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dupcubfin.html Duplication of the Cube : Darrell Mattingly, Cateryn Kiernan This triology of problems, the trisection of a given angle, the squaring of a circle, and the duplication of the cube, have since been proved impossible using exclusively the straight edge and the compass. This task set forth by the oracle originated the problem of the duplication of the cube and is often referred to as the "Delian Problem" (Boyer, 64). www.ms.uky.edu /~carl/ma330/projects/dupcubfin1.html   (1283 words)

Doubling the cube - Wikipedia, the free encyclopedia Doubling the cube is one of the three most famous geometric problems unsolvable by straightedge and compass alone. To double the cube means to be given a cube of some side length s and volume V, and to construct a new cube, larger than the first, with volume 2V and therefore side length False claims of doubling the cube with a compass and straightedge abound in mathematical crank literature. en.wikipedia.org /wiki/Doubling_the_cube   (357 words)

Archytas - Wikipedia, the free encyclopedia According to Eutocius Archytas solved the problem of duplicating the cube in his manner with a geometric construction. The Archytas curve, which he used in his solution of the doubling the cube problem, is named after him. Archytas was drowned in the Adriatic Sea; his body lay unburied on the shore till a sailor humanely cast a handful of sand on it, otherwise he would have had to wander on this side the Styx for a hundred years, such the virtue of a little dust, munera pulveris, as Horace calls it. en.wikipedia.org /wiki/Archytas   (279 words)

Ruler-and-compass construction   (Site not responding. Last check: 2007-11-03) Doubling the cube: using only ruler and compass, construct the side of a cube that has twice the volume of a cube with a given side. This is impossible because the cube root of 2, though algebraic, cannot be computed from integers by addition, subtraction, multiplication, division, and taking square roots. This requires taking the cube root of an arbitrary complex number with absolute value 1 and is likewise impossible. www.sciencedaily.com /encyclopedia/ruler_and_compass_construction   (1371 words)

Doubling the cube The origins of the problem of doubling the cube may be somewhat obscure as we have just seen, but there is no doubt that the Greeks had known for a long time how to solve the problem of doubling the square. Now often in articles on doubling the cube the argument of the last paragraph to prove the result of Hippocrates that (i) and (ii) are equivalent is given; see for example [3]. Although these many different methods were invented to double the cube and remarkable mathematical discoveries were made in the attempts, the ancient Greeks were never going to find the solution that they really sought, namely one which could be made with a ruler and compass construction. www-groups.dcs.st-and.ac.uk /~history/HistTopics/Doubling_the_cube.html   (2882 words)

Pappus of Alexandria - Wikipedia, the free encyclopedia It covers a wide range of topics, including geometry, recreational mathematics, doubling the cube, polygons and polyhedra. It may be divided into five sections: (I) On the famous problem of finding two mean proportionals between two given lines, which arose from that of duplicating the cube, reduced by Hippocrates to the former. The area of the surface included between this curve and its base is found the first known instance of a quadrature of a curved surface. en.wikipedia.org /wiki/Pappus   (1282 words)

An Incomplete Data Cube - Overview   (Site not responding. Last check: 2007-11-03) Incomplete, lazy, and semi-eager cubes also scale well, new dimensions can be added to the cube and existing dimensions can increase in size (i.e., a more precise measure can be added to the dimension) with no adjustment to the existing cube storage. In general, an incomplete cube is useful in situations where a complete, eager cube would be unnecessarily large, but where a lazy or semi-eager cube cannot be used because the source data is not available or expensive to query. One reason that data cubes are popular is that many data collections are characterised by the property that as data in the collection ages, each datum individually becomes less relevant, but remains relevant in aggregate. moab.eecs.wsu.edu /~cdyreson/pub/IncompleteDataCube/overview.htm   (850 words)

Math Forum: Ask Dr. Math FAQ: "Impossible" Geometric Constructions   (Site not responding. Last check: 2007-11-03) Three geometric construction problems from antiquity puzzled mathematicians for centuries: the trisection of an angle, squaring the circle, and duplicating the cube. A cube root is not a Euclidean number, and Lindemann showed that pi is a transcendental number, which means that it is not the root of an algebraic equation with integer coefficients, making it too non-Euclidean. Doubling a cube whose edge equals 1 yields the equation x^3 = 2, whose solution (the length of a side of the larger cube) is the cube root of 2. forum.swarthmore.edu /dr.math/faq/faq.impossible.construct.html   (688 words)

QUADRATRIX - LoveToKnow Article on QUADRATRIX The intercept on the axis of y is 2a/x; therefore, if it were possible to accurately construct the curve, the quadrature of the circle would be effected. The curve also permits the solution of the problems of duplicating a cube (q.v.) and trisecting an angle. The quadratrix of Tschirnhausen is constructed by dividing the arc and radius of a quadrant in the same number of equal parts as before. www.1911encyclopedia.org /Q/QU/QUADRATRIX.htm   (588 words)

