Doubling The Cube
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Doubling the cube, also known as the Delian problem, is an ancient
geometric Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ca ...
problem. Given the
edge Edge or EDGE may refer to: Technology Computing * Edge computing, a network load-balancing system * Edge device, an entry point to a computer network * Adobe Edge, a graphical development application * Microsoft Edge, a web browser developed by ...
of a
cube In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross. The cube is the only r ...
, the problem requires the construction of the edge of a second cube whose
volume Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). The de ...
is double that of the first. As with the related problems of
squaring the circle Squaring the circle is a problem in geometry first proposed in Greek mathematics. It is the challenge of constructing a square with the area of a circle by using only a finite number of steps with a compass and straightedge. The difficulty ...
and
trisecting the angle Angle trisection is a classical problem of straightedge and compass construction 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 a ...
, doubling the cube is now known to be impossible to construct by using only a
compass and straightedge In geometry, straightedge-and-compass construction – also known as ruler-and-compass construction, Euclidean construction, or classical construction – is the construction of lengths, angles, and other geometric figures using only an ideali ...
, but even in ancient times solutions were known that employed other tools. The
Egyptians Egyptians ( arz, المَصرِيُون, translit=al-Maṣriyyūn, ; arz, المَصرِيِين, translit=al-Maṣriyyīn, ; cop, ⲣⲉⲙⲛ̀ⲭⲏⲙⲓ, remenkhēmi) are an ethnic group native to the Nile, Nile Valley in Egypt. Egyptian ...
, Indians, and particularly the
Greeks The Greeks or Hellenes (; el, Έλληνες, ''Éllines'' ) are an ethnic group and nation indigenous to the Eastern Mediterranean and the Black Sea regions, namely Greece, Cyprus, Albania, Italy, Turkey, Egypt, and, to a lesser extent, oth ...
were 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 Pierre Laurent Wantzel (5 June 1814 in Paris – 21 May 1848 in Paris) was a French mathematician who proved that several ancient geometric problems were impossible to solve using only compass and straightedge. In a paper from 1837, Wantzel pr ...
in 1837. In algebraic terms, doubling a
unit cube A unit cube, more formally a cube of side 1, is a cube whose sides are 1 unit long.. See in particulap. 671. The volume of a 3-dimensional unit cube is 1 cubic unit, and its total surface area is 6 square units.. Unit hypercube The term '' ...
requires the construction of a
line segment In geometry, a line segment is a part of a straight line that is bounded by two distinct end points, and contains every point on the line that is between its endpoints. The length of a line segment is given by the Euclidean distance between ...
of length , where ; in other words, , the cube root of two. This is because a cube of side length 1 has a volume of , and a cube of twice that volume (a volume of 2) has a side length of the
cube root In mathematics, a cube root of a number is a number such that . 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. Fo ...
of 2. The impossibility of doubling the cube is therefore
equivalent Equivalence or Equivalent may refer to: Arts and entertainment *Album-equivalent unit, a measurement unit in the music industry * Equivalence class (music) *'' Equivalent VIII'', or ''The Bricks'', a minimalist sculpture by Carl Andre *''Equiva ...
to the statement that \sqrt /math> is not a
constructible number In geometry and algebra, a real number r is constructible if and only if, given a line segment of unit length, a line segment of length , r, can be constructed with compass and straightedge in a finite number of steps. Equivalently, r is con ...
. This is a consequence of the fact that the coordinates of a new point constructed by a compass and straightedge are roots of
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ...
s over the field generated by the coordinates of previous points, of no greater
degree Degree may refer to: As a unit of measurement * Degree (angle), a unit of angle measurement ** Degree of geographical latitude ** Degree of geographical longitude * Degree symbol (°), a notation used in science, engineering, and mathematics ...
than a quadratic. This implies that the
degree Degree may refer to: As a unit of measurement * Degree (angle), a unit of angle measurement ** Degree of geographical latitude ** Degree of geographical longitude * Degree symbol (°), a notation used in science, engineering, and mathematics ...
of the
field extension In mathematics, particularly in algebra, a field extension is a pair of fields E\subseteq F, such that the operations of ''E'' are those of ''F'' restricted to ''E''. In this case, ''F'' is an extension field of ''E'' and ''E'' is a subfield of ...
generated by a constructible point must be a power of 2. The field extension generated by \sqrt /math>, however, is of degree 3.


