Icositrigon

In geometry, an icositrigon (or icosikaitrigon) or 23-gon is a 23-sided polygon. The icositrigon has the distinction of being the smallest regular polygon that is not neusis constructible.

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

Regular icositrigon

A regular icositrigon is represented by Schläfli symbol {23}.

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

In addition to not being constructible with a compass and straightedge or angle trisection,[1] on account of the number 23 being neither a Fermat nor Pierpont prime, the regular icositrigon is the 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.[2]

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 of fields over ${\displaystyle \mathbb {Q} }$ such that ${\displaystyle \mathbb {Q} =K_{0}\subset K_{1}\subset \dots \subset K_{n}=K}$, 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-notched straightedge. Then ${\displaystyle \alpha }$ belongs to a field ${\displaystyle K}$ that lies in a tower of fields ${\displaystyle \mathbb {Q} =K_{0}\subset K_{1}\subset \dots \subset K_{n}=K}$ for which the index [${\displaystyle K_{j}}$ : K_j−1] at each step is 2, 3, 5, or 6. In particular, if ${\displaystyle N=}$[${\displaystyle K}$: ${\displaystyle \mathbb {Q} }$], 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 an irreducible polynomial of degree ${\displaystyle p}$ − ${\displaystyle 1}$. 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 ${\displaystyle \mathbb {Q} }$[${\displaystyle \zeta _{p}}$] is a subfield of ${\displaystyle K}$, so ${\displaystyle p}$ − ${\displaystyle 1}$ 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.[3]

It should also be noted that an icositrigon is not origami constructible either, given 23 is not a Pierpont prime, nor a power of two or three.[4] However, it can be made constructible via the use of the Quadratrix of Hippias, Archimedean spiral, and other auxiliary curves; yet this is true for all regular polygons.[5]

Approximate construction

A proximity construction animation

Based on the unit circle r = 1 [unit of length]

• Constructed side length of the icositrigon in GeoGebra (Display max 15 decimal places) ${\displaystyle a=0.272333298192493\;[unit\;of\;length]}$
• Side length of the icositrigon ${\displaystyle a_{target}=2\cdot \sin \left({\frac {180^{\circ }}{23}}\right)=0.272333298192493\ldots \;[unit\;of\;length]}$
• Absolute error of the constructed side length

Up to the max. displayed 15 decimal places is the absolute error${\displaystyle F_{a}=a-a_{target}=0.0\;[unit\;of\;length]}$

• Constructed central angle of the icositrigon in GeoGebra (display significant 13 decimal places, rounded) ${\displaystyle \mu =15.6521739130435^{\circ }}$
• Central angle of the icositrigon ${\displaystyle \mu _{target}={\frac {360^{\circ }}{23}}=15.65217391304347\ldots ^{\circ }}$
• Absolute error of the constructed central angle

Up to the rounded significant 13 decimal places is the absolute error ${\displaystyle F_{\mu }=\mu -\mu _{target}=0^{\circ }}$

Example to illustrate the error

At a circumscribed circle radius r = 1 billion km (the light needed for this distance about 55 minutes), the absolute error of the 1st side would be < 1 mm.

Related figures

Below is a table of five regular icositrigrams, or star 23-gons, labeled with their respective Schläfli symbol {23/q}, 2 ${\displaystyle \leq }$ q ${\displaystyle \leq }$ 11 where q is prime.

{23/2}

{23/3}

{23/5}

{23/7}

{23/11}

References

1. Tomahawk-nonconstructible n-gons OEIS; https://oeis.org/A048136
2. Benjamin, Elliot; Snyder, C. Mathematical Proceedings of the Cambridge Philosophical Society156.3 (May 2014): 409-424.; https://dx.doi.org/10.1017/S0305004113000753
3. Arthur Baragar (2002) Constructions Using a Compass and TwiceNotched Straightedge, The American Mathematical Monthly, 109:2, 151-164, doi:10.1080/00029890.2002.11919848
4. Young Lee, H. (2017) Origami-Constructible Numbers University of Georgia https://getd.libs.uga.edu/pdfs/lee_hwa-young_201712_ma.pdf
5. P. Milici, R. Dawson The equiangular compass December 1st, 2012, The Mathematical Intelligencer, Vol. 34, Issue 4 https://www.researchgate.net/profile/Pietro_Milici2/publication/257393577_The_Equiangular_Compass/links/5d4c687da6fdcc370a8725e0/The-Equiangular-Compass.pdf

External links

• Detection of Interesting Properties in Regular Polygons