constructible polygons

The Gauss-Wantzel theorem states that a regular polygon with $n$

sides can be constructed with ruler and compass if and only if $n$

can be written as $2^u\cdot p_1\cdot p_2\cdots p_m$ , where $u,m\ge0$ and the $p_i$

's are distinct Fermat primes, that is primes of the form $F_k = 2^{2^k}+1$ .

Note that only 5 Fermat primes are known, namely, $F_0=3$ , $F_1=5$ , $F_2 = 17$ , $F_3 = 257$ , and $F_4 = 65537$ .

A.M.Gleason has proved that if the use of the angle-trisector is allowed, then the constructible polygons have sides of the form $2^s\cdot3^t\cdot p_1\cdot p_2\cdots p_m$ , where the $p_i$ 's are distinct Pierpont primes.

The first $n$ such that a regular polygon with $n$ sides can be constructed with ruler and compass are 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 more terms

Pictorial representation of remainders (mod 2, 3, ...,11) frequency. For a table of values and more details click here

Useful links Wikipedia, Constructible polygon
Mathworld, Trigonometry Angles
OEIS, Sequence A003401

Constructible polygons can also be... (you may click on names or numbers and on + to get more values)

ABA 24 32 64 + 8796093022208 aban 10 12 15 + 960 abundant 12 20 24 + 44738560 admirable 12 20 24 + 393222 alternating 10 12 16 + 69632 amenable 12 16 17 + 858980352 apocalyptic 192 384 1088 + 26214 arithmetic 15 17 20 + 6684774 astonishing 15 204 balanced p. 257 Bell 15 betrothed 48 binomial 10 15 20 + 2147516416 brilliant 10 15 c.nonagonal 10 136 32896 2147516416 c.pentagonal 16 51 2176 c.square 85 c.triangular 10 64 85 + 514 cake 15 64 2048 82240 canyon 102 204 408 + 98304 Chen 17 257 65537 compositorial 24 192 congruent 15 20 24 + 8912896 cube 64 512 4096 + 281474976710656 Cunningham 10 15 17 + 17180131328 Curzon 30 510 8738 + 6684774 cyclic 15 17 51 + 5570645 D-number 15 51 771 196611 d-powerful 24 1542 2048 + 7864320 de Polignac 16843009 decagonal 10 85 deficient 10 15 16 + 8913032 dig.balanced 10 12 15 + 178257920 double fact. 15 48 384 3840 droll 240 52224 174080 Duffinian 16 32 64 + 8388608 eban 30 32 34 + 2056 economical 10 15 16 + 17895424 emirp 17 emirpimes 15 51 85 + 1114129 enlightened 256 2048 2176 + 274877906944 equidigital 10 15 16 + 17895424 eRAP 20 24 170 esthetic 10 12 32 34 Eulerian 120 evil 10 12 15 + 858993459 factorial 24 120 Fibonacci 34 Friedman 128 1024 1285 + 983040 frugal 128 256 512 + 855638016 gapful 120 160 170 + 86236206080 Giuga 30 good prime 17 257 65537 happy 10 32 68 + 8912896 Harshad 10 12 20 + 8589803520 heptagonal 34 hexagonal 15 120 32640 2147450880 highly composite 12 24 48 + 240 hoax 85 136 160 + 53478192 house 32 1285 hungry 17 iban 10 12 17 + 41120 iccanobiF 514 idoneal 10 12 15 + 408 impolite 16 32 64 + 562949953421312 inconsummate 272 771 816 + 983055 interprime 12 15 30 + 63160320 Jacobsthal 85 21845 1431655765 Jordan-Polya 12 16 24 + 844424930131968 junction 204 408 816 + 84215045 katadrome 10 20 30 + 960 Lehmer 15 51 85 + 4294967295 Leyland 17 32 320 + 281479271677952 lonely 120 lucky 15 51 1285 983055 Lynch-Bell 12 15 24 + 3264 magic 15 34 2056 8388736 magnanimous 12 16 20 + 512 metadrome 12 15 16 + 257 Motzkin 51 mountain 120 160 170 + 7864320 nialpdrome 10 20 30 + 7710 nonagonal 24 204 nude 12 15 24 + 26843136 O'Halloran 12 20 60 204 oban 10 12 15 + 960 octagonal 40 96 408 odious 16 32 64 + 536870912 palindromic 272 pancake 16 4096 panconsummate 10 12 15 + 257 pandigital 15 120 20560 30720 partition 15 30 pentagonal 12 51 pernicious 10 12 17 + 8912896 Perrin 10 12 17 + 68 Pierpont 17 257 65537 plaindrome 12 15 16 + 12336 Poulet 4369 16843009 power 16 32 64 + 35184372088832 powerful 16 32 64 + 562949953421312 practical 12 16 20 + 8947712 prim.abundant 12 20 30 + 393222 prime 17 257 65537 primorial 30 productive 12 16 136 256 pronic 12 20 30 + 4295032832 Proth 17 257 65537 pseudoperfect 12 20 24 + 4294901760 repunit 15 40 85 + 4294967295 Ruth-Aaron 15 16 24 Saint-Exupery 60 480 2040 + 281470681743360 self 20 64 255 + 855651072 semiprime 10 15 34 + 16843009 sliding 20 Smith 85 4369 32896 + 786444 sphenic 30 102 170 + 84215045 square 16 64 256 + 281474976710656 strobogrammatic 96 strong prime 17 65537 super Niven 10 12 20 + 4080 super-d 3084 3855 4369 + 8355840 superabundant 12 24 48 + 240 tau 12 24 40 + 858980352 tetrahedral 10 20 120 + 46912496107520 tetranacci 15 triangular 10 15 120 + 2147516416 tribonacci 24 trimorphic 24 51 truncatable prime 17 twin 17 65537 uban 10 12 15 + 96 Ulam 16 48 102 + 8421376 undulating 272 unprimeable 204 320 510 + 8947712 untouchable 96 120 408 + 986880 upside-down 64 8192 wasteful 12 20 24 + 8947712 Woodall 17 80 Zuckerman 12 15 24 + 2228224 Zumkeller 12 20 24 + 98688