional regions. Area can be defined through the use of axioms, defining it as a function of a collection of certain plane figures to the set of real numbers and it can be proved that such a function exists. An approach to defining what is meant by area is through axioms, area can be defined as a function from a collection M of special kind of plane figures to the set of real numbers which satisfies the following properties, For all S in M, a ≥0. If S and T are in M then so are S ∪ T and S ∩ T, if S and T are in M with S ⊆ T then T − S is in M and a = a − a. If a set S is in M and S is congruent to T then T is also in M, every rectangle R is in M. If the rectangle has length h and breadth k then a = hk, let Q be a set enclosed between two step regions S and T
8.
Dual polygon
–
In geometry, polygons are associated into pairs called duals, where the vertices of one correspond to the edges of the other. The dual of a polygon is an isotoxal polygon. For example, the rectangle and rhombus are duals, in a cyclic polygon, longer sides correspond to larger exterior angles in the dual, and shorter sides to smaller angles. Further, congruent sides in the original polygon yields congruent angles in the dual, for example, the dual of a highly acute isosceles triangle is an obtuse isosceles triangle. In the Dorman Luke construction, each face of a polyhedron is the dual polygon of the corresponding vertex figure. As an example of the duality of polygons we compare properties of the cyclic. This duality is perhaps more clear when comparing an isosceles trapezoid to a kite. The simplest qualitative construction of a polygon is a rectification operation. New edges are formed between these new vertices and that is, the polygon generated by applying it twice is in general not similar to the original polygon. As with dual polyhedra, one can take a circle and perform polar reciprocation in it. Combinatorially, one can define a polygon as a set of vertices, a set of edges, then the dual polygon is obtained by simply switching the vertices and edges. Thus for the triangle with vertices and edges, the triangle has vertices, and edges, where B connects AB & BC. This is not a particularly fruitful avenue, as combinatorially, there is a family of polygons, geometric duality of polygons is more varied. Dual curve Dual polyhedron Self-dual polygon Dual Polygon Applet by Don Hatch
9.
Rectangle
–
In Euclidean plane geometry, a rectangle is a quadrilateral with four right angles. It can also be defined as a quadrilateral, since equiangular means that all of its angles are equal. It can also be defined as a parallelogram containing a right angle, a rectangle with four sides of equal length is a square. The term oblong is occasionally used to refer to a non-square rectangle, a rectangle with vertices ABCD would be denoted as ABCD. The word rectangle comes from the Latin rectangulus, which is a combination of rectus and angulus, a crossed rectangle is a crossed quadrilateral which consists of two opposite sides of a rectangle along with the two diagonals. It is a case of an antiparallelogram, and its angles are not right angles. Other geometries, such as spherical, elliptic, and hyperbolic, have so-called rectangles with sides equal in length. Rectangles are involved in many tiling problems, such as tiling the plane by rectangles or tiling a rectangle by polygons, a convex quadrilateral with successive sides a, b, c, d whose area is 12. A rectangle is a case of a parallelogram in which each pair of adjacent sides is perpendicular. A parallelogram is a case of a trapezium in which both pairs of opposite sides are parallel and equal in length. A trapezium is a quadrilateral which has at least one pair of parallel opposite sides. A convex quadrilateral is Simple, The boundary does not cross itself, star-shaped, The whole interior is visible from a single point, without crossing any edge. De Villiers defines a more generally as any quadrilateral with axes of symmetry through each pair of opposite sides. This definition includes both right-angled rectangles and crossed rectangles, quadrilaterals with two axes of symmetry, each through a pair of opposite sides, belong to the larger class of quadrilaterals with at least one axis of symmetry through a pair of opposite sides. These quadrilaterals comprise isosceles trapezia and crossed isosceles trapezia, a rectangle is cyclic, all corners lie on a single circle. It is equiangular, all its corner angles are equal and it is isogonal or vertex-transitive, all corners lie within the same symmetry orbit. It has two lines of symmetry and rotational symmetry of order 2. The dual polygon of a rectangle is a rhombus, as shown in the table below, the figure formed by joining, in order, the midpoints of the sides of a rectangle is a rhombus and vice versa
10.
Convex polygon
–
A convex polygon is a simple polygon in which no line segment between two points on the boundary ever goes outside the polygon. Equivalently, it is a polygon whose interior is a convex set. In a convex polygon, all angles are less than or equal to 180 degrees. A simple polygon which is not convex is called concave, the following properties of a simple polygon are all equivalent to convexity, Every internal angle is less than or equal to 180 degrees. Every point on line segment between two points inside or on the boundary of the polygon remains inside or on the boundary. The polygon is contained in a closed half-plane defined by each of its edges. For each edge, the points are all on the same side of the line that the edge defines. The angle at each vertex contains all vertices in its edges. The polygon is the hull of its edges. Additional properties of convex polygons include, The intersection of two convex polygons is a convex polygon, a convex polygon may br triangulated in linear time through a fan triangulation, consisting in adding diagonals from one vertex to all other vertices. Hellys theorem, For every collection of at least three convex polygons, if the intersection of three of them is nonempty, then the whole collection has a nonempty intersection. Krein–Milman theorem, A convex polygon is the hull of its vertices. Thus it is defined by the set of its vertices. Hyperplane separation theorem, Any two convex polygons with no points in common have a separator line, if the polygons are closed and at least one of them is compact, then there are even two parallel separator lines. Inscribed triangle property, Of all triangles contained in a convex polygon, inscribing triangle property, every convex polygon with area A can be inscribed in a triangle of area at most equal to 2A. Equality holds for a parallelogram.5 × Area ≤ Area ≤2 × Area, the mean width of a convex polygon is equal to its perimeter divided by pi. So its width is the diameter of a circle with the perimeter as the polygon. Every polygon inscribed in a circle, if not self-intersecting, is convex, however, not every convex polygon can be inscribed in a circle
11.
Isotoxal figure
–
In geometry, a polytope, or a tiling, is isotoxal or edge-transitive if its symmetries act transitively on its edges. The term isotoxal is derived from the Greek τοξον meaning arc, an isotoxal polygon is an equilateral polygon, but not all equilateral polygons are isotoxal. The duals of isotoxal polygons are isogonal polygons, in general, an isotoxal 2n-gon will have Dn dihedral symmetry. A rhombus is a polygon with D2 symmetry. All regular polygons are isotoxal, having double the symmetry order. A regular 2n-gon is a polygon and can be marked with alternately colored vertices. An isotoxal polyhedron or tiling must be either isogonal or isohedral or both, regular polyhedra are isohedral, isogonal and isotoxal. Quasiregular polyhedra are isogonal and isotoxal, but not isohedral, their duals are isohedral and isotoxal, not every polyhedron or 2-dimensional tessellation constructed from regular polygons is isotoxal. An isotoxal polyhedron has the dihedral angle for all edges. There are nine convex isotoxal polyhedra formed from the Platonic solids,8 formed by the Kepler–Poinsot polyhedra, cS1 maint, Multiple names, authors list Coxeter, Harold Scott MacDonald, Longuet-Higgins, M. S. Miller, J. C. P. Uniform polyhedra, Philosophical Transactions of the Royal Society of London, mathematical and Physical Sciences,246, 401–450, doi,10. 1098/rsta.1954.0003, ISSN 0080-4614, JSTOR91532, MR0062446
12.
Euclidean geometry
–
Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry, the Elements. Euclids method consists in assuming a set of intuitively appealing axioms. Although many of Euclids results had been stated by earlier mathematicians, Euclid was the first to show how these propositions could fit into a comprehensive deductive and logical system. The Elements begins with plane geometry, still taught in school as the first axiomatic system. It goes on to the geometry of three dimensions. Much of the Elements states results of what are now called algebra and number theory, for more than two thousand years, the adjective Euclidean was unnecessary because no other sort of geometry had been conceived. Euclids axioms seemed so obvious that any theorem proved from them was deemed true in an absolute, often metaphysical. Today, however, many other self-consistent non-Euclidean geometries are known, Euclidean geometry is an example of synthetic geometry, in that it proceeds logically from axioms to propositions without the use of coordinates. This is in contrast to analytic geometry, which uses coordinates, the Elements is mainly a systematization of earlier knowledge of geometry. Its improvement over earlier treatments was recognized, with the result that there was little interest in preserving the earlier ones. There are 13 total books in the Elements, Books I–IV, Books V and VII–X deal with number theory, with numbers treated geometrically via their representation as line segments with various lengths. Notions such as numbers and rational and irrational numbers are introduced. The infinitude of prime numbers is proved, a typical result is the 1,3 ratio between the volume of a cone and a cylinder with the same height and base. Euclidean geometry is a system, in which all theorems are derived from a small number of axioms. To produce a straight line continuously in a straight line. To describe a circle with any centre and distance and that all right angles are equal to one another. Although Euclids statement of the only explicitly asserts the existence of the constructions. The Elements also include the five common notions, Things that are equal to the same thing are also equal to one another
13.
Simple polygon
–
In geometry a simple polygon /ˈpɒlɪɡɒn/ is a flat shape consisting of straight, non-intersecting line segments or sides that are joined pair-wise to form a closed path. If the sides then the polygon is not simple. The qualifier simple is frequently omitted, with the definition then being understood to define a polygon in general. The definition given above ensures the following properties, A polygon encloses a region which always has a measurable area, the line segments that make-up a polygon meet only at their endpoints, called vertices or less formally corners. Exactly two edges meet at each vertex, the number of edges always equals the number of vertices. Two edges meeting at a corner are usually required to form an angle that is not straight, otherwise, according to the definition in use, this boundary may or may not form part of the polygon itself. A polygon in the plane is simple if and only if it is equivalent to a circle. Its interior is topologically equivalent to a disk, if a collection of non-crossing line segments forms the boundary of a region of the plane that is topologically equivalent to a disk, then this boundary is called a weakly simple polygon. In the image on the left, ABCDEFGHJKLM is a simple polygon according to this definition. Referring to the image above, ABCM is a boundary of a planar region with a hole FGHJ. The cut ED connects the hole with the exterior and is traversed twice in the resulting weakly simple polygonal representation and this formalizes the notion that such a polygon allows segments to touch but not to cross. However, this type of weakly simple polygon does not need to form the boundary of a region, as its interior can be empty. For example, referring to the image above, the polygonal chain ABCBA is a simple polygon according to this definition. Point in polygon testing involves determining, for a simple polygon P, simple formulae are known for computing polygon area, that is, the area of the interior of the polygon. Polygon partition is a set of units, which do not overlap. A polygon partition problem is a problem of finding a partition which is minimal in some sense, for example, a special case of polygon partition is Polygon triangulation, dividing a simple polygon into triangles. Although convex polygons are easy to triangulate, triangulating a general polygon is more difficult because we have to avoid adding edges that cross outside the polygon. Nevertheless, Bernard Chazelle showed in 1991 that any simple polygon with n vertices can be triangulated in Θ time, the same algorithm may also be used for determining whether a closed polygonal chain forms a simple polygon
14.
