A
regular icositrigon is represented by
Schläfli symbol {23}. A regular icositrigon has
internal 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 41.8344\,a^2, where a is side length and r is the inradius, or
apothem. The regular icositrigon is not
constructible with a
compass and straightedge or
angle trisection, on account of the
number 23 being neither a
Fermat nor
Pierpont prime. In addition, the regular icositrigon is the
smallest 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 of
fields over \Q such that \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 \alpha in \Complex is constructible using a compass and twice-notched straightedge. Then \alpha belongs to a field K that lies in a tower of fields \Q = K_0 \subset K_1 \subset \dots \subset K_n = K for which the index [K_j: K_{j - 1}] at each step is 2, 3, 5, or 6. In particular, if N = [K : \Q], then the only primes dividing N are 2, 3, and 5. (Theorem 5.1) If we can construct the regular p-gon, then we can construct \zeta_p = e^\frac{2\pi i}{p}, which is the root of an
irreducible polynomial of degree p - 1. By Theorem 5.1, \zeta_p lies in a field K of degree N over \Q, where the only primes that divide N are 2, 3, and 5. But \Q[\zeta_p] is a subfield of K, so p - 1 divides N. In particular, for p = 23, N must be divisible by 11, and for p = 29,
N must be divisible by 7. It can be constructed using the
quadratrix of Hippias,
Archimedean spiral, and other
auxiliary curves; yet this is true for all regular polygons.{{cite journal ==Related figures==