Constructible polygon

In

of sides are known.

Conditions for constructibility

Some regular polygons are easy to construct with compass and straightedge; others are not. The ancient Greek mathematicians knew how to construct a regular polygon with 3, 4, or 5 sides,^{[1]: p. xi} and they knew how to construct a regular polygon with double the number of sides of a given regular polygon.^{[1]: pp. 49–50} This led to the question being posed: is it possible to construct all regular polygons with compass and straightedge? If not, which n-gons (that is, polygons with n edges) are constructible and which are not?

but never published his proof.

A full proof of necessity was given by

Fermat primes

. Here, a power of 2 is a number of the form ${\displaystyle 2^{m}}$, where m ≥ 0 is an integer. A Fermat prime is a prime number of the form ${\displaystyle 2^{(2^{m})}+1}$, where m ≥ 0 is an integer. The number of Fermat primes involved can be 0, in which case n is a power of 2.

In order to reduce a

${\displaystyle \cos(2\pi /n)}$ is a is constructible.

Detailed results by Gauss's theory

Restating the Gauss–Wantzel theorem:

A regular n-gon is constructible with straightedge and compass if and only if n = 2^kp₁p₂...p_t where k and t are non-negative integers, and the p_i's (when t > 0) are distinct Fermat primes.

The five known

are:

F₀ = 3, F₁ = 5, F₂ = 17, F₃ = 257, and F₄ = 65537 (sequence A019434 in the OEIS).

Since there are 31 nonempty subsets of the five known Fermat primes, there are 31 known constructible polygons with an odd number of sides.

The next twenty-eight Fermat numbers, F₅ through F₃₂, are known to be composite.^[3]

Thus a regular n-gon is constructible if

n =

120, 128, 136, 160, 170, 192, 204, 240, 255, 256, 257, 272, 320, 340, 384, 408, 480, 510, 512, 514, 544, 640, 680, 768, 771, 816, 960, 1020, 1024, 1028, 1088, 1280, 1285, 1360, 1536, 1542, 1632, 1920, 2040, 2048, ... (sequence A003401 in the OEIS

),

while a regular n-gon is not constructible with compass and straightedge if

n =

100, 101, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 121, 122, 123, 124, 125, 126, 127, ... (sequence A004169 in the OEIS

).

Connection to Pascal's triangle

Since there are 5 known Fermat primes, we know of 31 numbers that are products of distinct Fermat primes, and hence 31 constructible odd-sided regular polygons. These are 3, 5, 15, 17, 51, 85, 255, 257, 771, 1285, 3855, 4369, 13107, 21845, 65535,

.) This pattern breaks down after this, as the next Fermat number is composite (4294967297 = 641 × 6700417), so the following rows do not correspond to constructible polygons. It is unknown whether any more Fermat primes exist, and it is therefore unknown how many odd-sided constructible regular polygons exist. In general, if there are q Fermat primes, then there are 2^q−1 odd-sided regular constructible polygons.

General theory

In the light of later work on

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 ,

which is a

½ φ(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 a 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

, 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 complex roots, so in principle can be solved by geometric construction: this is 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.

Compass and straightedge constructions

, 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 coprime, there exists integers a and b such that ap + bq = 1. Then 2aπ/q + 2bπ/p = 2π/pq. From this, a pq-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. 150 AD).
• Although Gauss proved that the regular 17-gon is constructible, he did not actually show how to do it. The first construction is due to Erchinger, a few years after Gauss' work.
• The first explicit constructions of a regular 257-gon were given by Magnus Georg Paucker (1822)^[5] and Friedrich Julius Richelot (1832).^[6]
• A construction for a regular 65537-gon was first given by Johann Gustav Hermes (1894). The construction is very complex; Hermes spent 10 years completing the 200-page manuscript.^[7]

Gallery

From left to right, constructions of a 15-gon, 17-gon, 257-gon and 65537-gon. Only the first stage of the 65537-gon construction is shown; the constructions of the 15-gon, 17-gon, and 257-gon are given completely.

Other constructions

The concept of constructibility as discussed in this article applies specifically to

, for example, make use of a marked ruler. The constructions are a mathematical idealization and are assumed to be done exactly.

A regular polygon with n sides can be constructed with ruler, compass, and angle trisector if and only if ${\displaystyle n=2^{r}3^{s}p_{1}p_{2}\cdots p_{k},}$ where r, s, k ≥ 0 and where the p_i are distinct Pierpont primes greater than 3 (primes of the form ${\displaystyle 2^{t}3^{u}+1).}$^{[8]: Thm. 2} These polygons are exactly the regular polygons that can be constructed with Conic section, and the regular polygons that can be constructed with paper folding. The first numbers of sides of these polygons are:

3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 24, 26, 27, 28, 30, 32, 34, 35, 36, 37, 38, 39, 40, 42, 45, 48, 51, 52, 54, 56, 57, 60, 63, 64, 65, 68, 70, 72, 73, 74, 76, 78, 80, 81, 84, 85, 90, 91, 95, 96, 97, 102, 104, 105, 108, 109, 111, 112, 114, 117, 119, 120, 126, 128, 130, 133, 135, 136, 140, 144, 146, 148, 152, 153, 156, 160, 162, 163, 168, 170, 171, 180, 182, 185, 189, 190, 192, 193, 194, 195, 204, 208, 210, 216, 218, 219, 221, 222, 224, 228, 234, 238, 240, 243, 247, 252, 255, 256, 257, 259, 260, 266, 270, 272, 273, 280, 285, 288, 291, 292, 296, ... (sequence A122254 in the OEIS)

See also

• Polygon
• Carlyle circle

References

External links

• Duane W. DeTemple (1991). "Carlyle Circles and the Lemoine Simplicity of Polygonal Constructions".
  MR 1089454
  .
• Christian Gottlieb (1999). "The Simple and Straightforward Construction of the Regular 257-gon".
  S2CID 123567824
  .
• Regular Polygon Formulas, Ask Dr. Math FAQ.
• Carl Schick: Weiche Primzahlen und das 257-Eck : eine analytische Lösung des 257-Ecks. Zürich : C. Schick, 2008.
  ISBN 978-3-9522917-1-9
  .
• 65537-gon, exact construction for the 1st side, using the Quadratrix of Hippias and GeoGebra as additional aids, with brief description (German)