Suit (cards)
–
In playing cards, a suit is one of several categories into which the cards of a deck are divided. Most often, each bears one of several pips showing to which suit it belongs. The rank for each card is determined by the number of pips on it, ranking indicates which cards within a suit are better, higher or more valuable than others, whereas there is no order between the suits unless defined in the rules of a specific card game. Unless playing with multiple decks, there is one card of any given rank in any given suit. A deck may include cards that belong to no suit. Various languages have different terminology for such as colors, signs. Modern Western playing cards are divided into two or three general suit-systems. The older Latin suits are subdivided into the Italian and Spanish suit-systems, the younger Germanic suits are subdivided into the German and Swiss suit-systems. The French suits are a derivative of the German suits but are considered a separate system on its own. The Latin suits consist of coins, clubs, cups, and they are the earliest suit-system in Europe, having been adopted from the cards imported from Mamluk Egypt and Moorish Granada in the 1370s. These Turko-Arabic cards, called Kanjifa, employed the same suits, europeans changed that suit as polo was an obscure sport to them. Ultimately the suits can trace their roots back to China where playing cards were first invented, the earliest card games were trick-taking games and the invention of suits increased the level of strategy and depth in these games. A card of one suit cannot beat a card from another regardless of its rank, the concept of suits predate playing cards and can be found in Chinese dice and domino games such as Tien Gow. Chinese money-suited cards are believed to be the oldest ancestor to the Latin suit-system, the money-suit system is based on denominations of currency, Coins, Strings of Coins, Myriads of Strings, and Tens of Myriads. Old Chinese coins had holes in the middle to allow them to be strung together, a string of coins could easily be misinterpreted as a stick to those unfamiliar with them. The Mamluks called their suit of cups Myriads and this may have due to inverting the Chinese character for myriad. The Mamluk suit of swords may also have inspired by the Chinese numeral for Ten. Another clue linking these Chinese, Muslim, and European cards are the ranking of certain suits, in many early Chinese games like Madiao, the suit of coins was in reverse order so that the lower ones beat the higher ones
15.
Playing card
–
Playing cards are typically palm-sized for convenient handling. A complete set of cards is called a pack, deck, or set, a pack of cards may be used for playing a variety of card games, with varying elements of skill and chance, some of which are played for money. Playing cards are used for illusions, cardistry, building card structures, cartomancy. The front of each card carries markings that distinguish it from the cards in the pack. The back of each card is identical for all cards in any particular pack to create an imperfect information scenario, usually every card will be smooth, however, some packs have braille to allow blind people to read the card number and suit. Dedicated deck card games have sets that are used only for a specific game, the cards described in this article are used for many games and share a common origin stemming from the standards set in Mamluk Egypt. These sets divide their cards into four suits each consisting of three cards and numbered or pip cards. The scholarly consensus is that playing cards were invented in Imperial China and they first appeared as early as 9th century Tang China. The first known book on the game was called the Yezi Gexi and was allegedly written by a Tang woman. However, Ouyang claimed the leaves were pages of a book for a game played with dice. In any case, Ouyang asserted that the rules for the game were lost by 1067 and it may be that the first deck of cards ever printed was a 32-card Chinese domino pack, in whose cards all 21 combinations of a pair of dice are depicted. According to the Gui Tian Lu, an 11th-century Chinese text redacted, domino cards were printed during the Tang dynasty, there is difficulty distinguishing paper cards and gaming tiles in many early sources as the Chinese word pái is used to describe both. Playing cards are paper pái while tiles are called bone pái, paper playing cards and the woodblocks to print them are unambiguously attested in 1294. William Henry Wilkinson suggests that the first cards may have been actual paper currency which were both the tools of gaming and the stakes being played for, as in trading card games. As using paper money was inconvenient and risky, they were substituted by play money known as money-suited cards, one of the earliest games of which we know the rules is Madiao, a trick-taking game, which dates to the Ming Dynasty. 15th century scholar Lu Rong described it is as being played with 38 money-suited cards divided into four suits,9 in coins,9 in strings of coins,9 in myriads, and 11 in tens of myriads. The two latter suits had Water Margin characters instead of pips on them with Chinese ideograms to mark their rank, the pips were copied directly from Chinese banknotes such as the Songs Jiaozi or the Yuans Chao currencies. The suit of coins is in order with 9 of coins being the lowest going up to 1 of coins as the high card
16.
Octahedron
–
In geometry, an octahedron is a polyhedron with eight faces, twelve edges, and six vertices. A regular octahedron is a Platonic solid composed of eight equilateral triangles, a regular octahedron is the dual polyhedron of a cube. It is a square bipyramid in any of three orthogonal orientations and it is also a triangular antiprism in any of four orientations. An octahedron is the case of the more general concept of a cross polytope. A regular octahedron is a 3-ball in the Manhattan metric, the second and third correspond to the B2 and A2 Coxeter planes. The octahedron can also be represented as a tiling. This projection is conformal, preserving angles but not areas or lengths, straight lines on the sphere are projected as circular arcs on the plane. An octahedron with edge length √2 can be placed with its center at the origin and its vertices on the coordinate axes, the Cartesian coordinates of the vertices are then. In an x–y–z Cartesian coordinate system, the octahedron with center coordinates, additionally the inertia tensor of the stretched octahedron is I =. These reduce to the equations for the regular octahedron when x m = y m = z m = a 22, the interior of the compound of two dual tetrahedra is an octahedron, and this compound, called the stella octangula, is its first and only stellation. Correspondingly, an octahedron is the result of cutting off from a regular tetrahedron. One can also divide the edges of an octahedron in the ratio of the mean to define the vertices of an icosahedron. There are five octahedra that define any given icosahedron in this fashion, octahedra and tetrahedra can be alternated to form a vertex, edge, and face-uniform tessellation of space, called the octet truss by Buckminster Fuller. This is the only such tiling save the regular tessellation of cubes, another is a tessellation of octahedra and cuboctahedra. The octahedron is unique among the Platonic solids in having a number of faces meeting at each vertex. Consequently, it is the member of that group to possess mirror planes that do not pass through any of the faces. Using the standard nomenclature for Johnson solids, an octahedron would be called a square bipyramid, truncation of two opposite vertices results in a square bifrustum. The octahedron is 4-connected, meaning that it takes the removal of four vertices to disconnect the remaining vertices and it is one of only four 4-connected simplicial well-covered polyhedra, meaning that all of the maximal independent sets of its vertices have the same size
17.
Diamond
–
Diamond is a metastable allotrope of carbon, where the carbon atoms are arranged in a variation of the face-centered cubic crystal structure called a diamond lattice. Diamond is less stable than graphite, but the rate from diamond to graphite is negligible at standard conditions. Diamond is renowned as a material with superlative physical qualities, most of which originate from the covalent bonding between its atoms. In particular, diamond has the highest hardness and thermal conductivity of any bulk material and those properties determine the major industrial application of diamond in cutting and polishing tools and the scientific applications in diamond knives and diamond anvil cells. Because of its extremely rigid lattice, it can be contaminated by very few types of impurities, such as boron, small amounts of defects or impurities color diamond blue, yellow, brown, green, purple, pink, orange or red. Diamond also has relatively high optical dispersion, most natural diamonds are formed at high temperature and pressure at depths of 140 to 190 kilometers in the Earths mantle. Carbon-containing minerals provide the source, and the growth occurs over periods from 1 billion to 3.3 billion years. Diamonds are brought close to the Earths surface through deep volcanic eruptions by magma, Diamonds can also be produced synthetically in a HPHT method which approximately simulates the conditions in the Earths mantle. An alternative, and completely different growth technique is chemical vapor deposition, several non-diamond materials, which include cubic zirconia and silicon carbide and are often called diamond simulants, resemble diamond in appearance and many properties. Special gemological techniques have developed to distinguish natural diamonds, synthetic diamonds. The word is from the ancient Greek ἀδάμας – adámas unbreakable, the name diamond is derived from the ancient Greek αδάμας, proper, unalterable, unbreakable, untamed, from ἀ-, un- + δαμάω, I overpower, I tame. Diamonds have been known in India for at least 3,000 years, Diamonds have been treasured as gemstones since their use as religious icons in ancient India. Their usage in engraving tools also dates to early human history, later in 1797, the English chemist Smithson Tennant repeated and expanded that experiment. By demonstrating that burning diamond and graphite releases the same amount of gas, the most familiar uses of diamonds today are as gemstones used for adornment, a use which dates back into antiquity, and as industrial abrasives for cutting hard materials. The dispersion of light into spectral colors is the primary gemological characteristic of gem diamonds. In the 20th century, experts in gemology developed methods of grading diamonds, four characteristics, known informally as the four Cs, are now commonly used as the basic descriptors of diamonds, these are carat, cut, color, and clarity. A large, flawless diamond is known as a paragon and these conditions are met in two places on Earth, in the lithospheric mantle below relatively stable continental plates, and at the site of a meteorite strike. The conditions for diamond formation to happen in the mantle occur at considerable depth corresponding to the requirements of temperature and pressure
18.
