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Constructible polygons

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Constructible polygons (with straightedge and compass)

A constructible polygon is a regular polygon that can be constructed with straightedge and compass.

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 -gon can be constructed with straightedge and compass if is the product of a power of 2 and any number of distinct Fermat primes.

Gauss stated without proof that this condition was also necessary, but never published his proof. A full proof of necessity was given by Pierre Wantzel in 1837.

Constructible odd-sided polygons (with straightedge and compass)

Since there are 5 known Fermat primes, {, , , , } = {3, 5, 17, 257, 65537}, (Cf. A019434) then there are

known constructible odd-sided polygons (actually 31 since a polygon has at least 3 sides.)

Divisors of (polygons with an odd number of sides constructible with ruler and compass). (Cf. A004729)

{1, 3, 5, 15, 17, 51, 85, 255, 257, 771, 1285, 3855, 4369, 13107, 21845, 65535, 65537, 196611, 327685, 983055, 1114129, 3342387, 5570645, 16711935, 16843009, 50529027, 84215045, 252645135, 286331153, 858993459, 1431655765, 4294967295}

Odd values of for which a regular -gon can be constructed by compass and straightedge. (Cf. A045544)

{3, 5, 15, 17, 51, 85, 255, 257, 771, 1285, 3855, 4369, 13107, 21845, 65535, 65537, 196611, 327685, 983055, 1114129, 3342387, 5570645, 16711935, 16843009, 50529027, 84215045, 252645135, 286331153, 858993459, 1431655765, 4294967295, ?}

If one interprets the rows of Sierpinski's triangle as a binary number, the first 32 rows give the 32 products of distinct Fermat primes, 1 (empty product) for row 0 and the 31 non-empty products of distinct Fermat primes.

Sierpinski's triangle (Pascal's triangle mod 2) rows converted to decimal. (Cf. A001317)

{1, 3, 5, 15, 17, 51, 85, 255, 257, 771, 1285, 3855, 4369, 13107, 21845, 65535, 65537, 196611, 327685, 983055, 1114129, 3342387, 5570645, 16711935, 16843009, 50529027, 84215045, 252645135, 286331153, 858993459, 1431655765, 4294967295, 4294967297, ...}
Sierpinski's triangle rows converted to base 10
Factorization into Fermat numbers
0 1
1 3
2 5
3 15
4 17
5 51
6 85
7 255
8 257
9 771
10 1285
11 3855
12 4369
13 13107
14 21845
15 65535
16 65537
17 196611
18 327685
19 983055
20 1114129
21 3342387
22 5570645
23 16711935
24 16843009
25 50529027
26 84215045
27 252645135
28 286331153
29 858993459
30 1431655765
31 4294967295
32 4294967297* 4294967297*
* = 4294967297 = 641 * 6700417 is the first composite Fermat number.

Rows give , the th Fermat number.

For ( = 1 to 31) we get all the non-empty products of the 5 known Fermat primes, giving the number of sides of constructible odd-sided polygons. If there are no other Fermat primes, there are then no more constructible (with straightedge and compass) odd-sided polygons.

It can easily be proved by induction that the rows of Sierpinski's triangle interpreted as a binary number are products of distinct Fermat numbers.

Inductive proof:

  • Base case: For rows = 0 and 1, we get 1 (empty product) and 3 respectively, the products of distinct Fermat numbers in
  • Inductive hypothesis: Assume all rows from to are products of distinct Fermat numbers in .

If row is

then by the fractal property of Sierpinski's triangle, row is