Proof of impossibility

We begin with the unit line segment defined by points (0,0) and (1,0) in the
plane Plane(s) most often refers to: * Aero- or airplane, a powered, fixed-wing aircraft * Plane (geometry), a flat, 2-dimensional surface Plane or planes may also refer to: Biology * Plane (tree) or ''Platanus'', wetland native plant * ''Planes' ...
. We are required to construct a line segment defined by two points separated by a distance of \sqrt /math>. It is easily shown that compass and straightedge constructions would allow such a line segment to be freely moved to touch the
origin Origin(s) or The Origin may refer to: Arts, entertainment, and media Comics and manga * Origin (comics), ''Origin'' (comics), a Wolverine comic book mini-series published by Marvel Comics in 2002 * The Origin (Buffy comic), ''The Origin'' (Bu ...
,
parallel Parallel is a geometric term of location which may refer to: Computing * Parallel algorithm * Parallel computing * Parallel metaheuristic * Parallel (software), a UNIX utility for running programs in parallel * Parallel Sysplex, a cluster of ...
with the unit line segment - so equivalently we may consider the task of constructing a line segment from (0,0) to (\sqrt /math>, 0), which entails constructing the point (\sqrt /math>, 0). Respectively, the tools of a compass and straightedge allow us to create
circles A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is const ...
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 In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, their i ...
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 In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system. This contrasts with synthetic geometry. Analytic geometry is used in physics and engineerin ...
shows that in all three cases, both the - and -coordinates of the newly defined point satisfy a polynomial of degree no higher than a quadratic, with
coefficients In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or an expression; it is usually a number, but may be any expression (including variables such as , and ). When the coefficients are themselves var ...
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 - and -coordinates have minimal polynomials of degree at most 2 over the subfield of generated by the previous coordinates. Therefore, the
degree Degree may refer to: As a unit of measurement * Degree (angle), a unit of angle measurement ** Degree of geographical latitude ** Degree of geographical longitude * Degree symbol (°), a notation used in science, engineering, and mathematics ...
of the
field extension In mathematics, particularly in algebra, a field extension is a pair of fields E\subseteq F, such that the operations of ''E'' are those of ''F'' restricted to ''E''. In this case, ''F'' is an extension field of ''E'' and ''E'' is a subfield of ...
corresponding to each new coordinate is 2 or 1. So, given a coordinate of any constructed point, we may proceed inductively backwards through the - and -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 \mathbb 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 \mathbb of any coordinate of a constructed point is a
power of 2 A power of two is a number of the form where is an integer, that is, the result of exponentiation with number two as the base and integer  as the exponent. In a context where only integers are considered, is restricted to non-negative ...
. Now, is easily seen to be
irreducible In philosophy, systems theory, science, and art, emergence occurs when an entity is observed to have properties its parts do not have on their own, properties or behaviors that emerge only when the parts interact in a wider whole. Emergence ...
over \mathbb – any factorisation would involve a linear factor for some , and so must be a
root In vascular plants, the roots are the organs of a plant that are modified to provide anchorage for the plant and take in water and nutrients into the plant body, which allows plants to grow taller and faster. They are most often below the sur ...
of ; but also must divide 2, that is, or , and none of these are roots of . By Gauss's Lemma, is also irreducible over \mathbb, and is thus a minimal polynomial over \mathbb for \sqrt /math>. The field extension \mathbb (\sqrt :\mathbb is therefore of degree 3. But this is not a power of 2, so by the above, \sqrt /math> is not the coordinate of a constructible point, and thus a line segment of \sqrt /math> cannot be constructed, and the cube cannot be doubled.


