Constructible polygon

Construction of a regular pentagon

In mathematics is a constructible polygon is a regular polygon , which with compass and (unmarked) straightedge - - Euclidean tools designed can be. For example, the regular pentagon can be constructed, but the regular heptagon cannot.

Constructibility

Example of use: height theorem
By adding = 1 to the constructed number , it can be constructed using a Thales circle . Numerical example: ${\displaystyle a>0}$ ${\displaystyle |BC|}$ ${\displaystyle {\sqrt {a}}}$
${\displaystyle a=9;}$ ${\displaystyle \;1={\frac {a}{9}};}$ ${\displaystyle \;{\sqrt {a}}={\sqrt {9}}}$

In order to grasp the term “can be constructed with compass and ruler” mathematically precisely, it must be defined what is possible with these tools. We assume that two points are given at the beginning of every construction. With the ruler you can then construct a straight line through two points, with the compass a circle through a point around another point as the center point. In addition, the intersections of straight lines and circles can be constructed.

A number of other constructions can be derived from these basic constructions, such as the construction of a vertical center line or the felling of a perpendicular. A positive real number is called constructible if you can construct two points such that the Euclidean distance between them is equal to the absolute value of this number (whereby the distance between two given points is defined as 1). If, for example, the number can be constructed, you can use the height theorem to construct two points with a distance . If two numbers and can be constructed, then with the help of the theorem of rays their product and the reciprocal value , as well as their sum and difference by tapping a distance . → See also the article Construction with compasses and ruler for the algebraic operations . An angle is called constructible if the number is constructible; the meaning of this definition is quickly revealed by looking at the unit circle . ${\displaystyle a>0}$ ${\displaystyle {\sqrt {a}}}$ ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle ab}$ ${\displaystyle {\frac {1}{a}}}$ ${\displaystyle a+b}$ ${\displaystyle a-b}$ ${\displaystyle \alpha }$ ${\displaystyle \cos(\alpha )}$

In order to construct a regular corner, it is sufficient to construct the central angle , because once you have given the center of the corner and a corner, the next corner can be constructed based on the straight line connecting the center and corner. Conversely, if there is a regular corner, the central angle can be tapped. To answer the question whether the corner can be constructed, one has to go back to the case of deciding whether the central angle can be constructed. ${\displaystyle n}$ ${\displaystyle {\frac {2\pi }{n}}}$ ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle n}$

Constructibility of numbers

A number is called constructible with a compass and ruler if it is z. B. is a whole number , a decimal number with a finite number of decimal places or the positive square root of one of these numbers (see the example of the height theorem ), more precisely the length of a line that can be constructed as described here.

In synthetic geometry , points and numbers are also examined, which can be constructed in a more general way from an (almost) arbitrary default set of route lengths. For this purpose, body extensions of the rational numbers are considered, which are Euclidean bodies and thus coordinate bodies of a Euclidean plane (in the sense of synthetic geometry). The constructability of a number with a compass and ruler then means that it is a coordinate of a point in the plane that can be constructed from the specifications. → See also the article Euclidean solids on these concepts !

Criterion for constructibility

Carl Friedrich Gauß showed in 1796 that the regular seventeenth corner can be constructed. He also demonstrated that the number can be represented as an expression that only contains whole numbers, basic arithmetic operations and nested square roots. Through the theory developed in his Disquisitiones Arithmeticae , Gauß succeeded five years later in specifying a sufficient condition for the construction of regular polygons: ${\displaystyle \cos {\frac {2\pi }{17}}}$

If the product of a power of 2 with pairwise different Fermat prime numbers is, then the regular corner can be constructed. ${\displaystyle n}$ ${\displaystyle n}$

Although Gauss knew that the condition was also necessary, he did not publish his proof of this. Pierre-Laurent Wantzel made up for this in 1837.

One can show that a number is the product of a power of 2 with different Fermat prime numbers if and only if is a power of 2. Here denotes the Euler φ function . ${\displaystyle n\geq 2}$ ${\displaystyle \varphi (n)}$ ${\displaystyle \varphi }$

In summary: For a number the following statements are equivalent: ${\displaystyle n\geq 3}$

• The regular corner can be constructed with a compass and ruler. ${\displaystyle n}$
• ${\displaystyle n=2^{k}p_{1}\cdots p_{m}}$with and pairwise different Fermat prime numbers . ${\displaystyle k\in \mathbb {N} _{0}}$ ${\displaystyle m\in \mathbb {N} _{0}}$ ${\displaystyle p_{1},\dots ,p_{m}}$

The empty product resulting for m = 0 stands by definition for the number 1.