Lozenge
–
A lozenge, often referred to as a diamond, is a form of rhombus. The definition of lozenge is not strictly fixed, and it is used simply as a synonym for rhombus. Most often, though, lozenge refers to a thin rhombus—a rhombus with angles of less than 45°. The lozenge shape is used in parquetry and as decoration on ceramics, silverware. It also features in heraldry and playing cards, the lozenge motif dates as far back as the Neolithic and Paleolithic period in Eastern Europe and represents a sown field and female fertility. The ancient lozenge pattern often shows up in Diamond vault architecture, in dress patterns of Slavic peoples. The lozenge pattern also appears extensively in Celtic art, art from the Ottoman Empire, the lozenge symbolism is one of the main female symbols in Berber carpets. Common Berber jewelry from the Aurès Mountains or Kabylie in Algeria also uses this pattern as a female fertility sign and he suggested that ancient plantations were laid out in a lozenge pattern. Lozenges appear as symbols in ancient classic element systems, in amulets, in a suit of playing cards, diamonds is in the shape of a lozenge. The TeX command for the lozenge is \lozenge, in modal logic, the lozenge expresses that there is possibility. For example, the expression ◊ P expresses that it is possible that P is true, in axiomatic set theory, the lozenge refers to the principles known collectively as the diamond principle. On equipment, especially calculators, the lozenge is used to mark the subtotal key and it is standardized in ISO7000 as symbol ISO-7000-0650. Similar, the lozenge, part of the BCDIC code, was used on tabulation listings to indicate second level totals in banking installations in the 1960s. During the First World War, the Germans developed Lozenge-Tarnung and this camouflage was made up of colored polygons of four or five colors. The repeating patterns often used irregular four-, five- and six-sided polygons, because painting such a pattern was very time consuming, and the paint added considerably to the weight of the aircraft, the pattern was printed on fabric. This pre-printed fabric was used from 1916 until the end of the war, in various forms, the lozenge in heraldry is a diamond-shaped charge, usually somewhat narrower than it is tall. A mascle is a voided lozenge—that is, a lozenge with a hole in the middle—and the rarer rustre is a lozenge containing a circular hole. A field covered in a pattern of lozenges is described as lozengy, cough tablets have taken the name lozenge, based on their original shape
19.
Polyiamond
–
A polyiamond is a polyform whose base form is an equilateral triangle. The name was suggested by recreational mathematics writer Thomas H. OBeirne in New Scientist 1961 number 1, the basic combinatorial question is, How many different polyiamonds exist with a given number of cells. Like polyominoes, polyiamonds may be free or one-sided. Free polyiamonds are invariant under reflection as well as translation and rotation, the number of free n-iamonds for n =1,2,3, … is,1,1,1,3,4,12,24,66,160, …. Possible symmetries are mirror symmetry, 2-, 3-, and 6-fold rotational symmetry, 2-fold rotational symmetry with and without mirror symmetry requires at least 2 and 4 triangles, respectively. 6-fold rotational symmetry with and without mirror symmetry requires at least 6 and 18 triangles, asymmetry requires at least 5 triangles. 3-fold rotational symmetry without mirror symmetry requires at least 7 triangles, in the case of only mirror symmetry we can distinguish having the symmetry axis aligned with the grid or rotated 30°, ditto for 3-fold rotational symmetry, combined with mirror symmetry. Like polyominoes, but unlike polyhexes, polyiamonds have three-dimensional counterparts, however, polytetrahedra do not tile 3-space in the way polyiamonds can tile 2-space. Every polyiamond of order 6 or less tiles the plane, all but one of the heptiamonds tile the plane. Every polyiamond corresponds to a polyhex, as illustrated at right, conversely, every polyhex is also a polyiamond, because each hexagonal cell of a polyhex is the union of six adjacent equilateral triangles. Triangular tiling Rhombille tiling Sphinx tiling Weisstein, Eric W. Polyiamond, VERHEXT — a 1960s puzzle game by Heinz Haber based on hexiamonds
20.
Square
–
In geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles. It can also be defined as a rectangle in which two adjacent sides have equal length, a square with vertices ABCD would be denoted ◻ ABCD. e. A rhombus with equal diagonals a convex quadrilateral with sides a, b, c, d whose area is A =12 =12. Opposite sides of a square are both parallel and equal in length, all four angles of a square are equal. All four sides of a square are equal, the diagonals of a square are equal. The square is the n=2 case of the families of n-hypercubes and n-orthoplexes, a truncated square, t, is an octagon. An alternated square, h, is a digon, the perimeter of a square whose four sides have length ℓ is P =4 ℓ and the area A is A = ℓ2. In classical times, the power was described in terms of the area of a square. This led to the use of the square to mean raising to the second power. The area can also be calculated using the diagonal d according to A = d 22. In terms of the circumradius R, the area of a square is A =2 R2, since the area of the circle is π R2, in terms of the inradius r, the area of the square is A =4 r 2. Because it is a polygon, a square is the quadrilateral of least perimeter enclosing a given area. Dually, a square is the quadrilateral containing the largest area within a given perimeter. Indeed, if A and P are the area and perimeter enclosed by a quadrilateral, then the isoperimetric inequality holds,16 A ≤ P2 with equality if. The diagonals of a square are 2 times the length of a side of the square and this value, known as the square root of 2 or Pythagoras constant, was the first number proven to be irrational. A square can also be defined as a parallelogram with equal diagonals that bisect the angles, if a figure is both a rectangle and a rhombus, then it is a square. If a circle is circumscribed around a square, the area of the circle is π /2 times the area of the square, if a circle is inscribed in the square, the area of the circle is π /4 times the area of the square. A square has an area than any other quadrilateral with the same perimeter
21.
Greek language
–
Greek is an independent branch of the Indo-European family of languages, native to Greece and other parts of the Eastern Mediterranean. It has the longest documented history of any living language, spanning 34 centuries of written records and its writing system has been the Greek alphabet for the major part of its history, other systems, such as Linear B and the Cypriot syllabary, were used previously. The alphabet arose from the Phoenician script and was in turn the basis of the Latin, Cyrillic, Armenian, Coptic, Gothic and many other writing systems. Together with the Latin texts and traditions of the Roman world, during antiquity, Greek was a widely spoken lingua franca in the Mediterranean world and many places beyond. It would eventually become the official parlance of the Byzantine Empire, the language is spoken by at least 13.2 million people today in Greece, Cyprus, Italy, Albania, Turkey, and the Greek diaspora. Greek roots are used to coin new words for other languages, Greek. Greek has been spoken in the Balkan peninsula since around the 3rd millennium BC, the earliest written evidence is a Linear B clay tablet found in Messenia that dates to between 1450 and 1350 BC, making Greek the worlds oldest recorded living language. Among the Indo-European languages, its date of earliest written attestation is matched only by the now extinct Anatolian languages, the Greek language is conventionally divided into the following periods, Proto-Greek, the unrecorded but assumed last ancestor of all known varieties of Greek. The unity of Proto-Greek would have ended as Hellenic migrants entered the Greek peninsula sometime in the Neolithic era or the Bronze Age, Mycenaean Greek, the language of the Mycenaean civilisation. It is recorded in the Linear B script on tablets dating from the 15th century BC onwards, Ancient Greek, in its various dialects, the language of the Archaic and Classical periods of the ancient Greek civilisation. It was widely known throughout the Roman Empire, after the Roman conquest of Greece, an unofficial bilingualism of Greek and Latin was established in the city of Rome and Koine Greek became a first or second language in the Roman Empire. The origin of Christianity can also be traced through Koine Greek, Medieval Greek, also known as Byzantine Greek, the continuation of Koine Greek in Byzantine Greece, up to the demise of the Byzantine Empire in the 15th century. Much of the written Greek that was used as the language of the Byzantine Empire was an eclectic middle-ground variety based on the tradition of written Koine. Modern Greek, Stemming from Medieval Greek, Modern Greek usages can be traced in the Byzantine period and it is the language used by the modern Greeks, and, apart from Standard Modern Greek, there are several dialects of it. In the modern era, the Greek language entered a state of diglossia, the historical unity and continuing identity between the various stages of the Greek language is often emphasised. Greek speakers today still tend to regard literary works of ancient Greek as part of their own rather than a foreign language and it is also often stated that the historical changes have been relatively slight compared with some other languages. According to one estimation, Homeric Greek is probably closer to demotic than 12-century Middle English is to modern spoken English, Greek is spoken by about 13 million people, mainly in Greece, Albania and Cyprus, but also worldwide by the large Greek diaspora. Greek is the language of Greece, where it is spoken by almost the entire population
22.
Euclid
–
Euclid, sometimes called Euclid of Alexandria to distinguish him from Euclides of Megara, was a Greek mathematician, often referred to as the father of geometry. He was active in Alexandria during the reign of Ptolemy I, in the Elements, Euclid deduced the principles of what is now called Euclidean geometry from a small set of axioms. Euclid also wrote works on perspective, conic sections, spherical geometry, number theory, Euclid is the anglicized version of the Greek name Εὐκλείδης, which means renowned, glorious. Very few original references to Euclid survive, so little is known about his life, the date, place and circumstances of both his birth and death are unknown and may only be estimated roughly relative to other people mentioned with him. He is rarely mentioned by name by other Greek mathematicians from Archimedes onward, the few historical references to Euclid were written centuries after he lived by Proclus c.450 AD and Pappus of Alexandria c.320 AD. Proclus introduces Euclid only briefly in his Commentary on the Elements, Proclus later retells a story that, when Ptolemy I asked if there was a shorter path to learning geometry than Euclids Elements, Euclid replied there is no royal road to geometry. This anecdote is questionable since it is similar to a story told about Menaechmus, a detailed biography of Euclid is given by Arabian authors, mentioning, for example, a birth town of Tyre. This biography is generally believed to be completely fictitious, however, this hypothesis is not well accepted by scholars and there is little evidence in its favor. The only reference that historians rely on of Euclid having written the Elements was from Proclus, although best known for its geometric results, the Elements also includes number theory. The geometrical system described in the Elements was long known simply as geometry, today, however, that system is often referred to as Euclidean geometry to distinguish it from other so-called non-Euclidean geometries that mathematicians discovered in the 19th century. In addition to the Elements, at least five works of Euclid have survived to the present day and they follow the same logical structure as Elements, with definitions and proved propositions. Data deals with the nature and implications of information in geometrical problems. On Divisions of Figures, which only partially in Arabic translation. It is similar to a first-century AD work by Heron of Alexandria, catoptrics, which concerns the mathematical theory of mirrors, particularly the images formed in plane and spherical concave mirrors. The attribution is held to be anachronistic however by J J OConnor, phaenomena, a treatise on spherical astronomy, survives in Greek, it is quite similar to On the Moving Sphere by Autolycus of Pitane, who flourished around 310 BC. Optics is the earliest surviving Greek treatise on perspective, in its definitions Euclid follows the Platonic tradition that vision is caused by discrete rays which emanate from the eye. One important definition is the fourth, Things seen under a greater angle appear greater, proposition 45 is interesting, proving that for any two unequal magnitudes, there is a point from which the two appear equal. Other works are attributed to Euclid, but have been lost
23.
