Icositrigon: Difference between revisions

InThe additionregular toicositrigon notis beingnot [[Constructible polygon|constructible]] with a [[Straightedge and compass construction|compass and straightedge]] or [[angle trisection]],<ref>''{{Cite OEIS|A048136|Tomahawk-nonconstructible <math>n</math>-gons'' [[OEIS]]; https://oeis.org/A048136}}</ref> on account of the [[23 (number)|number 23]] being neither a [[Fermat prime|Fermat]] nor [[Pierpont prime#Polygon construction|Pierpont prime]]. In addition, the regular icositrigon is the [[Neusis construction#Use of the neusis|smallest regular polygon that is not constructible even with neusis]], after the discovery of neusis construction of the [[hendecagon]] by Elliot Benjamin and Chip Snyder in 2014.<ref>Benjamin, Elliot; Snyder, C. Mathematical Proceedings of the Cambridge Philosophical Society156.3 (May 2014): 409-424.; https://dx.doi.org/10.1017/S0305004113000753</ref>

If we can construct the regular p-gon, then we can construct <math>\zeta_p = e^\frac{2\pi i}{p}</math>, which is the root of an [[irreducible polynomial]] of degree <math>p - 1</math>. By Theorem 5.1, <math>\zeta_p</math> lies in a field <math>K</math> of degree <math>N</math> over <math>\Q</math>, where the only primes that divide <math>N</math> are 2, 3, and 5. But <math>\Q[\zeta_p]</math> is a subfield of <math>K</math>, so <math>p - 1</math> divides <math>N</math>. In particular, for <math>p = 23</math>, <math>N</math> must be divisible by 11, and for <math>p = 29</math>, ''N'' must be divisible by 7.<ref>Arthur{{cite journal
| last1=Baragar (| first1=Arthur
| date=2002)
| title=Constructions Using a Compass and TwiceNotchedTwice-Notched Straightedge,
| journal=The American Mathematical Monthly,
| volume=109:
| issue=2,
| pages=151-164,
| {{doi|=10.1080/00029890.2002.11919848}}</ref>

Below is a table of fiveten regular icositrigrams, or [[Star polygons|star]] 23-gons, labeled with their respective [[Schläfli symbol]] {23/q}, 2 ≤ q ≤ 11.