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Cylinder A cylinder (from Greek κύλινδρος – kulindros, "roller, tumbler"[1]), has traditionally been a threedimensional solid, one of the most basic of curvilinear geometric shapes. It is the idealized version of a solid physical tin can having lids on top and bottom. This traditional view is still used in elementary treatments of geometry, but the advanced mathematical viewpoint has shifted to the infinite curvilinear surface and this is how a cylinder is now defined in various modern branches of geometry and topology. The shift in the basic meaning (solid versus surface) has created some ambiguity with terminology. It is generally hoped that context makes the meaning clear [...More...] 


Tycho Brahe Planetarium The Tycho Brahe Tycho Brahe Planetarium Planetarium is located in Copenhagen, Denmark, at the southern end of Skt. Jørgens Sø. It is named after astronomer Tycho Brahe. It was designed by MAA Knud Munk and opened on November 1, 1989.[1] The planetarium is built where the theater Saltlageret was previously located. The foundation stone was placed on February 22, 1988, and the planetarium opened on November 1, 1989. The financial basis for building the planetarium was a 50,000,000 DKK donation by Bodil and Helge Petersen to the Urania foundation, which administered the construction of the planetarium. The planetarium has an IMAX IMAX theater, as well as a digital system which can show more than 10,000 stars. Before each show the guests are taken on a spacejourney using the new Digital Universe.[2] Exhibition[edit] The permanent exhibition 'The Active Universe' has information about astronomy, space and spacetravel [...More...] 


Real Number In mathematics, a real number is a value that represents a quantity along a line. The adjective real in this context was introduced in the 17th century by René Descartes, who distinguished between real and imaginary roots of polynomials. The real numbers include all the rational numbers, such as the integer −5 and the fraction 4/3, and all the irrational numbers, such as √2 (1.41421356..., the square root of 2, an irrational algebraic number). Included within the irrationals are the transcendental numbers, such as π (3.14159265...). Real numbers can be thought of as points on an infinitely long line called the number line or real line, where the points corresponding to integers are equally spaced. Any real number can be determined by a possibly infinite decimal representation, such as that of 8.632, where each consecutive digit is measured in units one tenth the size of the previous one [...More...] 


Semimajor Axis In geometry, the major axis of an ellipse is its longest diameter: a line segment that runs through the center and both foci, with ends at the widest points of the perimeter. The semimajor axis is one half of the major axis, and thus runs from the centre, through a focus, and to the perimeter [...More...] 


Greek Language Greek (Modern Greek: ελληνικά elliniká) is an independent branch of the IndoEuropean family of languages, native to Greece, Cyprus Cyprus and other parts of the Eastern Mediterranean and the Black Sea. It has the longest documented history of any living IndoEuropean language, spanning more than 3000 years of written records.[3] Its writing system has been the [...More...] 


Cavalieri's Principle In geometry, Cavalieri's principle, a modern implementation of the method of indivisibles, named after Bonaventura Cavalieri, is as follows:[1]2dimensional case: Suppose two regions in a plane are included between two parallel lines in that plane. If every line parallel to these two lines intersects both regions in line segments of equal length, then the two regions have equal areas. 3dimensional case: Suppose two regions in threespace (solids) are included between two parallel planes. If every plane parallel to these two planes intersects both regions in crosssections of equal area, then the two regions have equal volumes.Today Cavalieri's principle Cavalieri's principle is seen as an early step towards integral calculus, and while it is used in some forms, such as its generalization in Fubini's theorem, results using Cavalieri's principle can often be shown more directly via integration [...More...] 


Semimajor And Semiminor Axes In geometry, the major axis of an ellipse is its longest diameter: a line segment that runs through the center and both foci, with ends at the widest points of the perimeter. The semimajor axis is one half of the major axis, and thus runs from the centre, through a focus, and to the perimeter [...More...] 


Diameter In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints lie on the circle. It can also be defined as the longest chord of the circle. Both definitions are also valid for the diameter of a sphere. In more modern usage, the length of a diameter is also called the diameter. In this sense one speaks of the diameter rather than a diameter (which refers to the line itself), because all diameters of a circle or sphere have the same length, this being twice the radius r. d = 2 r ⇒ r = d 2 . displaystyle d=2rquad Rightarrow quad r= frac d 2 . For a convex shape in the plane, the diameter is defined to be the largest distance that can be formed between two opposite parallel lines tangent to its boundary, and the width is often defined to be the smallest such distance [...More...] 