Archimedes
–
Archimedes of Syracuse was a Greek mathematician, physicist, engineer, inventor, and astronomer. Although few details of his life are known, he is regarded as one of the scientists in classical antiquity. He was also one of the first to apply mathematics to physical phenomena, founding hydrostatics and statics and he is credited with designing innovative machines, such as his screw pump, compound pulleys, and defensive war machines to protect his native Syracuse from invasion. Archimedes died during the Siege of Syracuse when he was killed by a Roman soldier despite orders that he should not be harmed. Cicero describes visiting the tomb of Archimedes, which was surmounted by a sphere and a cylinder, unlike his inventions, the mathematical writings of Archimedes were little known in antiquity. Archimedes was born c.287 BC in the city of Syracuse, Sicily, at that time a self-governing colony in Magna Graecia. The date of birth is based on a statement by the Byzantine Greek historian John Tzetzes that Archimedes lived for 75 years, in The Sand Reckoner, Archimedes gives his fathers name as Phidias, an astronomer about whom nothing is known. Plutarch wrote in his Parallel Lives that Archimedes was related to King Hiero II, a biography of Archimedes was written by his friend Heracleides but this work has been lost, leaving the details of his life obscure. It is unknown, for instance, whether he married or had children. During his youth, Archimedes may have studied in Alexandria, Egypt and he referred to Conon of Samos as his friend, while two of his works have introductions addressed to Eratosthenes. Archimedes died c.212 BC during the Second Punic War, according to the popular account given by Plutarch, Archimedes was contemplating a mathematical diagram when the city was captured. A Roman soldier commanded him to come and meet General Marcellus but he declined, the soldier was enraged by this, and killed Archimedes with his sword. Plutarch also gives an account of the death of Archimedes which suggests that he may have been killed while attempting to surrender to a Roman soldier. According to this story, Archimedes was carrying mathematical instruments, and was killed because the thought that they were valuable items. General Marcellus was reportedly angered by the death of Archimedes, as he considered him a valuable asset and had ordered that he not be harmed. Marcellus called Archimedes a geometrical Briareus, the last words attributed to Archimedes are Do not disturb my circles, a reference to the circles in the mathematical drawing that he was supposedly studying when disturbed by the Roman soldier. This quote is given in Latin as Noli turbare circulos meos. The phrase is given in Katharevousa Greek as μὴ μου τοὺς κύκλους τάραττε
24.
Cone
–
A cone is a three-dimensional geometric shape that tapers smoothly from a flat base to a point called the apex or vertex. A cone is formed by a set of segments, half-lines, or lines connecting a common point. If the enclosed points are included in the base, the cone is a solid object, otherwise it is a two-dimensional object in three-dimensional space. In the case of an object, the boundary formed by these lines or partial lines is called the lateral surface, if the lateral surface is unbounded. In the case of line segments, the cone does not extend beyond the base, while in the case of half-lines, in the case of lines, the cone extends infinitely far in both directions from the apex, in which case it is sometimes called a double cone. Either half of a cone on one side of the apex is called a nappe. The axis of a cone is the line, passing through the apex. If the base is right circular the intersection of a plane with this surface is a conic section, in general, however, the base may be any shape and the apex may lie anywhere. Contrasted with right cones are oblique cones, in which the axis passes through the centre of the base non-perpendicularly, a cone with a polygonal base is called a pyramid. Depending on the context, cone may also mean specifically a convex cone or a projective cone, cones can also be generalized to higher dimensions. The perimeter of the base of a cone is called the directrix, the base radius of a circular cone is the radius of its base, often this is simply called the radius of the cone. The aperture of a circular cone is the maximum angle between two generatrix lines, if the generatrix makes an angle θ to the axis, the aperture is 2θ. A cone with a region including its apex cut off by a plane is called a cone, if the truncation plane is parallel to the cones base. An elliptical cone is a cone with an elliptical base, a generalized cone is the surface created by the set of lines passing through a vertex and every point on a boundary. The slant height of a circular cone is the distance from any point on the circle to the apex of the cone. It is given by r 2 + h 2, where r is the radius of the cirf the cone and this application is primarily useful in determining the slant height of a cone when given other information regarding the radius or height. The volume V of any conic solid is one third of the product of the area of the base A B and the height h V =13 A B h. In modern mathematics, this formula can easily be computed using calculus – it is, up to scaling, the integral ∫ x 2 d x =13 x 3
25.
Cross section (geometry)
–
In geometry and science, a cross section is the intersection of a body in three-dimensional space with a plane, or the analog in higher-dimensional space. Cutting an object into slices creates many parallel cross sections, conic sections – circles, ellipses, parabolas, and hyperbolas – are formed by cross-sections of a cone at various different angles, as seen in the diagram at left. Any planar cross-section passing through the center of an ellipsoid forms an ellipse on its surface, a cross-section of a cylinder is a circle if the cross-section is parallel to the cylinders base, or an ellipse with non-zero eccentricity if it is neither parallel nor perpendicular to the base. If the cross-section is perpendicular to the base it consists of two line segments unless it is just tangent to the cylinder, in which case it is a single line segment. A cross section of a polyhedron is a polygon, if instead the cross section is taken for a fixed value of the density, the result is an iso-density contour. For the normal distribution, these contours are ellipses, a cross section can be used to visualize the partial derivative of a function with respect to one of its arguments, as shown at left. In economics, a function f specifies the output that can be produced by various quantities x and y of inputs, typically labor. The production function of a firm or a society can be plotted in three-dimensional space, also in economics, a cardinal or ordinal utility function u gives the degree of satisfaction of a consumer obtained by consuming quantities w and v of two goods. Cross sections are used in anatomy to illustrate the inner structure of an organ. A cross section of a trunk, as shown at left, reveals growth rings that can be used to find the age of the tree. Cavalieris principle states that solids with corresponding sections of equal areas have equal volumes. The cross-sectional area of an object when viewed from an angle is the total area of the orthographic projection of the object from that angle. For example, a cylinder of height h and radius r has A ′ = π r 2 when viewed along its central axis, a sphere of radius r has A ′ = π r 2 when viewed from any angle. For a convex body, each ray through the object from the viewers perspective crosses just two surfaces, descriptive geometry Exploded view drawing Graphical projection Plans
26.
Diagonal
–
In geometry, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge. Informally, any sloping line is called diagonal, in matrix algebra, a diagonal of a square matrix is a set of entries extending from one corner to the farthest corner. There are also other, non-mathematical uses, diagonal pliers are wire-cutting pliers defined by the cutting edges of the jaws intersects the joint rivet at an angle or on a diagonal, hence the name. A diagonal lashing is a type of lashing used to bind spars or poles together applied so that the cross over the poles at an angle. In association football, the system of control is the method referees. As applied to a polygon, a diagonal is a line segment joining any two non-consecutive vertices, therefore, a quadrilateral has two diagonals, joining opposite pairs of vertices. For any convex polygon, all the diagonals are inside the polygon, in a convex polygon, if no three diagonals are concurrent at a single point, the number of regions that the diagonals divide the interior into is given by + =24. The number of regions is 1,4,11,25,50,91,154,246, in a polygon with n angles the number of diagonals is given by n ∗2. The number of intersections between the diagonals is given by, in the case of a square matrix, the main or principal diagonal is the diagonal line of entries running from the top-left corner to the bottom-right corner. For a matrix A with row index specified by i and column index specified by j, the off-diagonal entries are those not on the main diagonal. A diagonal matrix is one whose off-diagonal entries are all zero, a superdiagonal entry is one that is directly above and to the right of the main diagonal. Just as diagonal entries are those A i j with j = i and this plays an important part in geometry, for example, the fixed points of a mapping F from X to itself may be obtained by intersecting the graph of F with the diagonal. In geometric studies, the idea of intersecting the diagonal with itself is common, not directly and this is related at a deep level with the Euler characteristic and the zeros of vector fields. For example, the circle S1 has Betti numbers 1,1,0,0,0, a geometric way of expressing this is to look at the diagonal on the two-torus S1xS1 and observe that it can move off itself by the small motion to. Topics In Algebra, Waltham, Blaisdell Publishing Company, ISBN 978-1114541016 Nering, linear Algebra and Matrix Theory, New York, Wiley, LCCN76091646 Diagonals of a polygon with interactive animation Polygon diagonal from MathWorld. Diagonal of a matrix from MathWorld
27.
Internal and external angles
–
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°
28.