• ${\displaystyle \varphi (n)=2^{r}}$for a . ${\displaystyle r\in \mathbb {N} }$

If in particular and coprime and both the corner and the corner can be constructed, then the corner can also be constructed. For this fact, the geometric construction can also be given directly, because if and are coprime, there are two integers according to Bézout's lemma and by applying -time the central angle of the -gon and -time the central angle of the -gon , you have constructed the angle - and thus also the corner. ${\displaystyle m}$ ${\displaystyle n}$ ${\displaystyle m}$ ${\displaystyle n}$ ${\displaystyle \varphi (mn)=\varphi (m)\varphi (n)}$ ${\displaystyle mn}$ ${\displaystyle m}$ ${\displaystyle n}$ ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle 1=am+bn.}$ ${\displaystyle a}$ ${\displaystyle n}$ ${\displaystyle b}$ ${\displaystyle m}$ ${\displaystyle a{\frac {2\pi }{n}}+b{\frac {2\pi }{m}}={\frac {2\pi }{mn}}}$ ${\displaystyle mn}$

Concrete consequences of the criterion

Despite an intensive search, no more have been found beyond the five Fermat prime numbers 3, 5, 17, 257 and 65537 already known to Gauss. There is even a plausible assumption that there are no other Fermat prime numbers.

→ Main article : Fermat number

If there are actually only five Fermat prime numbers, then exactly the following 31 can be theoretically constructed among the polygons with an odd (!) Vertex number:

Number of corners Product Fermatscher prime numbers 3 ${\displaystyle 3}$ 5 ${\displaystyle 5}$ 15th ${\displaystyle 3\cdot 5}$ 17th ${\displaystyle 17}$ 51 ${\displaystyle 3\cdot 17}$ 85 ${\displaystyle 5\cdot 17}$ 255 ${\displaystyle 3\cdot 5\cdot 17}$

Number of corners Product Fermatscher prime numbers 257 ${\displaystyle 257}$ 771 ${\displaystyle 3\cdot 257}$ 1,285 ${\displaystyle 5\cdot 257}$ 3,855 ${\displaystyle 3\cdot 5\cdot 257}$ 4,369 ${\displaystyle 17\cdot 257}$ 13,107 ${\displaystyle 3\cdot 17\cdot 257}$ 21,845 ${\displaystyle 5\cdot 17\cdot 257}$ 65,535 ${\displaystyle 3\cdot 5\cdot 17\cdot 257}$

Number of corners Product Fermatscher prime numbers 65,537 ${\displaystyle 65537}$ 196.701 ${\displaystyle 3\cdot 65537}$ 327.835 ${\displaystyle 5\cdot 65537}$ 983.055 ${\displaystyle 3\cdot 5\cdot 65537}$ 1,114,639 ${\displaystyle 17\cdot 65537}$ 3,342,387 ${\displaystyle 3\cdot 17\cdot 65537}$ 5,570,645 ${\displaystyle 5\cdot 17\cdot 65537}$ 16,711,935 ${\displaystyle 3\cdot 5\cdot 17\cdot 65537}$

Number of corners Product Fermatscher prime numbers 16,850,719 ${\displaystyle 257\cdot 65537}$ 50,529,027 ${\displaystyle 3\cdot 257\cdot 65537}$ 84.215.045 ${\displaystyle 5\cdot 257\cdot 65537}$ 252.645.135 ${\displaystyle 3\cdot 5\cdot 257\cdot 65537}$ 286.331.153 ${\displaystyle 17\cdot 257\cdot 65537}$ 858.993.459 ${\displaystyle 3\cdot 17\cdot 257\cdot 65537}$ 1,431,655,765 ${\displaystyle 5\cdot 17\cdot 257\cdot 65537}$ 4,294,967,295 ${\displaystyle 3\cdot 5\cdot 17\cdot 257\cdot 65537}$

All other constructible polygons (then with an even number of corners) are the square or they result from (continued) doubling the number of corners.

Construction instructions are known for the triangle, pentagon, 17-sided and 257-sided, an allegedly existing construction instruction for the 65537-sided is - if it exists - neither accessible nor verified. This means that construction instructions are only available for the odd polygons up to the 65535 corner.

