Octadecagon

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Regular octadecagon
Regular polygon 18 annotated.svg
A regular octadecagon
Type Regular polygon
Edges and vertices 18
Schläfli symbol {18}
t{9}
Coxeter diagram CDel node 1.pngCDel 18.pngCDel node.png
CDel node 1.pngCDel 9.pngCDel node 1.png
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]

Construction[edit]

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

Uses[edit]

3.9.18 vertex.png
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 an alternating ring of the other two kinds of polygon.

Related figures[edit]

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.

n 2 3 4 5 6 7 8 9
Form Compounds Star polygon Compound Star polygon Compound
Image Regular star figure 2(9,1).svg
{18/2}
= 2{9}
Regular star figure 3(6,1).svg
{18/3}
= 3{6}
Regular star figure 2(9,2).svg
{18/4}
= 2{9/2}
Regular star polygon 18-5.svg
{18/5}
Regular star figure 6(3,1).svg
{18/6}
= 6{3}
Regular star polygon 18-7.svg
{18/7}
Regular star figure 2(9,4).svg
{18/8}
= 2{9/4}
Regular star figure 9(2,1).svg
{18/9}
= 9{2}

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
Regular polygon truncation 9 1.svg
t{9}={18}
Regular polygon truncation 9 2.svg Regular polygon truncation 9 3.svg Regular polygon truncation 9 4.svg Regular polygon truncation 9 5.svg Regular star polygon 9-4.svg
t{9/8}={18/8}
=2{9/4}
Regular star truncation 9-5 1.svg
t{9/5}={18/5}
Regular star truncation 9-5 2.svg Regular star truncation 9-5 3.svg Regular star truncation 9-5 4.svg Regular star truncation 9-5 5.svg Regular star polygon 9-2.svg
t{9/4}={18/4}
=2{9/2}
Regular star truncation 9-7 1.svg
t{9/7}={18/7}
Regular star truncation 9-7 2.svg Regular star truncation 9-7 3.svg Regular star truncation 9-7 4.svg Regular star truncation 9-7 5.svg Regular polygon 9.svg
t{9/2}={18/2}
=2{9}

Petrie polygons[edit]

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 t0.svg
17-simplex
9-cube t8.svg
9-orthoplex
9-cube t0.svg
9-cube
10-cube t9 B9.svg
711
10-demicube.svg
171
Up2 3 21 t0 E7.svg
321
Up2 2 31 t0 E7.svg
231
Up2 1 32 t0 E7.svg
132

References[edit]

  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[edit]