Octadecagon

Regular octadecagon

A regular octadecagon
Type Regular polygon
Edges and vertices 18
Schläfli symbol {18}
t{9}
Coxeter diagram
Symmetry group Dihedral (D18), order 2×18
Internal angle (degrees) 160°
Dual polygon self
Properties convex, cyclic, equilateral, isogonal, isotoxal

An octadecagon (or octakaidecagon[1]) is a polygon with 18 sides and 18 vertices.[2]

A regular octadecagon has a Schläfli symbol {18} and can be constructed as a quasiregular truncated enneagon, t{9}, which alternates two types of edges.

Construction

A regular octadecagon cannot be constructed using a compass and straightedge.[3] However, it is constructible using neusis, or an angle trisector.

Uses


A regular triangle, enneagon, and octadecagon can completely surround a point in the plane, one of 17 different combinations of regular polygons with this property.[4] However, this pattern cannot be extended to an Archimedean tiling of the plane: because the triangle and the enneagon both have an odd number of sides, neither of them can be completely surrounded by a ring alternating the other two kinds of polygon.

Related figures

An octadecagram is an 18-sided star polygon, represented by symbol {18/n}. There are two regular star polygons: {18/5} and {18/7}, using the same points, but connecting every fifth or seventh points. There are also five compounds: {18/2} is reduced to 2{9} or two 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.

n123456789
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}.[5]

Vertex-transitive truncations of enneagon and enneagrams
Quasiregular isogonal Quasiregular
Double covering

t{9}={18}
t{9/8}={18/8}
=2{9/4}

t{9/5}={18/5}
t{9/4}={18/4}
=2{9/2}

t{9/7}={18/7}
t{9/2}={18/2}
=2{9}

Petrie polygons

The regular octadecagon is the Petrie polygon for a number of higher-dimensional polytopes, shown in these skew orthogonal projections from Coxeter planes:

A17 B9 D10 E7

17-simplex
9-orthoplex
9-cube
711
171
321
231
132

References

  1. ↑ Kinsey, L. Christine; Moore, Teresa E. (2002), Symmetry, Shape, and Surfaces: An Introduction to Mathematics Through Geometry, Springer, p. 86, ISBN 9781930190092.
  2. ↑ Adams, Henry (1907), Cassell's Engineer's Handbook: Comprising Facts and Formulæ, Principles and Practice, in All Branches of Engineering, D. McKay, p. 528.
  3. ↑ Conway, John B. (2010), Mathematical Connections: A Capstone Course, American Mathematical Society, p. 31, ISBN 9780821849798.
  4. ↑ Dallas, Elmslie William (1855), The Elements of Plane Practical Geometry, Etc, John W. Parker & Son, p. 134.
  5. ↑ The Lighter Side of Mathematics: Proceedings of the Eugène Strens Memorial Conference on Recreational Mathematics and its History, (1994), Metamorphoses of polygons, Branko Grünbaum

External links