History

The problem owes its name to a story concerning the citizens of
Delos The island of Delos (; el, Δήλος ; Attic: , Doric: ), near Mykonos, near the centre of the Cyclades archipelago, is one of the most important mythological, historical, and archaeological sites in Greece. The excavations in the island are ...
, who consulted the oracle at
Delphi Delphi (; ), in legend previously called Pytho (Πυθώ), in ancient times was a sacred precinct that served as the seat of Pythia, the major oracle who was consulted about important decisions throughout the ancient classical world. The oracle ...
in order to learn how to defeat a plague sent by
Apollo Apollo, grc, Ἀπόλλωνος, Apóllōnos, label=genitive , ; , grc-dor, Ἀπέλλων, Apéllōn, ; grc, Ἀπείλων, Apeílōn, label=Arcadocypriot Greek, ; grc-aeo, Ἄπλουν, Áploun, la, Apollō, la, Apollinis, label= ...
.L. Zhmud ''The origin of the history of science in classical antiquity'', p.84
quoting
Plutarch Plutarch (; grc-gre, Πλούταρχος, ''Ploútarchos''; ; – after AD 119) was a Greek Middle Platonist philosopher, historian, biographer, essayist, and priest at the Temple of Apollo in Delphi. He is known primarily for his ''P ...
and
Theon of Smyrna Theon of Smyrna ( el, Θέων ὁ Σμυρναῖος ''Theon ho Smyrnaios'', ''gen.'' Θέωνος ''Theonos''; fl. 100 CE) was a Greek philosopher and mathematician, whose works were strongly influenced by the Pythagorean school of thought. His ...
According to
Plutarch Plutarch (; grc-gre, Πλούταρχος, ''Ploútarchos''; ; – after AD 119) was a Greek Middle Platonist philosopher, historian, biographer, essayist, and priest at the Temple of Apollo in Delphi. He is known primarily for his ''P ...
, however, the citizens of
Delos The island of Delos (; el, Δήλος ; Attic: , Doric: ), near Mykonos, near the centre of the Cyclades archipelago, is one of the most important mythological, historical, and archaeological sites in Greece. The excavations in the island are ...
consulted the
oracle An oracle is a person or agency considered to provide wise and insightful counsel or prophetic predictions, most notably including precognition of the future, inspired by deities. As such, it is a form of divination. Description The word '' ...
at
Delphi Delphi (; ), in legend previously called Pytho (Πυθώ), in ancient times was a sacred precinct that served as the seat of Pythia, the major oracle who was consulted about important decisions throughout the ancient classical world. The oracle ...
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 Plato ( ; grc-gre, Πλάτων ; 428/427 or 424/423 – 348/347 BC) was a Greek philosopher born in Athens during the Classical period in Ancient Greece. He founded the Platonist school of thought and the Academy, the first institution ...
, 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 The island of Delos (; el, Δήλος ; Attic: , Doric: ), near Mykonos, near the centre of the Cyclades archipelago, is one of the most important mythological, historical, and archaeological sites in Greece. The excavations in the island are ...
to occupy themselves with the study of geometry and mathematics in order to calm down their passions. According to
Plutarch Plutarch (; grc-gre, Πλούταρχος, ''Ploútarchos''; ; – after AD 119) was a Greek Middle Platonist philosopher, historian, biographer, essayist, and priest at the Temple of Apollo in Delphi. He is known primarily for his ''P ...
, Plato gave the problem to Eudoxus and
Archytas Archytas (; el, Ἀρχύτας; 435/410–360/350 BC) was an Ancient Greek philosopher, mathematician, music theorist, astronomer, statesman, and strategist. He was a scientist of the Pythagorean school and famous for being the reputed founder ...
and
Menaechmus :''There is also a Menaechmus in Plautus' play, ''The Menaechmi''.'' Menaechmus ( el, Μέναιχμος, 380–320 BC) was an ancient Greek mathematician, geometer and philosopher born in Alopeconnesus or Prokonnesos in the Thracian Chersonese, w ...
, 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 In Greek mythology, Sisyphus or Sisyphos (; Ancient Greek: Σίσυφος ''Sísyphos'') was the founder and king of Ancient Corinth, Ephyra (now known as Corinth). Hades punished him for cheating death twice by forcing him to roll an immense bo ...
'' (388e) as still unsolved. However another version of the story (attributed to
Eratosthenes Eratosthenes of Cyrene (; grc-gre, Ἐρατοσθένης ;  – ) was a Greek polymath: a mathematician, geographer, poet, astronomer, and music theorist. He was a man of learning, becoming the chief librarian at the Library of Alexandria ...
by
Eutocius of Ascalon Eutocius of Ascalon (; el, Εὐτόκιος ὁ Ἀσκαλωνίτης; 480s – 520s) was a Palestinian-Greek mathematician who wrote commentaries on several Archimedean treatises and on the Apollonian ''Conics''. Life and work Little is ...
) 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 Hippocrates of Chios ( grc-gre, Ἱπποκράτης ὁ Χῖος; c. 470 – c. 410 BC) was an ancient Greek mathematician, geometer, and astronomer. He was born on the isle of Chios, where he was originally a merchant. After some misadve ...
that it is equivalent to finding two mean proportionals between a line segment and another with twice the length.T.L. Heath ''A history of Greek mathematics'', Vol. 1] In modern notation, this means that given segments of lengths and , the duplication of the cube is equivalent to finding segments of lengths and so that :\frac = \frac = \frac . In turn, this means that :r=a\cdot\sqrt But
Pierre Wantzel Pierre Laurent Wantzel (5 June 1814 in Paris – 21 May 1848 in Paris) was a French mathematician who proved that several ancient geometric problems were impossible to solve using only compass and straightedge. In a paper from 1837, Wantzel pr ...
proved in 1837 that the
cube root In mathematics, a cube root of a number is a number such that . 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. Fo ...
of 2 is not constructible; that is, it cannot be constructed with straightedge and compass.


