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In mathematics, a constructible polygon is a regular
polygon that can be constructed with compass and
straightedge. For example, a regular pentagon is constructible with compass and
straightedge while a regular heptagon is not.
Conditions for constructibility
Some regular polygons are easy to construct with compass and straightedge; others are not. This led to the question being
posed: is it possible to construct all regular n-gons with compass and straightedge? If not, which
n-gons are constructible and which are not?
Carl Friedrich Gauss proved the constructibility of the
regular 17-gon in 1796. Five years later, he developed the theory of
Gaussian periods in his Disquisitiones Arithmeticae. This theory allowed
him to formulate a sufficient condition for the
constructibility of regular polygons:
- A regular n-gon can be constructed with compass and straightedge if n is the product of a power of 2 and
any number of distinct Fermat primes.
Gauss conjectured that this condition was also necessary, but he offered no proof of this fact, which was proved by
Pierre Wantzel in (1836). It seems very unlikely that Gauss had a correct proof, because by taking n = 9, one
can immediately deduce the impossibility of trisecting an angle of 120 degrees, a fact of which Gauss was certainly aware.
General theory
In the light of later work on Galois theory, the principles of these
proofs have been clarified. It is straightforward to show from analytic geometry that constructible lengths must come from base lengths by the solution of some sequence
of quadratic equations. In terms of field theory, such lengths must be contained in a
field extension generated by a tower of quadratic extensions.
It follows that a field generated by constructions will always have degree over the base field that is a power of two.
In the specific case of a regular n-gon, the question reduces to the question of constructing a length
- cos(2π/n).
This number lies in the n-th cyclotomic field —
and in fact in its real subfield, which is a totally real field
of degree over the rational numbers
- ½φ(n)
where φ(n) is Euler's totient
function. Wantzel's result comes down to a calculation showing that φ(n) is a power of 2 precisely in the cases
specified.
As for the construction of Gauss, when the Galois group is 2-group it follows that it has a sequence of subgroups of
orders
- 1, 2, 4, 8, ...
that are nested, each in the next (a composition series, in
group theory terms), something simple to prove by induction in this case of
an abelian group. Therefore there are subfields nested inside the
cyclotomic field, each of degree 2 over the one before. Generators for each such field can be written down by Gaussian period
theory. For example for n = 17 there is a period that is a sum of eight roots of unity, one that is a sum of four roots
of unity, and one that is the sum of two, which is
- cos(2π/17).
Each of those is a root of a quadratic equation in terms of the one before. Moreover these equations have real rather than
imaginary roots, so in principle can be solved by geometric construction: this because the work all goes on inside a totally real
field.
In this way the result of Gauss can be understood in current terms; for actual calculation of the equations to be solved, the
periods can be squared and compared with the 'lower' periods, in a quite feasible algorithm.
Detailed results in terms of Fermat primes
Only five Fermat primes are known:
- F0 = 3, F1 = 5, F2 = 17, F3 = 257, and
F4 = 65537
- (sequence A019434 in OEIS).
Thus an n-gon is constructible if
- n = 3, 4, 5, 6, 8, 10, 12, 15, 16, 17, 20, 24, ...
- (sequence A003401 in
OEIS),
while and an n-gon is not constructible with compass and straightedge if
- n = 7, 9, 11, 13, 14, 18, 19, 21, 22, 23, 25,...
- (sequence A004169 in
OEIS).
The first regular polygon for which the constructibility is unknown has
F33 = 2233+ 1 sides, because F33 is the first Fermat number of unknown primality (as of March 2004).
Compass and straightedge constructions
Compass and straightedge constructions are known for all constructible polygons. If
n = p·q with p = 2 or p and q relatively prime, an n-gon can be constructed from a
p-gon and a q-gon.
- If p = 2, draw a q-gon and bisect one of its central
angles. From this, a 2q-gon can be constructed.
- If p > 2, inscribe a p-gon and a q-gon in the same circle in such a way that they
share a vertex. Because p and q are relatively prime, there are two vertices a central angle
360°/(p·q) apart. From this, a p·q-gon can be constructed.
Thus one only has to find a compass and straightedge construction for n-gons where n is a Fermat prime.
- The construction for an equilateral triangle is simple and has been known since Antiquity. See equilateral triangle.
- Constructions for the regular pentagon were described both by Euclid (Elements, ca 300 BC), and by Ptolemy
(Almagest, ca AD 150). See pentagon.
- Although Gauss proved that the regular 17-gon is constructible, he didn't actually show how to do it. The
first construction is due to Erchinger, a few years after Gauss' work. See heptadecagon.
- A regular 257-gon is almost indistinguishable from a circle; the first explicit construction was given by F.J. Richelot
(1832).
- A construction for a regular 65537-gon was first given by J. Hermes (1894). The construction is very complex; Hermes spent 10
years completing the 200-page manuscript. (John Conway has cast doubt on the
validity of Hermes' construction, however.)
Other constructions
It should be stressed that the concept of constructibility as discussed in this article applies specifically to compass and straightedge constructions. More
constructions become possible if other tools are allowed. The so-called neusis constructions, for
example, make use of a marked ruler. The construction of a regular heptagon is then easy.
See also
External links
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