Orthodiagonal
–
In Euclidean geometry, an orthodiagonal quadrilateral is a quadrilateral in which the diagonals cross at right angles. In other words, it is a figure in which the line segments between non-adjacent vertices are orthogonal to each other. A kite is a quadrilateral in which one diagonal is a line of symmetry. The kites are exactly the orthodiagonal quadrilaterals that contain a circle tangent to all four of their sides, that is, a rhombus is an orthodiagonal quadrilateral with two pairs of parallel sides. A square is a case of both a kite and a rhombus. The square is one such quadrilateral, but there are many others. Conversely, any quadrilateral in which a2 + c2 = b2 + d2 must be orthodiagonal and this can be proved in a number of ways, including using the law of cosines, vectors, an indirect proof, and complex numbers. The diagonals of a quadrilateral are perpendicular if and only if the two bimedians have equal length. A convex quadrilateral is orthodiagonal if and only if its Varignon parallelogram is a rectangle, the center of this circle is the centroid of the quadrilateral. The quadrilateral formed by the feet of the maltitudes is called the principal orthic quadrilateral, a related characterization states that a convex quadrilateral is orthodiagonal if and only if RSTU is a rectangle whose sides are parallel to the diagonals of ABCD. There are several metric characterizations regarding the four triangles formed by the diagonal intersection P, denote by m1, m2, m3, m4 the medians in triangles ABP, BCP, CDP, DAP from P to the sides AB, BC, CD, DA respectively. A few metric characterizations of tangential quadrilaterals and orthodiagonal quadrilaterals are very similar in appearance, as can be seen in this table. The notations on the sides a, b, c, d, the circumradii R1, R2, R3, R4, and the altitudes h1, h2, h3, h4 are the same as above in both types of quadrilaterals. The area K of an orthodiagonal quadrilateral equals one half the product of the lengths of the p and q, K = p ⋅ q 2. Conversely, any convex quadrilateral where the area can be calculated with this formula must be orthodiagonal, the orthodiagonal quadrilateral has the biggest area of all convex quadrilaterals with given diagonals. Orthodiagonal quadrilaterals are the only quadrilaterals for which the sides and the angle formed by the diagonals do not uniquely determine the area, for example, two rhombi both having common side a, but one having a smaller acute angle than the other, have different areas. If squares are erected outward on the sides of any quadrilateral and this is called Van Aubels theorem. Then D2 = p 12 + p 22 + q 12 + q 22 = a 2 + c 2 = b 2 + d 2 where D is the diameter of the circumcircle and this holds because the diagonals are perpendicular chords of a circle
29.
Perpendicular
–
In elementary geometry, the property of being perpendicular is the relationship between two lines which meet at a right angle. The property extends to other related geometric objects, a line is said to be perpendicular to another line if the two lines intersect at a right angle. Perpendicularity can be shown to be symmetric, meaning if a first line is perpendicular to a second line, for this reason, we may speak of two lines as being perpendicular without specifying an order. Perpendicularity easily extends to segments and rays, in symbols, A B ¯ ⊥ C D ¯ means line segment AB is perpendicular to line segment CD. A line is said to be perpendicular to an if it is perpendicular to every line in the plane that it intersects. This definition depends on the definition of perpendicularity between lines, two planes in space are said to be perpendicular if the dihedral angle at which they meet is a right angle. Perpendicularity is one instance of the more general mathematical concept of orthogonality, perpendicularity is the orthogonality of classical geometric objects. Thus, in advanced mathematics, the perpendicular is sometimes used to describe much more complicated geometric orthogonality conditions. The word foot is used in connection with perpendiculars. This usage is exemplified in the top diagram, above, the diagram can be in any orientation. The foot is not necessarily at the bottom, step 2, construct circles centered at A and B having equal radius. Let Q and R be the points of intersection of two circles. Step 3, connect Q and R to construct the desired perpendicular PQ, to prove that the PQ is perpendicular to AB, use the SSS congruence theorem for and QPB to conclude that angles OPA and OPB are equal. Then use the SAS congruence theorem for triangles OPA and OPB to conclude that angles POA, to make the perpendicular to the line g at or through the point P using Thales theorem, see the animation at right. The Pythagorean Theorem can be used as the basis of methods of constructing right angles, for example, by counting links, three pieces of chain can be made with lengths in the ratio 3,4,5. These can be out to form a triangle, which will have a right angle opposite its longest side. This method is useful for laying out gardens and fields, where the dimensions are large, the chains can be used repeatedly whenever required. If two lines are perpendicular to a third line, all of the angles formed along the third line are right angles
30.
Bisection
–
In geometry, bisection is the division of something into two equal or congruent parts, usually by a line, which is then called a bisector. The most often considered types of bisectors are the segment bisector, in three-dimensional space, bisection is usually done by a plane, also called the bisector or bisecting plane. A line segment bisector passes through the midpoint of the segment, particularly important is the perpendicular bisector of a segment, which, according to its name, meets the segment at right angles. The perpendicular bisector of a segment also has the property that each of its points is equidistant from the segments endpoints, therefore, Voronoi diagram boundaries consist of segments of such lines or planes. In classical geometry, the bisection is a compass and straightedge construction, whose possibility depends on the ability to draw circles of equal radii. The segment is bisected by drawing intersecting circles of radius, whose centers are the endpoints of the segment. The line determined by the points of intersection of the two circles is the bisector of the segment, since it crosses the segment at its center. Brahmaguptas theorem states that if a cyclic quadrilateral is orthodiagonal, then the perpendicular to a side from the point of intersection of the diagonals bisects the opposite side. An angle bisector divides the angle into two angles with equal measures, an angle only has one bisector. Each point of an angle bisector is equidistant from the sides of the angle, the interior or internal bisector of an angle is the line, half-line, or line segment that divides an angle of less than 180° into two equal angles. The exterior or external bisector is the line divides the supplementary angle, formed by one side forming the original angle. To bisect an angle with straightedge and compass, one draws a circle whose center is the vertex, the circle meets the angle at two points, one on each leg. Using each of these points as a center, draw two circles of the same size, the intersection of the circles determines a line that is the angle bisector. The proof of the correctness of this construction is fairly intuitive and it is interesting to note that the trisection of an angle cannot be achieved with the compass and ruler alone. The internal and external bisectors of an angle are perpendicular, the interior angle bisectors of a triangle are concurrent in a point called the incenter of the triangle, as seen in the diagram at right. The bisectors of two angles and the bisector of the other interior angle are concurrent. Three intersection points, each of an angle bisector with the opposite extended side, are collinear. Three intersection points, two of them between an angle bisector and the opposite side, and the third between the other exterior angle bisector and the opposite side extended, are collinear
31.
Congruence (geometry)
–
In geometry, two figures or objects are congruent if they have the same shape and size, or if one has the same shape and size as the mirror image of the other. This means that either object can be repositioned and reflected so as to coincide precisely with the other object, so two distinct plane figures on a piece of paper are congruent if we can cut them out and then match them up completely. Turning the paper over is permitted, in elementary geometry the word congruent is often used as follows. The word equal is often used in place of congruent for these objects, two line segments are congruent if they have the same length. Two angles are congruent if they have the same measure, two circles are congruent if they have the same diameter. The related concept of similarity applies if the objects differ in size, for two polygons to be congruent, they must have an equal number of sides. Two polygons with n sides are congruent if and only if they each have identical sequences side-angle-side-angle-. for n sides. Congruence of polygons can be established graphically as follows, First, match, second, draw a vector from one of the vertices of the one of the figures to the corresponding vertex of the other figure. Translate the first figure by this vector so that two vertices match. Third, rotate the translated figure about the matched vertex until one pair of corresponding sides matches, fourth, reflect the rotated figure about this matched side until the figures match. If at any time the step cannot be completed, the polygons are not congruent, two triangles are congruent if their corresponding sides are equal in length, in which case their corresponding angles are equal in measure. SSS, If three pairs of sides of two triangles are equal in length, then the triangles are congruent, ASA, If two pairs of angles of two triangles are equal in measurement, and the included sides are equal in length, then the triangles are congruent. The ASA Postulate was contributed by Thales of Miletus, in most systems of axioms, the three criteria—SAS, SSS and ASA—are established as theorems. In the School Mathematics Study Group system SAS is taken as one of 22 postulates, AAS, If two pairs of angles of two triangles are equal in measurement, and a pair of corresponding non-included sides are equal in length, then the triangles are congruent. For American usage, AAS is equivalent to an ASA condition, RHS, also known as HL, If two right-angled triangles have their hypotenuses equal in length, and a pair of shorter sides are equal in length, then the triangles are congruent. The SSA condition which specifies two sides and a non-included angle does not by itself prove congruence, in order to show congruence, additional information is required such as the measure of the corresponding angles and in some cases the lengths of the two pairs of corresponding sides. The opposite side is longer when the corresponding angles are acute. This is the case and two different triangles can be formed from the given information, but further information distinguishing them can lead to a proof of congruence
32.
Incircle and excircles of a triangle
–
In geometry, the incircle or inscribed circle of a triangle is the largest circle contained in the triangle, it touches the three sides. The center of the incircle is a center called the triangles incenter. An excircle or escribed circle of the triangle is a circle lying outside the triangle, tangent to one of its sides, every triangle has three distinct excircles, each tangent to one of the triangles sides. The center of the incircle, called the incenter, can be found as the intersection of the three angle bisectors. The center of an excircle is the intersection of the internal bisector of one angle, the center of this excircle is called the excenter relative to the vertex A, or the excenter of A. Because the internal bisector of an angle is perpendicular to its external bisector, polygons with more than three sides do not all have an incircle tangent to all sides, those that do are called tangential polygons. See also Tangent lines to circles, suppose △ A B C has an incircle with radius r and center I. The distance from vertex A to the incenter I is, d = c sin cos = b sin cos The trilinear coordinates for a point in the triangle is the ratio of distances to the triangle sides. Because the Incenter is the distance of all sides the trilinear coordinates for the incenter are 1,1,1. The barycentric coordinates for a point in a triangle give weights such that the point is the average of the triangle vertex positions. The Cartesian coordinates of the incenter are an average of the coordinates of the three vertices using the side lengths of the triangle relative to the perimeter—i. e. Using the barycentric coordinates given above, normalized to sum to unity—as weights. If the three vertices are located at, and, and the sides opposite these vertices have corresponding lengths a, b, additionally, I A ⋅ I B ⋅ I C =4 R r 2, where R and r are the triangles circumradius and inradius respectively. The collection of triangle centers may be given the structure of a group under multiplication of trilinear coordinates, in this group. Then the incircle has the radius r = x y z x + y + z, the product of the incircle radius r and the circumcircle radius R of a triangle with sides a, b, and c is r R = a b c 2. Some relations among the sides, incircle radius, and circumcircle radius are, a b + b c + c a = s 2 + r, any line through a triangle that splits both the triangles area and its perimeter in half goes through the triangles incenter. There are either one, two, or three of these for any given triangle, the distance from any vertex to the incircle tangency on either adjacent side is half the sum of the vertexs adjacent sides minus half the opposite side. Thus for example for vertex B and adjacent tangencies TA and TC, the incircle radius is no greater than one-ninth the sum of the altitudes
33.
