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| '''Doubling the cube''' (also known as the '''Delian problem''') is one of [[Compass_and_straightedge#Impossible constructions|the three most famous geometric problems unsolvable by compass and straightedge construction]]. It was known to the [[Egyptian mathematics|Egyptians]], [[Greek mathematics|Greeks]], and [[Indian mathematics|Indians]].<ref>Lucye Guilbeau (1930). "The History of the Solution of the Cubic Equation", ''Mathematics News Letter'' '''5''' (4), pp. 8–12.</ref>
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| To "double the cube" means to be given a [[cube]] of some side length ''s'' and volume ''V''= ''s''<sup>3</sup>, and to construct the side of a new cube, larger than the first, with volume 2''V'' and therefore side length <math>s\cdot\sqrt[3]{2}</math>. The problem is known to be impossible to solve with only [[compass and straightedge]], because <math>\sqrt[3]{2}</math> ≈ 1.25992105... is not a [[constructible number]].
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| ==History==
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| The problem owes its name to a story concerning the citizens of [[Delos]], who consulted the oracle at [[Delphi]] in order to learn how to defeat a plague sent by [[Apollo]].<ref name="Zhmud">[http://books.google.com/books?id=oX28qf7LKdoC&pg=PA84 L. Zhmud ''The origin of the history of science in classical antiquity'', p.84], quoting [[Plutarch]] and [[Theon of Smyrna]]</ref> According to [[Plutarch]]<ref> [[Plutarch]], [http://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A2008.01.0243%3Asection%3D6 De E apud Delphos 386.E.4]</ref> it was the citizens of Delos who consulted the oracle at Delphi, seeking a solution for their internal political problems at the time, which had intensified relationships among the citizens. The oracle responded that they must double the size of the altar to Apollo, which was a regular cube. The answer seemed strange to the Delians and they consulted [[Plato]], who was able to interpret the oracle as the mathematical problem of doubling the volume of a given cube, thus explaining the oracle as the advice of Apollo for the citizens of Delos to occupy themselves with the study of geometry and mathematics in order to calm down their passions.<ref>[[Plutarch]], De genio Socratis 579.B</ref>
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| According to Plutarch, Plato gave the problem to [[Eudoxus of Cnidus|Eudoxus]] and [[Archytas]] and [[Menaechmus]], who solved the problem using mechanical means, earning a rebuke from Plato for not solving the problem using pure geometry (Plut., ''Quaestiones convivales'' [http://ebooks.adelaide.edu.au/p/plutarch/symposiacs/chapter8.html#section80 VIII.ii], 718ef). This may be why the problem is referred to in the 350s BC by the author of the pseudo-Platonic ''[[Sisyphus (dialogue)|Sisyphus]]'' (388e) as still unsolved.<ref>Carl Werner Müller, ''Die Kurzdialoge der Appendix Platonica'', Munich: Wilhelm Fink, 1975, pp. 105-106</ref> However another version of the story says that all three found solutions but they were too abstract to be of practical value{{Citation needed|date=November 2011}}.
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| A significant development in finding a solution to the problem was the discovery by [[Hippocrates of Chios]] that it is equivalent to finding two mean proportionals between a line segment and another with twice the length.<ref name="Heath">T.L. Heath ''A history of Greek mathematics'', Vol. 1]</ref> In modern notation, this means that given segments of lengths ''a'' and 2''a'', the duplication of the cube is equivalent to finding segments of lengths ''r'' and ''s'' so that
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| :<math>\frac{a}{r} = \frac{r}{s} = \frac{s}{2a} .\ </math>
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| In turn, this means that
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| :<math>r=a\cdot\sqrt[3]{2}</math>
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| But [[Pierre Wantzel]] proved in 1837 that the cube root of 2 is not [[constructible number|constructible]]; that is, it cannot be constructed with [[straightedge and compass]].
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| ==Solutions==
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| Menaechmus' original solution involves the intersection of two [[conic]] curves. Other more complicated methods of doubling the cube involve the [[cissoid of Diocles]], the [[Conchoid (mathematics)|conchoid of Nicomedes]], or the [[Philo line]]. [[Archytas]] solved the problem in the fourth century B.C. using geometric construction in three dimensions, determining a certain point as the intersection of three surfaces of revolution.
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| False claims of doubling the cube with compass and straightedge abound in mathematical [[Crank (person)|crank]] literature ([[pseudomathematics]]).
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| Origami may also be used to construct the [[Mathematics of origami#Doubling the cube|cube root of two by folding paper]].
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| ===Using a marked ruler===
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| [[File:Doubling the cube.svg|right|300px]]
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| There is a simple [[neusis construction]] using a marked ruler for a length which is the cube root of 2 times another length.<ref>{{cite book |title=100 Great Problems of Elementary Mathematics |author=Heinrich Dörrie |publisher=Dover |year=1965 |page=171 |isbn=0486-61348-8}}</ref>
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| *Mark a ruler with the given length, this will eventually be GH.
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| *Construct an equilateral triangle ABC with the given length as side.
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| *Extend AB an equal amount again to D.
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| *Extend the line BC forming the line CE.
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| *Extend the line DC forming the line CF
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| *Place the marked ruler so it goes through A and one end G of the marked length falls on CF and the other end of the marked length falls on ray CE. Thus GH is the given length.
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| The AG is the given length times the cube root of 2.
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| == References==
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| {{reflist}}
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| == External links ==
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| * {{springer|title=Duplication of the cube|id=p/d034200}}
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| * [http://www-history.mcs.st-and.ac.uk/HistTopics/Doubling_the_cube.html Doubling the cube]. J. J. O'Connor and E. F. Robertson in the MacTutor History of Mathematics archive.
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| * [http://mathforum.org/dr.math/faq/davies/cubedbl.htm To Double a Cube – The Solution of Archytas]. Excerpted with permission from A History of Greek Mathematics by Sir Thomas Heath.
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| * [http://www.cut-the-knot.org/Curriculum/Geometry/Delian.shtml Delian Problem Solved. Or Is It?] at [[cut-the-knot]].
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| {{Greek mathematics}}
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| [[Category:Euclidean plane geometry]]
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| [[Category:Mathematical problems]]
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| [[Category:History of geometry]]
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| [[Category:Compass and straightedge constructions]]
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Telecommunications Technician Dale Krejci from Boucherville, enjoys skateboarding, diet and rc model aircrafts. Has just finished a travel to Iguaçu National Park.
Visit my web blog: simple diet burn fat