Doubling the cube, also known as the Delian problem, is an ancient[a][1]: 9 geometric 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 to construct by using only a compass and straightedge, but even in ancient times solutions were known that employed other methods.
The Egyptians, Indians, and particularly the Greeks[2] were aware of the problem and made many futile attempts at solving what they saw as an obstinate but soluble problem.[3][b] 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 = , 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 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 , however, is of degree 3.
Cite error: There are <ref group=lower-alpha> tags or {{efn}} templates on this page, but the references will not show without a {{reflist|group=lower-alpha}} template or {{notelist}} template (see the help page).
^Kern, Willis F.; Bland, James R. (1934). Solid Mensuration With Proofs. New York: John Wiley & Sons.
^Guilbeau, Lucye (1930). "The History of the Solution of the Cubic Equation". Mathematics News Letter. 5 (4): 8–12. doi:10.2307/3027812. JSTOR 3027812.
Doublingthecube, also known as the Delian problem, is an ancient: 9 geometric problem. Given the edge of a cube, the problem requires the construction...
_{i=1}^{8}d_{i}^{2}}{8}}+{\frac {2R^{2}}{3}}\right)^{2}.} Doublingthecube, or the Delian problem, was the problem posed by ancient Greek mathematicians of using...
Aligning the two points on the two lines is another neusis construction as in the solution to doublingthecube. The problem of rigid origami, treating the folds...
and in the problem of finding the edge of a cube whose volume is twice that of a cube with a given edge (doublingthecube). In 1837 Pierre Wantzel proved...
the square root of n and the (Euclidean) length of the vector (1,1,1,....1,1) in n-dimensional space. Doublingthecube K-cell Robbins constant, the average...
trisecting an arbitrary angle or of doublingthe volume of a cube, based on the impossibility of constructing cube roots of lengths. He also showed that...
one-hundred percent Doublingthecube (i. e., hypothetical geometric construction of a cube with twice the volume of a given cube) Doubling time, the length of...
the conic sections and his solution to the problem of doublingthecube. Menaechmus likely discovered the conic sections, that is, the ellipse, the parabola...
are known. According to Eutocius, Archytas was the first to solve the problem of doublingthecube (the so-called Delian problem) with an ingenious geometric...
that they did. The problem of doublingthecube involves the simplest and oldest studied cubic equation, and one for which the ancient Egyptians did not believe...
devices, who lived probably during the 1st or 2nd century BC. Philo used the line to doublethecube; because doublingthecube cannot be done by a straightedge...
antiquity cannot be solved as they were stated (doublingthecube and trisecting the angle), and characterizing the regular polygons that are constructible (this...
doublingthecube and squaring the circle. In the case of doublingthecube, the impossibility of the construction originates from the fact that the compass...
approximate the problem accurately in few steps. Two other classical problems of antiquity, famed for their impossibility, were doublingthecube and trisecting...
proved that the problems of doublingthecube, and trisecting the angle are impossible to solve if one uses only compass and straightedge. In the same paper...
money play, thedoublingcube is used. At the start of each game, thedoublingcube is placed on the bar with the number 64 showing; thecube is then said...
as the mathematician to whom Plato sent those asking how to doublethecube. Perhaps on the basis of this mention of a mathematical Euclid roughly a century...
The greatest mathematician associated with the group, however, may have been Archytas (c. 435-360 BC), who solved the problem of doublingthecube, identified...
mathematics, Philo tackled the problem of doublingthecube. Thedoubling of thecube was necessitated by the following problem: given a catapult, construct...
drawing a square having the same area. Doublingthecube: Given any cube drawing a cube with twice its volume. Trisecting the angle: Given any angle dividing...
Two other classical problems—trisecting the general angle and doublingthecube—were also proved impossible in the 19th century, and all of these problems...
The positive square root of 2 is constructible. However, thecube root of 2 is not constructible; this is related to the impossibility of doubling the...