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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, ...}
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* |
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