Circumscribe 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 this circle is called the circumcenter and its radius is called the circumradius. A polygon which has a circumscribed circle is called a cyclic polygon (sometimes a concyclic polygon, because the vertices are concyclic). All regular simple polygons, all isosceles trapezoids, all triangles and all rectangles are cyclic. A related notion is the one of a minimum bounding circle, which is the smallest circle that completely contains the polygon within it [...More...] 


Rotation Of Axes In mathematics, a rotation of axes in two dimensions is a mapping from an xy Cartesian coordinate system Cartesian coordinate system to an x'y'Cartesian coordinate system in which the origin is kept fixed and the x' and y' axes are obtained by rotating the x and y axes counterclockwise through an angle θ displaystyle theta . A point P has coordinates (x, y) with respect to the original system and coordinates (x', y') with respect to the new system.[1] In the new coordinate system, the point P will appear to have been rotated in the opposite direction, that is, clockwise through the angle θ displaystyle theta [...More...] 


Rectangle In Euclidean plane geometry, a rectangle is a quadrilateral with four right angles. It can also be defined as an equiangular quadrilateral, since equiangular means that all of its angles are equal (360°/4 = 90°). 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 nonsquare rectangle.[1][2][3] A rectangle with vertices ABCD would be denoted as ABCD. The word rectangle comes from the Latin Latin rectangulus, which is a combination of rectus (as an adjective, right, proper) and angulus (angle). A crossed rectangle is a crossed (selfintersecting) quadrilateral which consists of two opposite sides of a rectangle along with the two diagonals.[4] It is a special case of an antiparallelogram, and its angles are not right angles [...More...] 


Translation Of Axes In mathematics, a translation of axes in two dimensions is a mapping from an xy Cartesian coordinate system Cartesian coordinate system to an x'y'Cartesian coordinate system in which the x' axis is parallel to the x axis and k units away, and the y' axis is parallel to the y axis and h units away. This means that the origin O' of the new coordinate system has coordinates (h, k) in the original system. The positive x' and y' directions are taken to be the same as the positive x and y directions [...More...] 


Cartesian Coordinates A Cartesian coordinate system Cartesian coordinate system is a coordinate system that specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular directed lines, measured in the same unit of length. Each reference line is called a coordinate axis or just axis (plural axes) of the system, and the point where they meet is its origin, at ordered pair (0, 0) [...More...] 


Plücker Conoid In geometry, Plücker’s conoid is a ruled surface named after the German mathematician Julius Plücker. It is also called a conical wedge or cylindroid; however, the latter name is ambiguous, as "cylindroid" may also refer to an elliptic cylinder. Plücker’s conoid is the surface defined by the function of two variables: z = 2 x y x 2 + y 2 . displaystyle z= frac 2xy x^ 2 +y^ 2 [...More...] 


Without Loss Of Generality Without loss of generality (often abbreviated to WOLOG, WLOG or w.l.o.g.; less commonly stated as without any loss of generality or with no loss of generality) is a frequently used expression in mathematics. The term is used before an assumption in a proof which narrows the premise to some special case; it implies that the proof for that case can be easily applied to all others, or that all other cases are equivalent or similar.[1] Thus, given a proof of the conclusion in the special case, it is trivial to adapt it to prove the conclusion in all other cases. This is often enabled by the presence of symmetry. For example, if some property P(x,y) of real numbers is known to be symmetrical in x and y, namely that P(x,y) is equivalent to P(y,x), then in proving that P(x,y) holds for every x and y, we may assume "without loss of generality" that x ≤ y [...More...] 


Apex (geometry) In geometry, an apex ( Latin Latin for 'summit, peak, tip, top, extreme end') is the vertex which is in some sense the "highest" of the figure to which it belongs. The term is typically used to refer to the vertex opposite from some "base." Isosceles triangles[edit] In an isosceles triangle, the apex is the vertex where the two sides of equal length meet, opposite the unequal third side.[1] Pyramids and cones[edit] In a pyramid or cone, the apex is the vertex at the "top" (opposite the base).[1] In a pyramid, the vertex is the point that is part of all the lateral faces, or where all the lateral edges meet.[2] References[edit]^ a b Weisstein, Eric W. "Apex". MathWorld. ^ Jacobs, Harold R. (2003). Geometry: Seeing, Doing, Understanding (Third ed.). New York City: W. H. Freeman and Company. pp. 647,655. ISBN 9780716743613. This Elementary geometry Elementary geometry related article is a stub [...More...] 
