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# Constructible polygons

## 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 ${\displaystyle \scriptstyle n\,}$-gon can be constructed with straightedge and compass if ${\displaystyle \scriptstyle n\,}$ 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, {${\displaystyle \scriptstyle {\rm {F_{0}}}\,}$, ${\displaystyle \scriptstyle {\rm {F_{1}}}\,}$, ${\displaystyle \scriptstyle {\rm {F_{2}}}\,}$, ${\displaystyle \scriptstyle {\rm {F_{3}}}\,}$, ${\displaystyle \scriptstyle {\rm {F_{4}}}\,}$} = {3, 5, 17, 257, 65537}, (Cf. A019434) then there are

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

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

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

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

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

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

If row ${\displaystyle \scriptstyle k,\,0\,\leq \,k\,\leq \,2^{n}-1,\,}$ is

${\displaystyle {\rm {row}}(k)=\prod _{i=0}^{n-1}{{\rm {F}}_{i}}^{\alpha _{i}},\,{\alpha _{i}}\in \{0,\,1\},\,}$

then by the fractal property of Sierpinski's triangle, row ${\displaystyle \scriptstyle 2^{n}+k,\,0\,\leq \,k\,\leq \,2^{n}-1,\,}$ is

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