Doubling the cube, also known as the Delian problem, is an ancientgeometric problem. Given the edge of a cube, the problem requires the construction of the edge of a second cube whose volume is double that of the first. As with the related problems of squaring the circle and trisecting the angle, doubling the cube is now known to be impossible using only a compass and straightedge, but even in ancient times solutions were known that employed other tools.
The Egyptians, Indians, and particularly the Greekswere aware of the problem and made many futile attempts at solving what they saw as an obstinate but soluble problem. However, the nonexistence of a compass-and-straightedge solution was finally proven by Pierre Wantzel in 1837.
In algebraic terms, doubling a unit cube requires the construction of a line segment of length x, where x3 = 2; in other words, x = 3√, 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 3√ is not a constructible number. This is a consequence of the fact that the coordinates of a new point constructed by a compass and straightedge are roots of polynomials over the field generated by the coordinates of previous points, of no greater degree than a quadratic. This implies that the degree of the field extension generated by a constructible point must be a power of 2. The field extension generated by 3√, however, is of degree 3.
We begin with the unit line segment defined by points (0,0) and (1,0) in the plane. We are required to construct a line segment defined by two points separated by a distance of 3√. It is easily shown that compass and straightedge constructions would allow such a line segment to be freely moved to touch the origin, parallel with the unit line segment - so equivalently we may consider the task of constructing a line segment from (0,0) to (3√, 0), which entails constructing the point (3√, 0).
Respectively, the tools of a compass and straightedge allow us to create circles centred on one previously defined point and passing through another, and to create lines passing through two previously defined points. Any newly defined point either arises as the result of the intersection of two such circles, as the intersection of a circle and a line, or as the intersection of two lines. An exercise of elementary analytic geometry shows that in all three cases, both the x- and y-coordinates of the newly defined point satisfy a polynomial of degree no higher than a quadratic, with coefficients that are additions, subtractions, multiplications, and divisions involving the coordinates of the previously defined points (and rational numbers). Restated in more abstract terminology, the new x- and y-coordinates have minimal polynomials of degree at most 2 over the subfield of ℝ generated by the previous coordinates. Therefore, the degree of the field extension corresponding to each new coordinate is 2 or 1.
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 ℚ 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 ℚ of any coordinate of a constructed point is a power of 2.
Now, p(x) = x3 − 2 = 0 is easily seen to be irreducible over ℤ – any factorisation would involve a linear factor (x − k) for some k ∈ ℤ, and so k must be a root of p(x); but also k must divide 2, that is, k = 1, 2, −1 or −2, and none of these are roots of p(x). By Gauss's Lemma, p(x) is also irreducible over ℚ, and is thus a minimal polynomial over ℚ for 3√. The field extension ℚ(3√):ℚ is therefore of degree 3. But this is not a power of 2, so by the above, 3√ is not the coordinate of a constructible point, and thus a line segment of 3√ cannot be constructed, and the cube cannot be doubled.
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.According to Plutarch 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.
According to Plutarch, Plato gave the problem to 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.This may be why the problem is referred to in the 350s BC by the author of the pseudo-Platonic Sisyphus (388e) as still unsolved. However another version of the story (attributed to Eratosthenes by Eutocius of Ascalon) says that all three found solutions but they were too abstract to be of practical value.
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. a and 2a, the duplication of the cube is equivalent to finding segments of lengths r and s so thatIn modern notation, this means that given segments of lengths
In turn, this means that
But Pierre Wantzel proved in 1837 that the cube root of 2 is not constructible; that is, it cannot be constructed with straightedge and compass.
Menaechmus' original solution involves the intersection of two conic curves. Other more complicated methods of doubling the cube involve neusis, the cissoid of Diocles, the conchoid of Nicomedes, or the Philo line. Pandrosion, a probably female mathematician of ancient Greece, found a numerically-accurate approximate solution using planes in three dimensions, but was heavily criticized by Pappus of Alexandria for not providing a proper mathematical proof.Archytas solved the problem in the 4th century BC using geometric construction in three dimensions, determining a certain point as the intersection of three surfaces of revolution.
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.
There is a simple neusis construction using a marked ruler for a length which is the cube root of 2 times another length.
Then AG is the given length times 3√.
In music theory, a natural analogue of doubling is the octave (a musical interval caused by doubling the frequency of a tone), and a natural analogue of a cube is dividing the octave into three parts, each the same interval. In this sense, the problem of doubling the cube is solved by the major third in equal temperament. This is a musical interval that is exactly one third of an octave. It multiplies the frequency of a tone by 24⁄12 = 21⁄3 = 3√, the side length of the Delian cube.
In geometry and algebra, a real number is constructible if and only if, given a line segment of unit length, a line segment of length can be constructed with compass and straightedge in a finite number of steps. Equivalently, is constructible if and only if there is a closed-form expression for using only the integers 0 and 1 and the operations for addition, subtraction, multiplication, division, and square roots.