M3210 Sample Exam II   (Site not responding. Last check: 2007-11-03) The problem is to construct a cube that has twice the volume of a given cube. A particular instance of this problem would be to construct a cube whose volume is twice that of the unit cube. This entails constructing a side of the larger cube, and in this case that means constructing a length equal to the cube root of 2. www-math.cudenver.edu /~wcherowi/courses/m3210/hgex2sam.html   (623 words)

presumed impossibilities In this construction, with AB the cube root, edge, of the first cube, AE is to be the cube root of the cube twice the size. First let me return to a previous cube duplication for which "insertion" is considered used, but the nature of which is not specified. My preceding solutions to trisection, cube duplication, and the heptagon, by using only straightedge and compass are difficult for the profession to acknowledge, because of the reluctance to recognize that any of its widely held convictions could be revealed wrong. www.vjecsner.net /presumed%20impossibilities.htm   (6350 words)

Duplicating the Cube The problem is to find a cube equal in volume to twice a given cube, or in other words, if the edge of the given cube is one, find a line equal in length to the cube root of two. That means if you use 63/50 to duplicate a cube a meter on a side, the duplicate will be accurate to within 0.008 cm or 0.08 millimeters, or 0.003 inches, about the thickness of a sheet of paper. If you mark a ruler you can trisect the angle and duplicate the cube, but this approach was not permitted under ancient Greek rules. www.uwgb.edu /dutchs/PSEUDOSC/DuplCube.HTM   (531 words)

Perplexing_Parabolas   (Site not responding. Last check: 2007-11-03) It could not be solved with geometric methods (using ruler and compass); Menaechmus furthered the studies into duplicating the cube by finding the intersection of two parabolas. This refers to forming a cube with double the volume of a given cube. The problem confronted ancient Athenians in 430 BC during the plague, when they were told by the oracle of Apollo at Delphi to double the size of their cubic altar. www.marymount.k12.ny.us /marynet/Studentwebwork00/conic%20sections/parabolas/html/history.htm   (235 words)

Math Lair - The Three Construction Problems of Antiquity   (Site not responding. Last check: 2007-11-03) These three problems are referred to as "duplicating the cube", "trisecting an angle", and "squaring the circle". Construct a cube having twice the volume of a given cube. The cube root of 2, however, is the root of a third-degree equation, and cannot be the solution of an equation of even degree. www.stormloader.com /ajy/construction.html   (300 words)

Cd Duplicating Machine   (Site not responding. Last check: 2007-11-03) Cell (biology) 217: DNA replication, or the process of duplicating a cell's genome, is required every time a cell di A boat was built at Piraeus duplicating as nearly as could be managed, Jason's vessel. Trade secret 28: a third party is not prevented from independently duplicating the secret information. www.referenceresearch.com /some/26176-cd-duplicating-machine.html   (523 words)

Trisect?   (Site not responding. Last check: 2007-11-03) In more modern terminology, give a length a, constuct a length b so that b/a is the cube root of 2. If you set the length of a to be one, then the problem is just to construct a line whose length is the cube root of 2. It was proved in the 19th century that a cube can not be duplicated with the Euclidean tools of straightedge and compass. aleph0.clarku.edu /~djoyce/java/Geometry/cube.html   (236 words)

Hippocrates   (Site not responding. Last check: 2007-11-03) Hippocrates also showed that a cube can be doubled if two mean proportionals can be determined between a number and its double. This had a major influence on attempts to duplicate the cube, all efforts after this being directed towards the mean proportionals problem. He was the first to write an Elements of Geometry and although his work is now lost it must have contained much of what Euclid later included in Books 1 and 2 of the Elements. www-groups.dcs.st-and.ac.uk /~history/Mathematicians/Hippocrates.html   (1237 words)

Cube Enhancements You can distribute cube data across multiple servers to provide more storage capacity, create linked cubes to distribute end-user access to information without duplicating cube data, create cubes that are updated in real time as data changes, and use a number of other new features to create cubes that address your specific business needs. Real-time cubes implement real-time OLAP by using ROLAP storage for partitions and dimensions, new SQL Server 2000 indexed views for aggregations, and automatic notification by the SQL Server 2000 relational engine when data changes. A cube can be stored on a single Analysis server and then defined as a linked cube on other Analysis servers. doc.ddart.net /mssql/sql2000/html/olapdmad/agwhatsnew_3ipf.htm   (751 words)

Ruler-and-compass construction   (Site not responding. Last check: 2007-11-03) The straightedge and compass give you the ability to produce ratios which are solutions to quadratic equations, but doubling the cube and trisecting the angle require ratios which are the solution to cubic equations, while squaring the circle requires a transcendental ratio. Without the constraint of requiring solution by ruler and compass alone, the problem is easily solvable by a wide variety of geometric and algebraic means, and has been solved many times in antiquity. Trisection, like many constructions impossible by ruler and compass, can easily be accomplished by the more powerful (but physically easy) operations of paper folding, or origami. www.worldhistory.com /wiki/R/Ruler-and-compass-construction.htm   (1570 words)