If you also allow an aid to divide an angle into three (trisection) , all regular polygons can be constructed with corner numbers of the shape , with different Pierpont prime numbers being greater than three of the shape . In this way, for example, the heptagon , the hexagon and the triangle can be constructed. ${\displaystyle n=2^{r}3^{s}p_{1}\cdots p_{k}}$ ${\displaystyle p_{1},\ldots ,p_{k}}$ ${\displaystyle k\in \mathbb {N} _{0}}$ ${\displaystyle 2^{t}3^{u}+1}$

Are as additional aids such. B. Hippias' quadratrix or the Archimedean spiral are accepted, which in addition to the three-part division also enable divisions with equal angles, as the example of the nineteenth- corner shows, theoretically all regular polygons can be constructed. ${\displaystyle n}$

It follows from this that if you only allow the three-part division of an angle (trisection) as an additional aid, the following table results for regular polygons up to 100 gons for the construction with compass and ruler (yes), or additionally trisection (Tr):

Number of corners	3	4th	5	6th	7th	8th	9	10	11	12	13	14th	15th	16
Constructible	Yes	Yes	Yes	Yes	Tr.	Yes	Tr.	Yes	No	Yes	Tr.	Tr.	Yes	Yes
Number of corners	17th	18th	19th	20th	21st	22nd	23	24	25th	26th	27	28	29	30th
Constructible	Yes	Tr.	Tr.	Yes	Tr.	No	No	Yes	No	Tr.	Tr.	Tr.	No	Yes
Number of corners	31	32	33	34	35	36	37	38	39	40	41	42	43	44
Constructible	No	Yes	No	Yes	Tr.	Tr.	Tr.	Tr.	Tr.	Yes	No	Tr.	No	No
Number of corners	45	46	47	48	49	50	51	52	53	54	55	56	57	58
Constructible	Tr.	No	No	Yes	No	No	Yes	Tr.	No	Tr.	No	Tr.	Tr.	No
Number of corners	59	60	61	62	63	64	65	66	67	68	69	70	71	72
Constructible	No	Yes	No	No	Tr.	Yes	Tr.	No	No	Yes	No	Tr.	No	Tr.
Number of corners	73	74	75	76	77	78	79	80	81	82	83	84	85	86
Constructible	Tr.	Tr.	No	Tr.	No	Tr.	No	Yes	Tr.	No	No	Tr.	Yes	No
Number of corners	87	88	89	90	91	92	93	94	95	96	97	98	99	100
Constructible	No	No	No	Tr.	Tr.	No	No	No	Tr.	Yes	Tr.	No	No	No

The following polygons (up to 1000) can be constructed in the classic way:

3 , 4 , 5 , 6 , 8 , 10 , 12 , 15 , 16 , 17 , 20 , 24, 30 , 32, 34, 40 , 48, 51 , 60, 64, 68, 80, 85, 96, 102, 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

Only with the help of at least three division (up to 1000) sequence A051913 in OEIS :

7 , 9 , 13 , 14 , 18 , 19 , 21 , 26, 27, 28, 35, 36, 37, 38, 39, 42, 45, 52, 54, 56, 57, 63, 65, 70, 72, 73, 74, 76, 78, 81, 84, 90, 91, 95, 97, 104, 105, 108, 109, 111, 112, 114, 117, 119, 126, 130, 133, 135, 140, 144, 146, 148, 152, 153, 156, 162, 163, 168, 171, 180, 182, 185, 189, 190, 193, 194, 195, 208, 210, 216, 218, 219, 221, 222, 224, 228, 234, 238, 243, 247, 252, 259, 260, 266, 270, 273, 280, 285, 288, 291, 292, 296, 304, 306, 312, 315, 323, 324, 326, 327, 333, 336, 342, 351, 357, 360, 364, 365, 370, 378, 380, 386, 388, 390, 399, 405, 416, 420, 432, 433, 436, 438, 442, 444, 448, 456, 459, 468, 476, 481, 485, 486, 487, 489, 494, 504, 511, 513, 518, 520, 532, 540, 545, 546, 555, 560, 567, 570, 576, 577, 579, 582, 584, 585, 592, 608, 612, 624, 629, 630, 646, 648, 652, 654, 657, 663, 666, 672, 679, 684, 702, 703, 714, 720, 728, 729, 730, 740, 741, 756, 760, 763, 765, 769, 772, 776, 777, 780, 798, 810, 815, 819, 832, 840, 855, 864, 866, 872, 873, 876, 884, 888, 896, 912, 918, 936, 945, 949, 952, 962, 965, 969, 970, 972, 974, 978, 981, 988, 999

Corner numbers of constructible polygons can also be found in sequence A003401 in OEIS , corner numbers of non-classically constructible polygons in sequence A004169 in OEIS .

