Regular icositrigon
A regular icositrigon
Type	Regular polygon
Edges andvertices	23
Schläfli symbol	{23}
Coxeter–Dynkin diagrams
Symmetry group	Dihedral (D23), order 2×23
Internal angle (degrees)	≈164.348°
Properties	Convex,cyclic,equilateral,isogonal,isotoxal
Dual polygon	Self

Ingeometry, anicositrigon (oricosikaitrigon) or 23-gon is a 23-sidedpolygon. The icositrigon has the distinction of being the smallest regular polygon that is notneusis constructible.

Aregular icositrigon is represented bySchläfli symbol {23}.

A regular icositrigon hasinternal angles of$\frac{3780}{23}$ degrees, with an area of$A=\frac{23}{4}{a}^2\cot \frac{\pi }{23}=23r^2\tan \frac{\pi }{23}\simeq 41.8344a^2,$ where${\displaystyle a}$ is side length and${\displaystyle r}$ is the inradius, orapothem.

The regular icositrigon is notconstructible with acompass and straightedge orangle trisection,^[1] on account of thenumber 23 being neither aFermat norPierpont prime. In addition, the regular icositrigon is thesmallest regular polygon that is not constructible even with neusis.

Concerning the nonconstructability of the regular icositrigon, A. Baragar (2002) showed it is not possible to construct a regular 23-gon using only a compass and twice-notched straightedge by demonstrating that every point constructible with said method lies in a tower offields over${\displaystyle \mathbb{Q}}$ such that${\displaystyle \mathbb{Q}=K_0\subset K_1\subset \cdots \subset K_n=K}$, being a sequence of nested fields in which the degree of the extension at each step is 2, 3, 5, or 6.

Suppose${\displaystyle \alpha }$ in${\displaystyle \mathbb{C}}$ is constructible using a compass and twice-notchedstraightedge. Then${\displaystyle \alpha }$ belongs to a field${\displaystyle K}$ that lies in a tower of fields${\displaystyle \mathbb{Q}=K_0\subset K_1\subset \cdots \subset K_n=K}$for which the index${\displaystyle [K_j:K_{j-1}]}$ at each step is 2, 3, 5, or 6. In particular, if${\displaystyle N=[K:\mathbb{Q}]}$, then the only primes dividing${\displaystyle N}$ are 2, 3, and 5. (Theorem 5.1)

If we can construct the regular p-gon, then we can construct${\displaystyle {\zeta }_p=e^{\frac{2\pi i}{p}}}$, which is the root of anirreducible polynomial of degree${\displaystyle p-1}$. By Theorem 5.1,${\displaystyle {\zeta }_p}$ lies in a field${\displaystyle K}$ of degree${\displaystyle N}$ over${\displaystyle \mathbb{Q}}$, where the only primes that divide${\displaystyle N}$ are 2, 3, and 5. But${\displaystyle \mathbb{Q}[{\zeta }_p]}$ is a subfield of${\displaystyle K}$, so${\displaystyle p-1}$ divides${\displaystyle N}$. In particular, for${\displaystyle p=23}$,${\displaystyle N}$ must be divisible by 11, and for${\displaystyle p=29}$,N must be divisible by 7.^[2]

This result establishes, considering prime-power regular polygons less than the 100-gon, that it is impossible to construct the 23-, 29-, 43-, 47-, 49-, 53-, 59-, 67-, 71-, 79-, 83-, and 89-gons with neusis. But it is not strong enough to decide the cases of the11-, 25-, 31-, 41-, and 61-gons. Elliot Benjamin and Chip Snyder discovered in 2014 that the regular hendecagon (11-gon) is neusis constructible; the remaining cases are still open.^[3]

An icositrigon is notorigami constructible either, because 23 is not a Pierpont prime, nor apower of two orthree.^[4] It can be constructed using thequadratrix of Hippias,Archimedean spiral, and otherauxiliary curves; yet this is true for all regular polygons.^[5]

Below is a table of ten regular icositrigrams, orstar 23-gons, labeled with their respectiveSchläfli symbol {23/q}, 2 ≤ q ≤ 11.

{23/2}	{23/3}	{23/4}	{23/5}	{23/6}
{23/7}	{23/8}	{23/9}	{23/10}	{23/11}

• Kovács, Zoltán (2020)."Automated Detection of Interesting Properties in Regular Polygons".Mathematics in Computer Science.14:727–755.doi:10.1007/s11786-020-00491-z.

• Polygons by the number of sides

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