Triangle
–
A triangle is a polygon with three edges and three vertices. It is one of the shapes in geometry. A triangle with vertices A, B, and C is denoted △ A B C, in Euclidean geometry any three points, when non-collinear, determine a unique triangle and a unique plane. This article is about triangles in Euclidean geometry except where otherwise noted, triangles can be classified according to the lengths of their sides, An equilateral triangle has all sides the same length. An equilateral triangle is also a polygon with all angles measuring 60°. An isosceles triangle has two sides of equal length, some mathematicians define an isosceles triangle to have exactly two equal sides, whereas others define an isosceles triangle as one with at least two equal sides. The latter definition would make all equilateral triangles isosceles triangles, the 45–45–90 right triangle, which appears in the tetrakis square tiling, is isosceles. A scalene triangle has all its sides of different lengths, equivalently, it has all angles of different measure. Hatch marks, also called tick marks, are used in diagrams of triangles, a side can be marked with a pattern of ticks, short line segments in the form of tally marks, two sides have equal lengths if they are both marked with the same pattern. In a triangle, the pattern is no more than 3 ticks. Similarly, patterns of 1,2, or 3 concentric arcs inside the angles are used to indicate equal angles, triangles can also be classified according to their internal angles, measured here in degrees. A right triangle has one of its interior angles measuring 90°, the side opposite to the right angle is the hypotenuse, the longest side of the triangle. The other two sides are called the legs or catheti of the triangle, special right triangles are right triangles with additional properties that make calculations involving them easier. One of the two most famous is the 3–4–5 right triangle, where 32 +42 =52, in this situation,3,4, and 5 are a Pythagorean triple. The other one is a triangle that has 2 angles that each measure 45 degrees. Triangles that do not have an angle measuring 90° are called oblique triangles, a triangle with all interior angles measuring less than 90° is an acute triangle or acute-angled triangle. If c is the length of the longest side, then a2 + b2 > c2, a triangle with one interior angle measuring more than 90° is an obtuse triangle or obtuse-angled triangle. If c is the length of the longest side, then a2 + b2 < c2, a triangle with an interior angle of 180° is degenerate
34.
Mathematical proof
–
In mathematics, a proof is an inferential argument for a mathematical statement. In the argument, other previously established statements, such as theorems, in principle, a proof can be traced back to self-evident or assumed statements, known as axioms, along with accepted rules of inference. Axioms may be treated as conditions that must be met before the statement applies, Proofs are examples of exhaustive deductive reasoning or inductive reasoning and are distinguished from empirical arguments or non-exhaustive inductive reasoning. A proof must demonstrate that a statement is true, rather than enumerate many confirmatory cases. An unproved proposition that is believed to be true is known as a conjecture, Proofs employ logic but usually include some amount of natural language which usually admits some ambiguity. In fact, the vast majority of proofs in mathematics can be considered as applications of rigorous informal logic. Purely formal proofs, written in language instead of natural language, are considered in proof theory. The distinction between formal and informal proofs has led to examination of current and historical mathematical practice, quasi-empiricism in mathematics. The philosophy of mathematics is concerned with the role of language and logic in proofs, the word proof comes from the Latin probare meaning to test. Related modern words are the English probe, probation, and probability, the Spanish probar, Italian provare, the early use of probity was in the presentation of legal evidence. A person of authority, such as a nobleman, was said to have probity, whereby the evidence was by his relative authority, plausibility arguments using heuristic devices such as pictures and analogies preceded strict mathematical proof. It is likely that the idea of demonstrating a conclusion first arose in connection with geometry, the development of mathematical proof is primarily the product of ancient Greek mathematics, and one of the greatest achievements thereof. Thales proved some theorems in geometry, eudoxus and Theaetetus formulated theorems but did not prove them. Aristotle said definitions should describe the concept being defined in terms of other concepts already known and his book, the Elements, was read by anyone who was considered educated in the West until the middle of the 20th century. Further advances took place in medieval Islamic mathematics, while earlier Greek proofs were largely geometric demonstrations, the development of arithmetic and algebra by Islamic mathematicians allowed more general proofs that no longer depended on geometry. In the 10th century CE, the Iraqi mathematician Al-Hashimi provided general proofs for numbers as he considered multiplication, division and he used this method to provide a proof of the existence of irrational numbers. An inductive proof for arithmetic sequences was introduced in the Al-Fakhri by Al-Karaji, alhazen also developed the method of proof by contradiction, as the first attempt at proving the Euclidean parallel postulate. Modern proof theory treats proofs as inductively defined data structures, there is no longer an assumption that axioms are true in any sense, this allows for parallel mathematical theories built on alternate sets of axioms
35.
Symmetry
–
Symmetry in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, symmetry has a precise definition, that an object is invariant to any of various transformations. Although these two meanings of symmetry can sometimes be told apart, they are related, so they are discussed together. The opposite of symmetry is asymmetry, a geometric shape or object is symmetric if it can be divided into two or more identical pieces that are arranged in an organized fashion. This means that an object is symmetric if there is a transformation that moves individual pieces of the object, an object has rotational symmetry if the object can be rotated about a fixed point without changing the overall shape. An object has symmetry if it can be translated without changing its overall shape. An object has symmetry if it can be simultaneously translated and rotated in three-dimensional space along a line known as a screw axis. An object has symmetry if it does not change shape when it is expanded or contracted. Fractals also exhibit a form of symmetry, where small portions of the fractal are similar in shape to large portions. Other symmetries include glide reflection symmetry and rotoreflection symmetry, a dyadic relation R is symmetric if and only if, whenever its true that Rab, its true that Rba. Thus, is the age as is symmetrical, for if Paul is the same age as Mary. Symmetric binary logical connectives are and, or, biconditional, nand, xor, the set of operations that preserve a given property of the object form a group. In general, every kind of structure in mathematics will have its own kind of symmetry, examples include even and odd functions in calculus, the symmetric group in abstract algebra, symmetric matrices in linear algebra, and the Galois group in Galois theory. In statistics, it appears as symmetric probability distributions, and as skewness, symmetry in physics has been generalized to mean invariance—that is, lack of change—under any kind of transformation, for example arbitrary coordinate transformations. This concept has one of the most powerful tools of theoretical physics. See Noethers theorem, and also, Wigners classification, which says that the symmetries of the laws of physics determine the properties of the found in nature. Important symmetries in physics include continuous symmetries and discrete symmetries of spacetime, internal symmetries of particles, in biology, the notion of symmetry is mostly used explicitly to describe body shapes. Bilateral animals, including humans, are more or less symmetric with respect to the plane which divides the body into left
36.
Angle
–
In planar geometry, an angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle. Angles formed by two rays lie in a plane, but this plane does not have to be a Euclidean plane, Angles are also formed by the intersection of two planes in Euclidean and other spaces. Angles formed by the intersection of two curves in a plane are defined as the angle determined by the tangent rays at the point of intersection. Similar statements hold in space, for example, the angle formed by two great circles on a sphere is the dihedral angle between the planes determined by the great circles. Angle is also used to designate the measure of an angle or of a rotation and this measure is the ratio of the length of a circular arc to its radius. In the case of an angle, the arc is centered at the vertex. In the case of a rotation, the arc is centered at the center of the rotation and delimited by any other point and its image by the rotation. The word angle comes from the Latin word angulus, meaning corner, cognate words are the Greek ἀγκύλος, meaning crooked, curved, both are connected with the Proto-Indo-European root *ank-, meaning to bend or bow. Euclid defines a plane angle as the inclination to each other, in a plane, according to Proclus an angle must be either a quality or a quantity, or a relationship. In mathematical expressions, it is common to use Greek letters to serve as variables standing for the size of some angle, lower case Roman letters are also used, as are upper case Roman letters in the context of polygons. See the figures in this article for examples, in geometric figures, angles may also be identified by the labels attached to the three points that define them. For example, the angle at vertex A enclosed by the rays AB, sometimes, where there is no risk of confusion, the angle may be referred to simply by its vertex. However, in geometrical situations it is obvious from context that the positive angle less than or equal to 180 degrees is meant. Otherwise, a convention may be adopted so that ∠BAC always refers to the angle from B to C. Angles smaller than an angle are called acute angles. An angle equal to 1/4 turn is called a right angle, two lines that form a right angle are said to be normal, orthogonal, or perpendicular. Angles larger than an angle and smaller than a straight angle are called obtuse angles. An angle equal to 1/2 turn is called a straight angle, Angles larger than a straight angle but less than 1 turn are called reflex angles
37.