Galois theory

The development of Galois theory led to a deeper insight into the problem. The set of constructible numbers forms a field in which the square root can also be drawn from positive numbers. In particular, intersecting straight lines is equivalent to solving a linear equation, and intersecting a straight line with a circle or intersecting two circles is equivalent to solving a quadratic equation. In the language of body extensions, the following fact is:

Is a constructible number, there is a body tower such that and for a . ${\displaystyle a}$ ${\displaystyle \mathbb {Q} \subsetneq M_{0}\subsetneq M_{1}\subsetneq \dots \subsetneq M_{m}}$ ${\displaystyle a\in M_{m}}$ ${\displaystyle M_{i+1}=M_{i}[{\sqrt {\gamma _{i}}}]}$ ${\displaystyle \gamma _{i}\in M_{i}}$

Conversely, every number can of course also be constructed from . So if it is constructible, then it is algebraic and it is a power of 2. ${\displaystyle M_{m}}$ ${\displaystyle a}$ ${\displaystyle a}$ ${\displaystyle [\mathbb {Q} [a]:\mathbb {Q} ]=2^{m}}$

To clarify the construction of regular - corners with one considers a circle dividing body as a body extension over , where the -th denotes the root of the unit . The -th roots of unit are the corners of a regular corner lying on the unit circle . It suffices to construct the real number . ${\displaystyle n}$ ${\displaystyle n\geq 3}$ ${\displaystyle \mathbb {Q} [\zeta _{n}]}$ ${\displaystyle \mathbb {Q} }$ ${\displaystyle \zeta _{n}:=\exp {\frac {2\pi \mathrm {i} }{n}}}$ ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle \alpha _{n}:=\zeta _{n}+\zeta _{n}^{-1}\in \mathbb {R} }$

Are for example and coprime, so is . If the - and the - corner can be constructed, then the - corner can also be constructed. ${\displaystyle m}$ ${\displaystyle n}$ ${\displaystyle \mathbb {Q} [\zeta _{mn}]=\mathbb {Q} [\zeta _{m},\zeta _{n}]}$ ${\displaystyle m}$ ${\displaystyle n}$ ${\displaystyle mn}$

In order to be able to apply the above arguments, some degrees of body extension must be determined. Since the circular division polynomials are irreducible, is . Because is , therefore is , and with it . ${\displaystyle [\mathbb {Q} [\zeta _{n}]:\mathbb {Q} ]=\varphi (n)}$ ${\displaystyle \zeta _{n}\not \in \mathbb {R} }$ ${\displaystyle [\mathbb {Q} [\zeta _{n}]:\mathbb {Q} [\alpha _{n}]]>1}$ ${\displaystyle \operatorname {MinPol} _{\mathbb {Q} [\alpha _{n}]}(\zeta _{n})=X^{2}-\alpha _{n}X+1}$ ${\displaystyle [\mathbb {Q} [\zeta _{n}]:\mathbb {Q} [\alpha _{n}]]=2}$

The central angle is in the regular corner . If the regular corner can be constructed, so can a segment of length . Because of this, this number can also be constructed, so it must be a power of 2. With that then . ${\displaystyle n}$ ${\displaystyle \beta :={\frac {2\pi }{n}}}$ ${\displaystyle n}$ ${\displaystyle |{\cos \beta }|=\cos {\frac {2\pi }{n}}}$ ${\displaystyle \alpha _{n}=2\cos {\frac {2\pi }{n}}}$ ${\displaystyle [\mathbb {Q} [\alpha _{n}]:\mathbb {Q} ]=2^{m}}$ ${\displaystyle \varphi (n)=[\mathbb {Q} [\zeta _{n}]:\mathbb {Q} ]=[\mathbb {Q} [\zeta _{n}]:\mathbb {Q} [\alpha _{n}]]\cdot [\mathbb {Q} [\alpha _{n}]:\mathbb {Q} ]=2^{m+1}}$

