The following approximate construction is very similar to that of the enneagon, as an octadecagon can be constructed as a truncated enneagon. It is also feasible with exclusive use of compass and straightedge.
Downsize the angle AMC (also 60°) with four angle bisectors and make a thirds of circular arc MON with an approximate solution between angle bisectors w3 and w4.
Straight auxiliary line g aims over the point O to the point N (virtually a ruler at the points O and N applied), between O and N, therefore no auxiliary line.
Thus, the circular arc MON is freely accessible for the later intersection point R.
AMR = 19.999999994755615...°
360° ÷ 18 = 20°
AMR - 20° = -5.244...E-9°
Example to illustrate the error:
At a circumscribed circle radius r = 100,000 km, the absolute error of the 1st side would be approximately -9 mm.
Dih18 symmetry, order 36. There are 5 subgroup dihedral symmetries: Dih9, (Dih6, Dih3), and (Dih2 Dih1), and 6 cyclic group
symmetries: (Z18, Z9), (Z6, Z3), and (Z2, Z1).
These 15 symmetries can be seen in 12 distinct symmetries on the octadecagon. John Conway labels these by a letter and group order.[4] Full symmetry of the regular form is r36 and no symmetry is labeled a1. The dihedral symmetries are divided depending on whether they pass through vertices (d for diagonal) or edges (p for perpendiculars), and i when reflection lines path through both edges and vertices. Cyclic symmetries in the middle column are labeled as g for their central gyration orders.
Each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only the g18 subgroup has no degrees of freedom but can be seen as
directed edges
.
Dissection
Coxeter states that every zonogon (a 2m-gon whose opposite sides are parallel and of equal length) can be dissected into m(m-1)/2 parallelograms.[6]
In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. For the regular octadecagon, m=9, and it can be divided into 36: 4 sets of 9 rhombs. This decomposition is based on a Petrie polygon projection of a 9-cube, with 36 of 4608 faces. The list OEIS: A006245 enumerates the number of solutions as 112018190, including up to 18-fold rotations and chiral forms in reflection.
Dissection into 36 rhombs
Uses
Archimedean tiling
of the plane: because the triangle and the nonagon both have an odd number of sides, neither of them can be completely surrounded by a ring alternating the other two kinds of polygon.
The regular octadecagon can tessellate the plane with concave hexagonal gaps. And another tiling mixes in nonagons and octagonal gaps. The first tiling is related to a truncated hexagonal tiling, and the second the truncated trihexagonal tiling.
Related figures
An octadecagram is an 18-sided star polygon, represented by symbol {18/n}. There are two regular
enneagons, {18/3} is reduced to 3{6} or three hexagons, {18/4} and {18/8} are reduced to 2{9/2} and 2{9/4} or two enneagrams, {18/6} is reduced to 6{3} or 6 equilateral triangles, and finally {18/9} is reduced to 9{2} as nine digons
.
Compounds and star polygons
n
1
2
3
4
5
6
7
8
9
Form
Convex polygon
Compounds
Star polygon
Compound
Star polygon
Compound
Image
{18/1} = {18}
{18/2} = 2{9}
{18/3} = 3{6}
{18/4} = 2{9/2}
{18/5}
{18/6} = 6{3}
{18/7}
{18/8} = 2{9/4}
{18/9} = 9{2}
Interior angle
160°
140°
120°
100°
80°
60°
40°
20°
0°
Deeper truncations of the regular enneagon and enneagrams can produce isogonal (
vertex-transitive) intermediate octadecagram forms with equally spaced vertices and two edge lengths. Other truncations form double coverings: t{9/8}={18/8}=2{9/4}, t{9/4}={18/4}=2{9/2}, t{9/2}={18/2}=2{9}.[8]
Vertex-transitive truncations of enneagon and enneagrams
^The Lighter Side of Mathematics: Proceedings of the Eugène Strens Memorial Conference on Recreational Mathematics and its History, (1994), Metamorphoses of polygons, Branko Grünbaum