Editing Constructible polygon (section)

==Other constructions==

The concept of constructibility as discussed in this article applies specifically to [[compass and straightedge]] constructions. More constructions become possible if other tools are allowed. The so-called [[neusis construction]]s, for example, make use of a ''marked'' ruler. The constructions are a mathematical idealization and are assumed to be done exactly.

A regular polygon with ''n'' sides can be constructed with ruler, compass, and [[angle trisection|angle trisector]] if and only if <math>n=2^r3^sp_1p_2\cdots p_k,</math> where ''r, s, k'' ≥ 0 and where the ''p''<sub>''i''</sub> are distinct [[Pierpont prime]]s greater than 3 (primes of the form <math>2^t3^u +1).</math><ref name="Gleason">{{cite journal|last=Gleason|first=Andrew M.|authorlink=Andrew M. Gleason|title=Angle trisection, the heptagon, and the triskaidecagon |journal=[[American Mathematical Monthly]]|date=March 1988|volume=95|issue=3 |pages=185–194 |doi= 10.2307/2323624|jstor=2323624 }}</ref>{{rp|Thm. 2}} These polygons are exactly the regular polygons that can be constructed with [[conic section]]s, and the regular polygons that can be constructed with [[Mathematics of paper folding|paper folding]]. The first numbers of sides of these polygons are:
:3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 24, 26, 27, 28, 30, 32, 34, 35, 36, 37, 38, 39, 40, 42, 45, 48, 51, 52, 54, 56, 57, 60, 63, 64, 65, 68, 70, 72, 73, 74, 76, 78, 80, 81, 84, 85, 90, 91, 95, 96, 97, 102, 104, 105, 108, 109, 111, 112, 114, 117, 119, 120, 126, 128, 130, 133, 135, 136, 140, 144, 146, 148, 152, 153, 156, 160, 162, 163, 168, 170, 171, 180, 182, 185, 189, 190, 192, 193, 194, 195, 204, 208, 210, 216, 218, 219, 221, 222, 224, 228, 234, 238, 240, 243, 247, 252, 255, 256, 257, 259, 260, 266, 270, 272, 273, 280, 285, 288, 291, 292, 296, ... {{OEIS|A122254}}