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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 \scriptstyle n \,-gon can be constructed with straightedge and compass if \scriptstyle n \, 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, {\scriptstyle {\rm F_0} \,, \scriptstyle {\rm F_1} \,, \scriptstyle {\rm F_2} \,, \scriptstyle {\rm F_3} \,, \scriptstyle {\rm F_4} \,} = {3, 5, 17, 257, 65537}, (Cf. A019434) then there are

\binom{5}{0} + \binom{5}{1} + \binom{5}{2} + \binom{5}{3} + \binom{5}{4} + \binom{5}{5} = 1 + 5 + 10 + 10 + 5 + 1 = 32 = 2^5 \,

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

Divisors of \scriptstyle 2^{32} - 1 \, (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 \scriptstyle n \, for which a regular \scriptstyle n \,-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\scriptstyle (n),\, n \,\ge\, 0 \,)

{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
n \, [{\rm Row}(n)]_{10} \, Factorization into Fermat numbers
0 1
1 3 \scriptstyle 3 \,=\, {\rm F}_0 \,
2 5 \scriptstyle 5 \,=\, {\rm F}_1 \,
3 15 \scriptstyle 3 \,\cdot\, 5 \,=\, {\rm F}_0 {\rm F}_1 \,
4 17 \scriptstyle 17 \,=\, {\rm F}_2 \,
5 51 \scriptstyle 3 \,\cdot\, 17 \,=\, {\rm F}_0 {\rm F}_2 \,
6 85 \scriptstyle 5 \,\cdot\, 17 \,=\, {\rm F}_1 {\rm F}_2 \,
7 255 \scriptstyle 3 \,\cdot\, 5 \,\cdot\, 17 \,=\, {\rm F}_0 {\rm F}_1 {\rm F}_2 \,
8 257 \scriptstyle 257 \,=\, {\rm F}_3 \,
9 771 \scriptstyle 3 \,\cdot\, 257 \,=\, {\rm F}_0 {\rm F}_3 \,
10 1285 \scriptstyle 5 \,\cdot\, 257 \,=\, {\rm F}_1 {\rm F}_3 \,
11 3855 \scriptstyle 3 \,\cdot\, 5 \,\cdot\, 257 \,=\, {\rm F}_0 {\rm F}_1 {\rm F}_3 \,
12 4369 \scriptstyle 17 \,\cdot\, 257 \,=\, {\rm F}_2 {\rm F}_3 \,
13 13107 \scriptstyle 3 \,\cdot\, 17 \,\cdot\, 257 \,=\, {\rm F}_0 {\rm F}_2 {\rm F}_3 \,
14 21845 \scriptstyle 5 \,\cdot\, 17 \,\cdot\, 257 \,=\, {\rm F}_1 {\rm F}_2 {\rm F}_3 \,
15 65535 \scriptstyle 3 \,\cdot\, 5 \,\cdot\, 17 \,\cdot\, 257 \,=\, {\rm F}_0 {\rm F}_1 {\rm F}_2 {\rm F}_3 \,
16 65537 \scriptstyle 65537 \,=\, {\rm F}_4 \,
17 196611 \scriptstyle 3 \,\cdot\, 65537 \,=\, {\rm F}_0 {\rm F}_4 \,
18 327685 \scriptstyle 5 \,\cdot\, 65537 \,=\, {\rm F}_1 {\rm F}_4 \,
19 983055 \scriptstyle 3 \,\cdot\, 5 \,\cdot\, 65537 \,=\, {\rm F}_0 {\rm F}_1 {\rm F}_4 \,
20 1114129 \scriptstyle 17 \,\cdot\, 65537 \,=\, {\rm F}_2 {\rm F}_4 \,
21 3342387 \scriptstyle 3 \,\cdot\, 17 \,\cdot\, 65537 \,=\, {\rm F}_0 {\rm F}_2 {\rm F}_4 \,
22 5570645 \scriptstyle 5 \,\cdot\, 17 \,\cdot\, 65537 \,=\, {\rm F}_1 {\rm F}_2 {\rm F}_4 \,
23 16711935 \scriptstyle 3 \,\cdot\, 5 \,\cdot\, 17 \,\cdot\, 65537 \,=\, {\rm F}_0 {\rm F}_1 {\rm F}_2 {\rm F}_4 \,
24 16843009 \scriptstyle 257 \,\cdot\, 65537 \,=\, {\rm F}_3 {\rm F}_4 \,
25 50529027 \scriptstyle 3 \,\cdot\, 257 \,\cdot\, 65537 \,=\, {\rm F}_0 {\rm F}_3 {\rm F}_4 \,
26 84215045 \scriptstyle 5 \,\cdot\, 257 \,\cdot\, 65537 \,=\, {\rm F}_1 {\rm F}_3 {\rm F}_4 \,
27 252645135 \scriptstyle 3 \,\cdot\, 5 \,\cdot\, 257 \,\cdot\, 65537 \,=\, {\rm F}_0 {\rm F}_1 {\rm F}_3 {\rm F}_4 \,
28 286331153 \scriptstyle 17 \,\cdot\, 257 \,\cdot\, 65537 \,=\, {\rm F}_2 {\rm F}_3 {\rm F}_4 \,
29 858993459 \scriptstyle 3 \,\cdot\, 17 \,\cdot\, 257 \,\cdot\, 65537 \,=\, {\rm F}_0 {\rm F}_2 {\rm F}_3 {\rm F}_4 \,
30 1431655765 \scriptstyle 5 \,\cdot\, 17 \,\cdot\, 257 \,\cdot\, 65537 \,=\, {\rm F}_1 {\rm F}_2 {\rm F}_3 {\rm F}_4 \,
31 4294967295 \scriptstyle 3 \,\cdot\, 5 \,\cdot\, 17 \,\cdot\, 257 \,\cdot\, 65537 \,=\, {\rm F}_0 {\rm F}_1 {\rm F}_2 {\rm F}_3 {\rm F}_4 \,
32 4294967297* 4294967297* \scriptstyle \,=\, {\rm F}_5 \,
* \scriptstyle {\rm F}_5 \, = 4294967297 = 641 * 6700417 is the first composite Fermat number.

Rows \scriptstyle 2^n,\, n \,\ge\, 0, \, give \scriptstyle F_{n} \,, the \scriptstyle n \,th Fermat number.

For (\scriptstyle n \, = 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 \scriptstyle n \, = 0 and 1, we get 1 (empty product) and 3 respectively, the products of distinct Fermat numbers in \scriptstyle \{ F_0 \} \,=\, \{ 3 \}; \,
  • Inductive hypothesis: Assume all rows from \scriptstyle n \,=\, 0 \, to \scriptstyle 2^n - 1 \, are products of distinct Fermat numbers in \scriptstyle \{ {\rm F}_0, \ldots, {\rm F}_{n-1} \}; \,.

If row \scriptstyle k,\, 0 \,\le\, k \,\le\, 2^n - 1, \, is

{\rm row}(k) = \prod_{i=0}^{n-1} {{\rm F}_i}^{\alpha_i},\, {\alpha_i} \in \{ 0,\, 1 \}, \,

then by the fractal property of Sierpinski's triangle, row \scriptstyle 2^n + k,\, 0 \,\le\, k \,\le\, 2^n - 1, \, is

{\rm row}(2^n + k) = {\rm row}(k) ~ {\rm F}_n. \,
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