Orthodiagonal quadrilateral
–
In Euclidean geometry, an orthodiagonal quadrilateral is a quadrilateral in which the diagonals cross at right angles. In other words, it is a figure in which the line segments between non-adjacent vertices are orthogonal to each other. A kite is a quadrilateral in which one diagonal is a line of symmetry. The kites are exactly the orthodiagonal quadrilaterals that contain a circle tangent to all four of their sides, that is, a rhombus is an orthodiagonal quadrilateral with two pairs of parallel sides. A square is a case of both a kite and a rhombus. The square is one such quadrilateral, but there are many others. Conversely, any quadrilateral in which a2 + c2 = b2 + d2 must be orthodiagonal and this can be proved in a number of ways, including using the law of cosines, vectors, an indirect proof, and complex numbers. The diagonals of a quadrilateral are perpendicular if and only if the two bimedians have equal length. A convex quadrilateral is orthodiagonal if and only if its Varignon parallelogram is a rectangle, the center of this circle is the centroid of the quadrilateral. The quadrilateral formed by the feet of the maltitudes is called the principal orthic quadrilateral, a related characterization states that a convex quadrilateral is orthodiagonal if and only if RSTU is a rectangle whose sides are parallel to the diagonals of ABCD. There are several metric characterizations regarding the four triangles formed by the diagonal intersection P, denote by m1, m2, m3, m4 the medians in triangles ABP, BCP, CDP, DAP from P to the sides AB, BC, CD, DA respectively. A few metric characterizations of tangential quadrilaterals and orthodiagonal quadrilaterals are very similar in appearance, as can be seen in this table. The notations on the sides a, b, c, d, the circumradii R1, R2, R3, R4, and the altitudes h1, h2, h3, h4 are the same as above in both types of quadrilaterals. The area K of an orthodiagonal quadrilateral equals one half the product of the lengths of the p and q, K = p ⋅ q 2. Conversely, any convex quadrilateral where the area can be calculated with this formula must be orthodiagonal, the orthodiagonal quadrilateral has the biggest area of all convex quadrilaterals with given diagonals. Orthodiagonal quadrilaterals are the only quadrilaterals for which the sides and the angle formed by the diagonals do not uniquely determine the area, for example, two rhombi both having common side a, but one having a smaller acute angle than the other, have different areas. If squares are erected outward on the sides of any quadrilateral and this is called Van Aubels theorem. Then D2 = p 12 + p 22 + q 12 + q 22 = a 2 + c 2 = b 2 + d 2 where D is the diameter of the circumcircle and this holds because the diagonals are perpendicular chords of a circle
38.
Parallelogram law
–
In mathematics, the simplest form of the parallelogram law belongs to elementary geometry. It states that the sum of the squares of the lengths of the four sides of a parallelogram equals the sum of the squares of the lengths of the two diagonals, using the notation in the diagram on the right, the sides are. For the general quadrilateral with four sides not necessarily equal,2 +2 +2 +2 =2 +2 +4 x 2, where x is the length of the line segment joining the midpoints of the diagonals. It can be seen from the diagram that, for a parallelogram, x =0, and the general formula simplifies to the parallelogram law. In a normed space, the statement of the law is an equation relating norms,2 ∥ x ∥2 +2 ∥ y ∥2 = ∥ x + y ∥2 + ∥ x − y ∥2. In an inner space, the norm is determined using the inner product. Most real and complex normed vector spaces do not have inner products, for example, a commonly used norm is the p-norm, ∥ x ∥ p =1 / p, where the x i are the components of vector x. Given a norm, one can evaluate both sides of the law above. A remarkable fact is that if the law holds, then the norm must arise in the usual way from some inner product. In particular, it holds for the if and only if p =2. For any norm satisfying the law, the inner product generating the norm is unique as a consequence of the polarization identity. Commutative property Inner product space Normed vector space Polarization identity Weisstein, Eric W. Parallelogram Law
39.
Tangential quadrilateral
–
In Euclidean geometry, a tangential quadrilateral or circumscribed quadrilateral is a convex quadrilateral whose sides are all tangent to a single circle within the quadrilateral. This circle is called the incircle of the quadrilateral or its inscribed circle, its center is the incenter, Tangential quadrilaterals are a special case of tangential polygons. Due to the risk of confusion with a quadrilateral that has a circumcircle, all triangles have an incircle, but not all quadrilaterals do. An example of a quadrilateral that cannot be tangential is a non-square rectangle, the section characterizations below states what necessary and sufficient conditions a quadrilateral must satisfy to have an incircle. Examples of tangential quadrilaterals are the kites, which include the rhombi, the kites are exactly the tangential quadrilaterals that are also orthodiagonal. A right kite is a kite with a circumcircle, if a quadrilateral is both tangential and cyclic, it is called a bicentric quadrilateral, and if it is both tangential and a trapezoid, it is called a tangential trapezoid. In a tangential quadrilateral, the four angle bisectors meet at the center of the incircle, conversely, a convex quadrilateral in which the four angle bisectors meet at a point must be tangential and the common point is the incenter. Conversely a convex quadrilateral in which a + c = b + d must be tangential, the second of these is almost the same as one of the equalities in Urquharts theorem. The only differences are the signs on both sides, in Urquharts theorem there are instead of differences. Another necessary and sufficient condition is that a convex quadrilateral ABCD is tangential if, a characterization regarding the angles formed by diagonal BD and the four sides of a quadrilateral ABCD is due to Iosifescu. Several more characterizations are known in the four subtriangles formed by the diagonals, the eight tangent lengths of a tangential quadrilateral are the line segments from a vertex to the points where the incircle is tangent to the sides. From each vertex there are two congruent tangent lengths, the two tangency chords of a tangential quadrilateral are the line segments that connect points on opposite sides where the incircle is tangent to these sides. These are also the diagonals of the contact quadrilateral, the area K of a tangential quadrilateral is given by K = r ⋅ s, where s is the semiperimeter and r is the inradius. Another formula is K =12 p 2 q 2 −2 which gives the area in terms of the p, q. The area can also be expressed in terms of just the four tangent lengths, if these are e, f, g, h, then the tangential quadrilateral has the area K =. Furthermore, the area of a quadrilateral can be expressed in terms of the sides a, b, c, d. Since eg = fh if and only if the quadrilateral is also cyclic and hence bicentric. For given side lengths, the area is maximum when the quadrilateral is also cyclic, then K = a b c d since opposite angles are supplementary angles
40.
Inscribed figure
–
In geometry, an inscribed planar shape or solid is one that is enclosed by and fits snugly inside another geometric shape or solid. To say that figure F is inscribed in figure G means precisely the same thing as figure G is circumscribed about figure F, a circle or ellipse inscribed in a convex polygon is tangent to every side or face of the outer figure. Familiar examples of inscribed figures include circles inscribed in triangles or regular polygons, a circle inscribed in any polygon is called its incircle, in which case the polygon is said to be a tangential polygon. A polygon inscribed in a circle is said to be a polygon. The inradius or filling radius of a given outer figure is the radius of the circle or sphere. The definition given above assumes that the objects concerned are embedded in two- or three-dimensional Euclidean space, but can easily be generalized to higher dimensions and other metric spaces. For an alternative usage of the term inscribed, see the inscribed square problem, every circle has an inscribed triangle with any three given angle measures, and every triangle can be inscribed in some circle. Every triangle has a circle, called the incircle. Every circle has a regular polygon of n sides, for any n≥3. Every regular polygon has a circle, and every circle can be inscribed in some regular polygon of n sides. Not every polygon with more than three sides has a circle, those polygons that do are called tangential polygons. Not every polygon with more than three sides is a polygon of a circle, those polygons that are so inscribed are called cyclic polygons. Every triangle can be inscribed in an ellipse, called its Steiner circumellipse or simply its Steiner ellipse, whose center is the triangles centroid, every triangle has an infinitude of inscribed ellipses. One of them is a circle, and one of them is the Steiner inellipse which is tangent to the triangle at the midpoints of the sides, every acute triangle has three inscribed squares. In a right triangle two of them are merged and coincide with other, so there are only two distinct inscribed squares. An obtuse triangle has an inscribed square, with one side coinciding with part of the triangles longest side. A Reuleaux triangle, or more generally any curve of constant width, circumconic and inconic Cyclic quadrilateral Inscribed and circumscribed figures
41.
Semiperimeter
–
In geometry, the semiperimeter of a polygon is half its perimeter. Although it has such a derivation from the perimeter, the semiperimeter appears frequently enough in formulas for triangles. When the semiperimeter occurs as part of a formula, it is denoted by the letter s. The semiperimeter is used most often for triangles, the formula for the semiperimeter of a triangle with side lengths a, b, the three splitters concur at the Nagel point of the triangle. A cleaver of a triangle is a segment that bisects the perimeter of the triangle and has one endpoint at the midpoint of one of the three sides. So any cleaver, like any splitter, divides the triangle into two each of whose length equals the semiperimeter. The three cleavers concur at the center of the Spieker circle, which is the incircle of the medial triangle, a line through the triangles incenter bisects the perimeter if and only if it also bisects the area. A triangles semiperimeter equals the perimeter of its medial triangle, by the triangle inequality, the longest side length of a triangle is less than the semiperimeter. The area A of any triangle is the product of its inradius and its semiperimeter, the area of a triangle can also be calculated from its semiperimeter and side lengths a, b, c using Herons formula, A = s. The circumradius R of a triangle can also be calculated from the semiperimeter and side lengths and this formula can be derived from the law of sines. The law of cotangents gives the cotangents of the half-angles at the vertices of a triangle in terms of the semiperimeter, the sides, and the inradius. The length of the internal bisector of the angle opposite the side of length a is t a =2 b c s b + c, in a right triangle, the radius of the excircle on the hypotenuse equals the semiperimeter. The semiperimeter is the sum of the inradius and twice the circumradius, the area of the right triangle is where a and b are the legs. The formula for the semiperimeter of a quadrilateral with side lengths a, b, c and d is s = a + b + c + d 2. The simplest form of Brahmaguptas formula for the area of a quadrilateral has a form similar to that of Herons formula for the triangle area. Bretschneiders formula generalizes this to all convex quadrilaterals, K = − a b c d ⋅ cos 2 , the four sides of a bicentric quadrilateral are the four solutions of a quartic equation parametrized by the semiperimeter, the inradius, and the circumradius. The area of a regular polygon is the product of its semiperimeter
42.
