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Encyclopedia > Doubling the cube

Doubling the cube (also known as The Delian Problem) is one of the three most famous geometric problems unsolvable by compass and straightedge construction. It was known to the Egyptians, Greeks, and Indians.[1] For other uses, see Geometry (disambiguation). ... Creating a regular hexagon with a ruler and compass Construction of a regular pentagon Compass and straightedge or ruler-and-compass construction is the construction of lengths or angles using only an idealized ruler and compass. ...


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 scdotsqrt[3]{2}. The problem is known to be impossible to solve with only compass and straightedge, because sqrt[3]{2} is not a constructible number. A cube[1] is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. ... A point in the Euclidean plane is a constructible point if, given a fixed coordinate system (or a fixed line segment of unit length), one can construct the point with unruled straightedge and compass. ...

Contents

History

According to legend, the citizens of Athens consulted the oracle of Apollo at Delos in 430 BC, in order to learn how to defeat a plague which was ravaging their lands. The oracle responded that to stop the plague, they must double the size of their altar. The Athenians dutifully doubled each side of the altar, and the plague increased. The correct interpretation was that they must double the volume of their altar, not merely its side length; this proved to be a most difficult problem indeed, but was solved in 350 BC to the efforts of Menaechmus. The only problem was that the plague was finished several decades before. It is due to this legend that the problem is often known as the Delian problem.[2] This article is about the capital of Greece. ... This article is about prophetic oracles in various cultures. ... For other uses, see Apollo (disambiguation). ... The island of Delos, Carl Anton Joseph Rottmann, 1847 The island of Delos (Greek: Δήλος, Dhilos), isolated in the centre of the roughly circular ring of islands called the Cyclades, near Mykonos, had a position as a holy sanctuary for a millennium before Olympian Greek mythology made it the birthplace of... Bubonic plague is the best-known manifestation of the bacterial disease plague, caused by the bacterium Yersinia pestis. ... Greek mathematician and geometer said to have been the tutor of Alexander the Great. ...


Solutions

An illustration of the ruler-and-compass method
An illustration of the ruler-and-compass method

There are many ways to construct sqrt[3]{2} which involve tools other than compass and straightedge. In fact, some of these tools can themselves be constructed using compass and straightedge, but must be cut out of a sheet of paper before they can be used. For example, following sir Isaac Newton, construct a ruler with a single unit distance marked on it. Construct an equilateral triangle ABC with side length 1, and extend side overline{AB} by one unit to form the line segment overline{ABD}. Extend side overline{BC} to form the ray overrightarrow{BCE}, and draw the ray overrightarrow{DCF}. Now take the ruler and place it so that it passes through vertex A and intersects overline{DCF} at G and overline{BCE} at H, such that the distance GH is exactly 1. The distance AG will then be precisely sqrt[3]{2}. Image File history File links An image showing the construction of the cube root of 2 using only straightedge, compass, and a unit ruler. ... Image File history File links An image showing the construction of the cube root of 2 using only straightedge, compass, and a unit ruler. ... Sir Isaac Newton FRS (4 January 1643 – 31 March 1727) [ OS: 25 December 1642 – 20 March 1727][1] was an English physicist, mathematician, astronomer, natural philosopher, and alchemist. ...


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 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. The cissoid of Diocles is an unbounded plane curve with a single cusp, which is symmetric about the line of tangency of the cusp, and whose pair of symmetrical branches both approach the same asymptote (but in opposite directions) as a point moving along the cissoid moves farther away from... A conchoid is a curve derived from a fixed point O, another curve, and a length d. ... In geometry, the Philo line is a line segment defined from an angle and a point. ... Archytas Archytas (428 BC - 347 BC) was a Greek philosopher, mathematician, astronomer, statesman, strategist and commander-in-chief. ...


False claims of doubling the cube with compass and straightedge abound in mathematical crank literature (Pseudomathematics). Crank is a pejorative term for a person who holds some belief which the vast majority of his contemporaries would consider false, clings to this belief in the face of all counterarguments or evidence presented to him. ... Pseudomathematics is a form of mathematics-like activity that does not work within the framework, definitions, rules, or rigor of formal mathematical models. ...


References

  1. ^ Lucye Guilbeau (1930). "The History of the Solution of the Cubic Equation", Mathematics News Letter 5 (4), p. 8-12.
  2. ^ (1982) Famous problems of geometry and how to solve them. Dover Publications, 29-30. ISBN 0486242978. 

External links

  • Doubling the cube. J. J. O'Connor and E. F. Robertson in the MacTutor History of Mathematics archive.
  • To Double a Cube -- The Solution of Archytas. Excerpted with permission from A History of Greek Mathematics by Sir Thomas Heath.
  • Delian Problem Solved. Or Is It? at cut-the-knot.
cut-the-knot is an educational website maintained by Alexander Bogomolny and devoted to popular exposition of a great variety of topics in mathematics. ...

  Results from FactBites:
 
GammonEmpire - Doubling Cube Rules (593 words)
The doubling cube is the very heart of backgammon and makes it the exciting game that it is. Without it the game might well have died off in the 1920`s.
At the commencement of play, the doubling cube rests to one side of the board, in the centre between the two players with a displayed value of 64 (there is no 1 on the doubling cube so 64 serves as 1 at the start of the game).
Doubling is far more complex than this because cube ownership, psychology, gammons and backgammons (and in tournaments the match score) all play a part in doubling decisions but that is beyond the scope of this first article on the topic.
PlanetMath: classical problems of constructibility (553 words)
Doubling the cube: Given an arbitrary cube, can a cube with double the volume be constructed?
The discovery that doubling the cube using only compass and straightedge is impossible is also attributed to Pierre Wantzel.
Theorem 2 (Wantzel)   Doubling the cube is impossible using only compass and straightedge.
  More results at FactBites »

 
 

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