If the reverse is true , then there is a finite Abelian group of order . According to the main theorem about finitely generated Abelian groups, there then exists a chain of successive normal subdivisions with . With the main theorem of Galois theory one then obtains from this as a fixed body from a body tower with , therefore is for , and thus and thus also the regular corner can be constructed. ${\displaystyle \varphi (n)=2^{m}}$ ${\displaystyle \operatorname {Gal} (\mathbb {Q} [\zeta _{n}]:\mathbb {Q} )\cong \left(\mathbb {Z} /n\mathbb {Z} \right)^{\times }}$ ${\displaystyle 2^{m}}$ ${\displaystyle \left(\mathbb {Z} /n\mathbb {Z} \right)^{\times }=H_{0}\triangleright H_{1}\triangleright \dots \triangleright H_{m}=\{1\}}$ ${\displaystyle H_{i}}$ ${\displaystyle |H_{i}|=2^{m-i}}$ ${\displaystyle \mathbb {Q} [\zeta _{n}]}$ ${\displaystyle \mathbb {Q} =M_{0}\subseteq M_{1}\subseteq \dots \subseteq M_{m}=\mathbb {Q} [\zeta _{n}]}$ ${\displaystyle [M_{i+1}:M_{i}]=|H_{i}/H_{i+1}|=2}$ ${\displaystyle M_{i+1}=M_{i}[{\sqrt {\gamma _{i}}}]}$ ${\displaystyle \gamma _{i}\in M_{i}}$ ${\displaystyle \zeta _{n}}$ ${\displaystyle n}$

Be for example . Then is a power of 2 and , since 2 is a primitive root modulo 5. One possible chain of normal parts is . The associated body tower is . It is because it is normalized and canceled and is irreducible with reduction modulo 2. After solving the equation we get . Now you could already construct the first corner by constructing the point at a distance from the center point on an axis and then the perpendicular falls through this point. Solving the results in . With this algebraic expression, the first corner can alternatively be constructed by drawing in a real and an imaginary axis and using them to construct the point . ${\displaystyle n=5}$ ${\displaystyle \varphi (n)=2^{2}}$ ${\displaystyle \operatorname {Gal} (\mathbb {Q} [\zeta _{5}]:\mathbb {Q} )\cong \left(\mathbb {Z} /5\mathbb {Z} \right)^{\times }=\langle 2\rangle }$ ${\displaystyle H_{0}:=\langle 2\rangle \triangleright H_{1}:=\langle 4\rangle \triangleright H_{2}:=\langle 1\rangle }$ ${\displaystyle M_{0}:=\mathbb {Q} \subseteq M_{1}:=\mathbb {Q} [\zeta _{5}+\zeta _{5}^{-1}]\subseteq M_{2}:=\mathbb {Q} [\zeta _{5}]}$ ${\displaystyle \operatorname {MinPol} _{\mathbb {Q} }(\alpha _{5})=X^{2}+X-1}$ ${\displaystyle \alpha _{5}}$ ${\displaystyle x^{2}+x-1=0}$ ${\displaystyle \alpha _{5}=-{\frac {1}{2}}+{\frac {1}{2}}{\sqrt {5}}}$ ${\displaystyle \alpha _{5}}$ ${\displaystyle x^{2}-\alpha _{5}x+1=0}$ ${\displaystyle \zeta _{5}=-{\frac {1}{4}}\left(-1+{\sqrt {5}}+\mathrm {i} {\sqrt {10+2{\sqrt {5}}}}\right)}$ ${\displaystyle \zeta _{5}}$

Individual evidence

1. ↑ Edmund Weitz : The regular 17-sided. In: YouTube. 2017, accessed August 27, 2020 . 
2. ↑ Follow A045544 in OEIS
3. ↑ Andrew Gleason : Angle Trisection, the Heptagon, and the Triskaidecagon . In: The American Mathematical Monthly . tape 95 , no. 3 , 1988, pp. 185–194 ( page 186, Fig. 1. Construction of a regular heptagon (heptagon). PDF and page 193, Fig. 4. Construction of a regular triskaidecagon (three-corner). PDF ( Memento from December 19, 2015 in the Internet Archive ) ). Angle Trisection, the Heptagon, and the Triskaidecagon ( Memento of the original dated February 2, 2016 in the Internet Archive ) Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice. Original regenerated from the archive on January 31, 2016  

See also

• 257 corner
• 65537-corner
• 4294967295-corner

Web links

Wikiversity: A Lecture on Constructible Polygons  - Course Materials

Wikibooks: Scheme for the approximate construction of regular polygons  - learning and teaching materials

• Construction of a decimal number
• Eric W. Weisstein : Constructible Polygon . In: MathWorld (English).
• Triskaidecagon, Dreizehneck (Wikipedia)