Bivector
–
In mathematics, a bivector or 2-vector is a quantity in exterior algebra or geometric algebra that extends the idea of scalars and vectors. If a scalar is considered an order zero quantity, and a vector is an order one quantity, bivectors have applications in many areas of mathematics and physics. They are related to numbers in two dimensions and to both pseudovectors and quaternions in three dimensions. They can be used to generate rotations in any number of dimensions and they also are used in physics, tying together a number of otherwise unrelated quantities. Bivectors are generated by the product on vectors, given two vectors a and b, their exterior product a ∧ b is a bivector, as is the sum of any bivectors. Not all bivectors can be generated as an exterior product. More precisely, a bivector that can be expressed as a product is called simple. The exterior product of two vectors is alternating, so b ∧ a is the negation of the bivector a ∧ b, producing the opposite orientation, and a ∧ a is the zero bivector. Geometrically, a bivector can be interpreted as an oriented plane segment. In layman terms, any surface is the bivector, if it has the same area, same orientation. The bivector was first defined in 1844 by German mathematician Hermann Grassmann in exterior algebra as the result of the product of two vectors. Around the same time in 1843 in Ireland William Rowan Hamilton discovered quaternions, around this time Josiah Willard Gibbs and Oliver Heaviside developed vector calculus, which included separate cross product and dot products that were derived from quaternion multiplication. Gibbs used vectors to fill the role of bivectors in three dimensions, and used bivector to describe an unrelated quantity, a use that has sometimes been copied. Today the bivector is largely studied as a topic in geometric algebra and its resurgence was led by David Hestenes who, along with others, applied geometric algebra to a range of new applications in physics. For this article the bivector will be considered only in real geometric algebras and this in practice is not much of a restriction, as all useful applications are drawn from such algebras. Also unless otherwise stated, all examples have a Euclidean metric, the bivector arises from the definition of the geometric product over a vector space. For a space with Euclidean metric ϵa is 1 so can be omitted, and the condition becomes, a 2 = | a |2 From associativity a = a2b. When b is not parallel to and hence not a multiple of a
43.
Determinant
–
In linear algebra, the determinant is a useful value that can be computed from the elements of a square matrix. The determinant of a matrix A is denoted det, detA and it can be viewed as the scaling factor of the transformation described by the matrix. In the case of a 2 ×2 matrix, the formula for the determinant. Each determinant of a 2 ×2 matrix in this equation is called a minor of the matrix A, the same sort of procedure can be used to find the determinant of a 4 ×4 matrix, the determinant of a 5 ×5 matrix, and so forth. The use of determinants in calculus includes the Jacobian determinant in the change of rule for integrals of functions of several variables. Determinants are also used to define the characteristic polynomial of a matrix, in analytical geometry, determinants express the signed n-dimensional volumes of n-dimensional parallelepipeds. Sometimes, determinants are used merely as a notation for expressions that would otherwise be unwieldy to write down. When the entries of the matrix are taken from a field, it can be proven that any matrix has an inverse if. There are various equivalent ways to define the determinant of a square matrix A, i. e. one with the number of rows. Another way to define the determinant is expressed in terms of the columns of the matrix and these properties mean that the determinant is an alternating multilinear function of the columns that maps the identity matrix to the underlying unit scalar. These suffice to uniquely calculate the determinant of any square matrix, provided the underlying scalars form a field, the definition below shows that such a function exists, and it can be shown to be unique. Assume A is a matrix with n rows and n columns. The entries can be numbers or expressions, the definition of the determinant depends only on the fact that they can be added and multiplied together in a commutative manner. The determinant of a 2 ×2 matrix is defined by | a b c d | = a d − b c. If the matrix entries are numbers, the matrix A can be used to represent two linear maps, one that maps the standard basis vectors to the rows of A. In either case, the images of the vectors form a parallelogram that represents the image of the unit square under the mapping. The parallelogram defined by the rows of the matrix is the one with vertices at. The absolute value of ad − bc is the area of the parallelogram, the absolute value of the determinant together with the sign becomes the oriented area of the parallelogram
44.
Law of cosines
–
In trigonometry, the law of cosines relates the lengths of the sides of a triangle to the cosine of one of its angles. The law of cosines is useful for computing the third side of a triangle when two sides and their enclosed angle are known, and in computing the angles of a triangle if all three sides are known. Though the notion of the cosine was not yet developed in his time, Euclids Elements, dating back to the 3rd century BC, the cases of obtuse triangles and acute triangles are treated separately, in Propositions 12 and 13 of Book 2. Using notation as in Fig.2, Euclids statement can be represented by the formula A B2 = C A2 + C B2 +2 and this formula may be transformed into the law of cosines by noting that CH = cos = − cos γ. Proposition 13 contains an analogous statement for acute triangles. In the 15th century, Jamshīd al-Kāshī provided the first explicit statement of the law of cosines in a suitable for triangulation. In France, the law of cosines is still referred to as the theorem of Al-Kashi, the theorem was popularized in the Western world by François Viète in the 16th century. At the beginning of the 19th century, modern algebraic notation allowed the law of cosines to be written in its current symbolic form, the theorem is used in triangulation, for solving a triangle or circle, i. e. These formulas produce high round-off errors in floating point calculations if the triangle is very acute and it is even possible to obtain a result slightly greater than one for the cosine of an angle. The third formula shown is the result of solving for a the quadratic equation a2 − 2ab cos γ + b2 − c2 =0 and this equation can have 2,1, or 0 positive solutions corresponding to the number of possible triangles given the data. It will have two positive solutions if b sin γ < c < b, only one positive solution if c = b sin γ and these different cases are also explained by the side-side-angle congruence ambiguity. Consider a triangle with sides of length a, b, c and this triangle can be placed on the Cartesian coordinate system by plotting the following points, as shown in Fig.4, A =, B =, and C =. By the distance formula, we have c =2 +2, an advantage of this proof is that it does not require the consideration of different cases for when the triangle is acute vs. right vs. obtuse. Drop the perpendicular onto the c to get c = a cos β + b cos α. Multiply through by c to get c 2 = a c cos β + b c cos α. By considering the other perpendiculars obtain a 2 = a c cos β + a b cos γ, b 2 = b c cos α + a b cos γ. Adding the latter two equations gives a 2 + b 2 = a c cos β + b c cos α +2 a b cos γ and this proof uses trigonometry in that it treats the cosines of the various angles as quantities in their own right. It uses the fact that the cosine of an angle expresses the relation between the two sides enclosing that angle in any right triangle
45.
Inscribed
–
In geometry, an inscribed planar shape or solid is one that is enclosed by and fits snugly inside another geometric shape or solid. To say that figure F is inscribed in figure G means precisely the same thing as figure G is circumscribed about figure F, a circle or ellipse inscribed in a convex polygon is tangent to every side or face of the outer figure. Familiar examples of inscribed figures include circles inscribed in triangles or regular polygons, a circle inscribed in any polygon is called its incircle, in which case the polygon is said to be a tangential polygon. A polygon inscribed in a circle is said to be a polygon. The inradius or filling radius of a given outer figure is the radius of the circle or sphere. The definition given above assumes that the objects concerned are embedded in two- or three-dimensional Euclidean space, but can easily be generalized to higher dimensions and other metric spaces. For an alternative usage of the term inscribed, see the inscribed square problem, every circle has an inscribed triangle with any three given angle measures, and every triangle can be inscribed in some circle. Every triangle has a circle, called the incircle. Every circle has a regular polygon of n sides, for any n≥3. Every regular polygon has a circle, and every circle can be inscribed in some regular polygon of n sides. Not every polygon with more than three sides has a circle, those polygons that do are called tangential polygons. Not every polygon with more than three sides is a polygon of a circle, those polygons that are so inscribed are called cyclic polygons. Every triangle can be inscribed in an ellipse, called its Steiner circumellipse or simply its Steiner ellipse, whose center is the triangles centroid, every triangle has an infinitude of inscribed ellipses. One of them is a circle, and one of them is the Steiner inellipse which is tangent to the triangle at the midpoints of the sides, every acute triangle has three inscribed squares. In a right triangle two of them are merged and coincide with other, so there are only two distinct inscribed squares. An obtuse triangle has an inscribed square, with one side coinciding with part of the triangles longest side. A Reuleaux triangle, or more generally any curve of constant width, circumconic and inconic Cyclic quadrilateral Inscribed and circumscribed figures
46.
Circumscribed circle
–
In geometry, the circumscribed circle or circumcircle of a polygon is a circle which passes through all the vertices of the polygon. The center of circle is called the circumcenter and its radius is called the circumradius. A polygon which has a circle is called a cyclic polygon. All regular simple polygons, all isosceles trapezoids, all triangles, a related notion is the one of a minimum bounding circle, which is the smallest circle that completely contains the polygon within it. All triangles are cyclic, i. e. every triangle has a circumscribed circle and this can be proven on the grounds that the general equation for a circle with center and radius r in the Cartesian coordinate system is 2 +2 = r 2. Since this equation has three parameters only three points coordinate pairs are required to determine the equation of a circle, since a triangle is defined by its three vertices, and exactly three points are required to determine a circle, every triangle can be circumscribed. The circumcenter of a triangle can be constructed by drawing any two of the three perpendicular bisectors, the center is the point where the perpendicular bisectors intersect, and the radius is the length to any of the three vertices. This is because the circumcenter is equidistant from any pair of the triangles vertices, in coastal navigation, a triangles circumcircle is sometimes used as a way of obtaining a position line using a sextant when no compass is available. The horizontal angle between two landmarks defines the circumcircle upon which the observer lies, in the Euclidean plane, it is possible to give explicitly an equation of the circumcircle in terms of the Cartesian coordinates of the vertices of the inscribed triangle. Suppose that A = B = C = are the coordinates of points A, B, using the polarization identity, these equations reduce to the condition that the matrix has a nonzero kernel. Thus the circumcircle may alternatively be described as the locus of zeros of the determinant of this matrix, a similar approach allows one to deduce the equation of the circumsphere of a tetrahedron. A unit vector perpendicular to the containing the circle is given by n ^ = × | × |. An equation for the circumcircle in trilinear coordinates x, y, z is a/x + b/y + c/z =0, an equation for the circumcircle in barycentric coordinates x, y, z is a2/x + b2/y + c2/z =0. The isogonal conjugate of the circumcircle is the line at infinity, given in coordinates by ax + by + cz =0. Additionally, the circumcircle of a triangle embedded in d dimensions can be using a generalized method. Let A, B, and C be d-dimensional points, which form the vertices of a triangle and we start by transposing the system to place C at the origin, a = A − C, b = B − C. The circumcenter, p0, is given by p 0 = ×2 ∥ a × b ∥2 + C, the Cartesian coordinates of the circumcenter are U x =1 D U y =1 D with D =2. Without loss of generality this can be expressed in a form after translation of the vertex A to the origin of the Cartesian coordinate systems