Doubling the cube
Doubling the cube, also known as the Delian problem, is an ancient
The
In algebraic terms, doubling a unit cube requires the construction of a line segment of length x, where x3 = 2; in other words, x = , the cube root of two. This is because a cube of side length 1 has a volume of 13 = 1, and a cube of twice that volume (a volume of 2) has a side length of the cube root of 2. The impossibility of doubling the cube is therefore equivalent to the statement that is not a
Proof of impossibility
We begin with the unit line segment defined by
Respectively, the tools of a compass and straightedge allow us to create
So, given a coordinate of any constructed point, we may proceed inductively backwards through the x- and y-coordinates of the points in the order that they were defined until we reach the original pair of points (0,0) and (1,0). As every field extension has degree 2 or 1, and as the field extension over of the coordinates of the original pair of points is clearly of degree 1, it follows from the tower rule that the degree of the field extension over of any coordinate of a constructed point is a
Now, p(x) = x3 − 2 = 0 is easily seen to be irreducible over – any
History
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.[4][1]: 9 According to Plutarch,[5] however, the citizens of Delos consulted the oracle at Delphi to find 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.[6]
According to
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.[10] In modern notation, this means that given segments of lengths a and 2a, the duplication of the cube is equivalent to finding segments of lengths r and s so that
In turn, this means that
But
Solutions via means other than compass and straightedge
Menaechmus' original solution involves the intersection of two
Descartes theory of geometric solution of equations uses a parabola to introduce cubic equations, in this way it is possible to set up an equation whose solution is a cube root of two. Note that the parabola itself is not constructible except by three dimensional methods.
False claims of doubling the cube with compass and straightedge abound in mathematical crank literature (pseudomathematics).
Origami may also be used to construct the cube root of two by folding paper.
Using a marked ruler
There is a simple neusis construction using a marked ruler for a length which is the cube root of 2 times another length.[13]
- Mark a ruler with the given length; this will eventually be GH.
- Construct an equilateral triangle ABC with the given length as side.
- Extend AB an equal amount again to D.
- Extend the line BC forming the line CE.
- Extend the line DC forming the line CF.
- Place the marked ruler so it goes through A and one end, G, of the marked length falls on ray CF and the other end of the marked length, H, falls on ray CE. Thus GH is the given length.
Then AG is the given length times .
In music theory
In
Explanatory notes
References
- ^ a b Kern, Willis F.; Bland, James R. (1934). Solid Mensuration With Proofs. New York: John Wiley & Sons.
- JSTOR 3027812.
- ^ Stewart, Ian. Galois Theory. p. 75.
- ^ L. Zhmud The origin of the history of science in classical antiquity, p.84, quoting Plutarch and Theon of Smyrna
- ^ Plutarch, De E apud Delphos 386.E.4
- ^ Plutarch, De genio Socratis 579.B
- ^ (Plut., Quaestiones convivales VIII.ii, 718ef)
- ^ Carl Werner Müller, Die Kurzdialoge der Appendix Platonica, Munich: Wilhelm Fink, 1975, pp. 105–106
- ISBN 9780486675329.
- ^ T.L. Heath A History of Greek Mathematics, Vol. 1
- .
- ISBN 9780817633875.
- ISBN 0486-61348-8.
- ProQuest 7191936
External links
- Frédéric Beatrix, Peter Katzlinger: A pretty accurate solution to the Delian problem. In: Parabola Volume 59 (2023) Issue 1, online magazine (ISSN 1446-9723) published by the School of Mathematics and Statistics University of New South Wales
- Doubling the cube, proximity construction as animation (side = 1.259921049894873)—Wikimedia Commons
- "Duplication of the cube", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
- 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. Excerpt from A History of Greek Mathematics by Sir Thomas Heath.
- Delian Problem Solved. Or Is It? at cut-the-knot.
- Mathologer video: "2000 years unsolved: Why is doubling cubes and squaring circles impossible?"