Straightedge and compass construction, also known as ruler-and-compass construction or classical construction, is the construction of lengths, angles, and other geometric figures using only an idealized ruler and a pair of compasses.
Angle trisection is a classical problem of compass and straightedge constructions of ancient Greek mathematics. It concerns construction of an angle equal to one third of a given arbitrary angle, using only two tools: an unmarked straightedge and a compass.
In algebra, a cubic equation in one variable is an equation of the form
Squaring the circle is a problem proposed by ancient geometers. It is the challenge of constructing a square with the same area as a given circle by using only a finite number of steps with compass and straightedge. The difficulty of the problem raised the question of whether specified axioms of Euclidean geometry concerning the existence of lines and circles implied the existence of such a square.
The discipline of origami or paper folding has received a considerable amount of mathematical study. Fields of interest include a given paper model's flat-foldability, and the use of paper folds to solve up-to cubic mathematical equations.
The Huzita–Justin axioms or Huzita–Hatori axioms are a set of rules related to the mathematical principles of origami, describing the operations that can be made when folding a piece of paper. The axioms assume that the operations are completed on a plane, and that all folds are linear. These are not a minimal set of axioms but rather the complete set of possible single folds.
Archytas was an Ancient Greek philosopher, mathematician, astronomer, statesman, and strategist. He was a scientist of the Pythagorean school and famous for being the reputed founder of mathematical mechanics, as well as a good friend of Plato.
In mathematics, a cube root of a number x is a number y such that y3 = x. All nonzero real numbers, have exactly one real cube root and a pair of complex conjugate cube roots, and all nonzero complex numbers have three distinct complex cube roots. For example, the real cube root of 8, denoted , is 2, because 23 = 8, while the other cube roots of 8 are and . The three cube roots of −27i are
In the branch of mathematics known as Euclidean geometry, the Poncelet–Steiner theorem is one of several results concerning compass and straightedge constructions with additional restrictions imposed. This result states that whatever can be constructed by straightedge and compass together can be constructed by straightedge alone, provided that a single circle and its centre are given. This theorem is related to the rusty compass equivalence.
In geometry, the cissoid of Diocles is a cubic plane curve notable for the property that it can be used to construct two mean proportionals to a given ratio. In particular, it can be used to double a cube. It can be defined as the cissoid of a circle and a line tangent to it with respect to the point on the circle opposite to the point of tangency. In fact, the curve family of cissoids is named for this example and some authors refer to it simply as the cissoid. It has a single cusp at the pole, and is symmetric about the diameter of the circle which is the line of tangency of the cusp. The line is an asymptote. It is a member of the conchoid of de Sluze family of curves and in form it resembles a tractrix.
In mathematics, the Mohr–Mascheroni theorem states that any geometric construction that can be performed by a compass and straightedge can be performed by a compass alone.
Elementary mathematics consists of mathematics topics frequently taught at the primary or secondary school levels.
Menaechmus was an ancient Greek mathematician, geometer and philosopher born in Alopeconnesus or Prokonnesos in the Thracian Chersonese, who was known for his friendship with the renowned philosopher Plato and for his apparent discovery of conic sections and his solution to the then-long-standing problem of doubling the cube using the parabola and hyperbola.
In mathematics, a quadratrix is a curve having ordinates which are a measure of the area of another curve. The two most famous curves of this class are those of Dinostratus and E. W. Tschirnhaus, which are both related to the circle.
A proof of impossibility, also known as negative proof, proof of an impossibility theorem, or negative result, is a proof demonstrating that a particular problem cannot be solved as described in the claim, or that a particular set of problems cannot be solved in general. Proofs of impossibility often put decades or centuries of work attempting to find a solution to rest. To prove that something is impossible is usually much harder than the opposite task, as it is often necessary to develop a theory. Impossibility theorems are usually expressible as negative existential propositions, or universal propositions in logic.
The neusis is a geometric construction method that was used in antiquity by Greek mathematicians.
In algebra, casus irreducibilis is one of the cases that may arise in attempting to solve polynomials of degree 3 or higher with integer coefficients, to obtain roots that are expressed with radicals. It shows that many algebraic numbers are real-valued but cannot be expressed in radicals without introducing complex numbers. The most notable occurrence of casus irreducibilis is in the case of cubic polynomials that are irreducible over the rational numbers and have three real roots, which was proven by Pierre Wantzel in 1843. One can decide whether a given irreducible cubic polynomial is in casus irreducibilis using the discriminant Δ, via Cardano's formula. Let the cubic equation be given by
A timeline of algebra and geometry
The Ancient Tradition of Geometric Problems is a book on ancient Greek mathematics, focusing on three problems now known to be impossible if one uses only the straightedge and compass constructions favored by the Greek mathematicians: squaring the circle, doubling the cube, and trisecting the angle. It was written by Wilbur Knorr (1945–1997), a historian of mathematics, and published in 1986 by Birkhäuser. Dover Publications reprinted it in 1993.