Constructible polygon

In mathematics, a konstruierbares polygon is a regular polygon with a compass and ( unlabeled ) ruler -. Absorbed the Euclidean tools - can be constructed. For example, the regular pentagon regular heptagon is constructible, is not.

Constructability

To capture the notion of " constructible with compass and straightedge " mathematically precise, it must be defined what is possible with these tools. We assume that at the beginning of each construction are given two points. Using the ruler, you can then construct a line through two points, with the compasses a circle through a point to a point other than the center. In addition, the points of intersection of lines and circles are constructible.

From these basic constructions, a number of other constructions can be derived, such as the construction of a perpendicular bisector or dropping a perpendicular. It is then called a positive real number constructible if one can construct two points, so that the Euclidean distance between them is the value of this number is equal to (the distance between two given points is defined as 1). For example, if the number constructible, then one can construct using the height set two points with distance. Are two numbers and constructible, so using the intercept theorem their product and the inverse, as well as by tapping a distance their sum and difference. An angle hot constructible, if the number is constructible; the meaning of this definition reveals itself quickly by looking at the unit circle.

To construct now a regular -gon, it is sufficient to construct the central angle, because if you have been the center of the pentagon and a corner, can be based on the connecting straight through the center and corner of the next vertex construct. Conversely, if a regular -gon given, so you can tap the central angle. To answer the question whether the corner is constructible, we are thus reduced to the case, to decide if the central angle is constructible.

Constructability of numbers

A number is said to be constructible with compass and straightedge, if it is the length of a line, which can be constructed as described here.

In the synthetic geometry, points and numbers are investigated, which can be constructed from an ( almost) any default set of route lengths slightly more general. For this purpose, field extensions of the rational numbers are considered, the Euclidean body and thus coordinates the body of a Euclidean plane (in the sense of synthetic geometry). The constructibility with ruler and compasses a number then means that it is a coordinate of a constructible from the specifications point in the plane. → Refer to these conceptions the article Euclidean body!

Criterion for constructability

Carl Friedrich Gauss in 1796 showed that regular seventeen- is constructible. To this end, he showed that the number can be represented as an expression that contains only integers, basic arithmetic operations and nested square roots. By developed in his Disquisitiones Arithmeticae theory succeeded Gauss five years later, specify a sufficient condition for the construction of regular polygons:

Although Gauss knew that the condition is also necessary, however, has not published his proof of this. Pierre -Laurent Wantzel got this 1837.

It can be shown that a number of well then the product of a power of 2 with different Fermat primes is when a power of 2. Here denotes the Euler φ - function.

In summary: For a number of the following statements are equivalent:

• The regular -gon is constructible with compass and straightedge.
• And with pairwise distinct Fermat primes.

• For one.

In particular, if and are relatively prime and both the corner and the corner - constructible, so is constructible because even the corner. This fact can also directly specify the geometric construction, because if and are relatively prime, then there is by the lemma of Bézout two integers and with By now times applies the central angles of the pentagon and times the central angle of the pentagon, one has the angle - and therefore also the corner - constructed.

Concrete consequences of the criterion

Despite an intensive search primes 3, 5, 17, 257 and 65537 were also found to date no further about the five already known Gauss Fermat. There is even a plausible conjecture that there are no further Fermat primes.

Should it actually be only five Fermat primes, then under the polygons with an odd number of vertices exactly the following, in principle, be constructed:

All other constructible polygons ( then with even number of vertices ) are due by ( continued ) doubling the number of corners.

It is allowed for the construction of an additional tool to three-division of an angle ( Trisection ), so all of regular polygons can be constructed with area numbers of the form, wherein more than three of the mold with different Pierpont primes. In this way, the heptagon, the Neuneck and Dreizehneck example, are constructible, but not the Elfeck.

From the following table for the construction by ruler and compass arises for regular polygons up to 100 -Eck (yes ), or in addition Trisection (Tr ):

Classic constructible polygons are as follows (up to 1000):

Only with the help of at least a three-division (up to 1000):

Corner numbers constructible polygons can also be found in the sequence A003401 in OEIS, corner numbers not classically constructible polygons in the sequence A004169 in OEIS.

Galois theory

Through development of Galois theory we came to a deeper insight into the problem. The amount of constructible numbers namely forms a body in which in addition also of positive numbers, the square root can be drawn. In particular, the cutting of straight line corresponds to solving a linear equation and the cutting of a line and a circle or two circles cutting the solving a quadratic equation. In the language of field extensions is the following fact:

Conversely, naturally, every number of constructible. So is constructible, so is algebraic and it is a power of 2

To clarify the structure of regular corners with considering cyclotomic field over a field extension, where the- th root of unity called. The roots of unity are lying on the unit circle corners of a regular - gon. It is sufficient to construct the real number.

For example, if and are relatively prime, then. Are then the - and - corner constructible, as well as the corner is constructible.

To be able to apply the above arguments, some field extension degrees must be determined. Since the cyclotomic polynomials are irreducible, is. Because is so, and so.

In the regular -gon is the central angle. Thus, if the regular -gon constructible, as well as a range of length. Because even this number is constructible then so must be a power of 2. This is then.

Conversely, it is a finite abelian group of order. After the main theorem on finitely generated abelian groups then exist with a chain of successive normal subgroups. The main theorem of Galois theory then one obtains as a fixed field of a body with a tower, and therefore is, and thus and thus also the regular -gon constructible.

Be instance. Then a power of 2, and, as 2 is a primitive root mod 5. A possible chain of normal subgroups is. The corresponding field tower is. It is because it is normalized and canceled and is irreducible with reduction modulo 2. After solving the equation arises. Now you might already construct the first corner by. The point with distance from the center on an axis from constructed and then the solder falls through this point By dissolving arises. By this algebraic expression can alternatively be the first corner constructed by inscribing a real and an imaginary axis, and by means of which the designed point.