Solutions via means other than compass and straightedge

Menaechmus' original solution involves the intersection of two
conic In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a special ...
curves. Other more complicated methods of doubling the cube involve
neusis In geometry, the neusis (; ; plural: grc, νεύσεις, neuseis, label=none) is a geometric construction method that was used in antiquity by Greek mathematics, Greek mathematicians. Geometric construction The neusis construction consists ...
, the
cissoid of Diocles In geometry, the cissoid of Diocles (; named for 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 de ...
, the
conchoid of Nicomedes In geometry, a conchoid is a curve derived from a fixed point , another curve, and a length . It was invented by the ancient Greek mathematician Nicomedes. Description For every line through that intersects the given curve at the two points ...
, or the Philo line.
Pandrosion Pandrosion of Alexandria () was a mathematician in fourth-century-AD Alexandria, discussed in the ''Mathematical Collection'' of Pappus of Alexandria and known for developing an approximate method for doubling the cube. Although there is disagreem ...
, 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 Pappus of Alexandria (; grc-gre, Πάππος ὁ Ἀλεξανδρεύς; AD) was one of the last great Greek mathematicians of antiquity known for his ''Synagoge'' (Συναγωγή) or ''Collection'' (), and for Pappus's hexagon theorem i ...
for not providing a proper
mathematical proof A mathematical proof is an inferential argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use other previously established statements, such as theorems; but every proo ...
.
Archytas Archytas (; el, Ἀρχύτας; 435/410–360/350 BC) was an Ancient Greek philosopher, mathematician, music theorist, astronomer, statesman, and strategist. He was a scientist of the Pythagorean school and famous for being the reputed founder ...
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 Pseudomathematics, or mathematical crankery, is a mathematics-like activity that does not adhere to the framework of rigor of formal mathematical practice. Common areas of pseudomathematics are solutions of problems proved to be unsolvable or r ...
). 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 In geometry, the neusis (; ; plural: grc, νεύσεις, neuseis, label=none) is a geometric construction method that was used in antiquity by Greek mathematicians. Geometric construction The neusis construction consists of fitting a line e ...
using a marked ruler for a length which is the cube root of 2 times another length. #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 \sqrt /math>.


In music theory

In
music theory Music theory is the study of the practices and possibilities of music. ''The Oxford Companion to Music'' describes three interrelated uses of the term "music theory". The first is the "rudiments", that are needed to understand music notation (ke ...
, a natural analogue of doubling is the
octave In music, an octave ( la, octavus: eighth) or perfect octave (sometimes called the diapason) is the interval between one musical pitch and another with double its frequency. The octave relationship is a natural phenomenon that has been refer ...
(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 classical music, a third is a musical interval encompassing three staff positions (see Interval number for more details), and the major third () is a third spanning four semitones. Forte, Allen (1979). ''Tonal Harmony in Concept and P ...
in
equal temperament An equal temperament is a musical temperament or tuning system, which approximates just intervals by dividing an octave (or other interval) into equal steps. This means the ratio of the frequencies of any adjacent pair of notes is the same, wh ...
. This is a musical interval that is exactly one third of an octave. It multiplies the frequency of a tone by 2^=2^=\sqrt /math>, the side length of the Delian cube.


Explanatory notes


References


External links


Doubling the cube, proximity construction as animation (side = 1.259921049894873)
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*
Doubling the cube
J. J. O'Connor and E. F. Robertson in the MacTutor History of Mathematics archive.

Excerpted with permission from ''A History of Greek Mathematics'' by Sir Thomas Heath.
Delian Problem Solved. Or Is It?
at
cut-the-knot Alexander Bogomolny (January 4, 1948 July 7, 2018) was a Soviet-born Israeli-American mathematician. He was Professor Emeritus of Mathematics at the University of Iowa, and formerly research fellow at the Moscow Institute of Electronics and Math ...
.
Mathologer video: "2000 years unsolved: Why is doubling cubes and squaring circles impossible?"
{{Authority control Compass and straightedge constructions Cubic irrational numbers Euclidean plane geometry History of geometry Unsolvable puzzles