69th AB - Abstract Two of the three classical Greek problems, which cannot be solved by using ruler and compasses alone, are to trisect a general angle and to duplicate a cube. The aim of this section is to show that, in fact, a ruler and compasses is enough if one strange pre-step is allowed, namely the following: given ruler, compasses and a segment of unit length, mark (using compasses) two points on the ruler 1 unit apart, as shown on the picture. This shows that a general angle can be trisected by using compasses and a "marked ruler", the same set of instruments as in the case of duplicating the cube. www.karlin.mff.cuni.cz /~rokyta/api/doc/abstr069.htm   (1197 words)

Menaechmus   (Site not responding. Last check: 2007-11-03) This he achieved and therefore solved the problem of the duplicating the cube using these conic sections. What has come to be known as Plato's solution to the problem of duplicating the cube is widely recognised as not due to Plato since it involves a mechanical instrument. Menaechmus's construction for the duplication of the cube www-gap.dcs.st-and.ac.uk /~history/Mathematicians/Menaechmus.html   (1008 words)

Duplicating   (Site not responding. Last check: 2007-11-03) Steven Dutch, Natural and Applied Sciences, Duplicating the Cube is one of the three classic unsolved problems of antiquity. The Duplicating Center is the least expensive alternative for printing cost effective to plan ahead and use the services of the Duplicating Center. The major use of duplicating a file descriptor is to implement but there are also convenient functions dup and dup2 for duplicating descriptors. call-center.allworldsites.com /q/call-center-duplicating.htm   (862 words)

dBforums - Analysis Server Error: Connection to the server is lost (after partitioning cube) I have a cube that started out with 2 million fact table rows, but the need for more duplicating portions of the cube with the others. > duplicating portions of the cube with the others. dbforums.com /t430477.html   (573 words)

Omar Khayyam on Cubics   (Site not responding. Last check: 2007-11-03) One of the accomplishments of the Persian mathematician Omar Khayyam was to give geometrical constructions for the roots of a cubic as the intersections of two conics. Of course, this approach had been used earlier by Menaechmus and others to solve certain special cubics (notably in relation to the problem of "duplicating the cube"), but Khayyam generalized it to cover essentially all cubics (albeit with many individual cases so as to avoid negative numbers). For the solution we need conic sections." Here we might credit Khayyam with anticipating the eventual proof of the unsolvability of the Delian problem (duplicating the cube) by straight-edge and compass, but it seems to me this comment may also shed some light on his statement that the cubic cannot be "solved algebraically". www.mathpages.com /home/kmath448.htm   (386 words)

MathsNet: Geometric Construction Course - classic problems of geometry It is impossible to construct a cube whose volume is twice that of a given volume. In 1837 Pierre Laurent Wantzel (born: 1814 in Paris, France; died: 1848 in Paris) published proofs on the means of deciding if a geometric problem can be solved with ruler and compasses. Gauss had stated that the problems of duplicating a cube and trisecting an angle could not be solved with ruler and compasses but he gave no proofs. www.mathsnet.net /campus/classic2.html   (107 words)

Sporus   (Site not responding. Last check: 2007-11-03) His solution of the problem of duplicating the cube is similar to that of Diocles and in fact Pappus also followed a similar construction. Not only did Sporus work on squaring the circle and duplicating the cube but he also constructively criticised others work in these areas. One of his contributions, which is described by Pappus, was to criticise the method of squaring the circle using the quadratrix of Hippias. www-history.mcs.st-andrews.ac.uk /history/Mathematicians/Sporus.html   (296 words)

Hippocrates of Chios, Greek mathematician born in Chios In Hippocrates attempt to square the circle, he was able to find the areas of lunes, certain crescent-shaped figures, using his theory that the ratio of the areas of two circles is the same as the ratio of the squares of their radii. Hippocrates proofed that if between a number and its double, two mean proportionals can be found that the cube can be doubled. This finding had a major influence and changed the attempts on duplicating the cube. www.fragrant-chios.com /info/people_hippocrates.php   (226 words)

Conic sections   (Site not responding. Last check: 2007-11-03) If a cylinder is sliced by a plane a number of curves arise depending on the angle of the plane with respect to the cylinder axis, these are called conic sections. Conic sections were studied extensively by the Greeks as early as 350 BC in an attempt to solve the great geometric problems of the day, namely, squaring the circle, duplicating the cube, and trisecting an angle. They were also studied extensively in relation to Keplers laws of planetary motion by Descarte and Fermat. astronomy.swin.edu.au /~pbourke/curves/conic   (339 words)

Conchoid The name means shell form and was studied by the Greek mathematician Nicomedes in about 200 BC in relation to the problem of duplication of the cube. It was a favourite with 17 Century mathematicians and could be used, as Nicomedes had intended, to solve the problems of duplicating the cube and trisecting an angle. Newton said it should be a 'standard' curve. www-groups.dcs.st-andrews.ac.uk /%7Ehistory/Curves/Conchoid.html   (168 words)

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