Octadecagon
Regular octadecagon | |
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![]() A regular octadecagon | |
Type | Regular polygon |
Edgesandvertices | 18 |
Schläfli symbol | {18}, t{9} |
Coxeter–Dynkin diagrams | ![]() ![]() ![]() ![]() ![]() ![]() |
Symmetry group | Dihedral(D18), order 2×18 |
Internal angle(degrees) | 160° |
Properties | Convex,cyclic,equilateral,isogonal,isotoxal |
Dual polygon | Self |
Ingeometry,anoctadecagon(oroctakaidecagon[1]) or 18-gon is an eighteen-sidedpolygon.[2]
Regular octadecagon
[edit]![](https://upload.wikimedia.org/wikipedia/commons/thumb/e/ee/Achtzehneck_mit_Diagonalen.svg/220px-Achtzehneck_mit_Diagonalen.svg.png)
Aregularoctadecagonhas aSchläfli symbol{18} and can be constructed as a quasiregulartruncatedenneagon,t{9}, which alternates two types of edges.
Construction
[edit]As 18 = 2 × 32,a regular octadecagon cannot beconstructedusing acompass and straightedge.[3]However, it is constructible usingneusis,or anangle trisectionwith atomahawk.
![](https://upload.wikimedia.org/wikipedia/commons/thumb/c/cf/01-Achtzehneck-Tomahawk.gif/400px-01-Achtzehneck-Tomahawk.gif)
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.
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Symmetry
[edit]![](https://upload.wikimedia.org/wikipedia/commons/thumb/3/33/Symmetries_of_octadecagon.png/320px-Symmetries_of_octadecagon.png)
Theregular octadecagonhasDih18symmetry,order 36. There are 5 subgroup dihedral symmetries: Dih9,(Dih6,Dih3), and (Dih2Dih1), and 6cyclic groupsymmetries: (Z18,Z9), (Z6,Z3), and (Z2,Z1).
These 15 symmetries can be seen in 12 distinct symmetries on the octadecagon.John Conwaylabels these by a letter and group order.[4]Full symmetry of the regular form isr36and no symmetry is labeleda1.The dihedral symmetries are divided depending on whether they pass through vertices (dfor diagonal) or edges (pfor perpendiculars), andiwhen reflection lines path through both edges and vertices. Cyclic symmetries in the middle column are labeled asgfor their central gyration orders.
Each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only theg18subgroup has no degrees of freedom but can be seen asdirected edges.
Dissection
[edit]![](https://upload.wikimedia.org/wikipedia/commons/thumb/2/2f/18-gon_rhombic_dissection-size2.svg/220px-18-gon_rhombic_dissection-size2.svg.png)
![](https://upload.wikimedia.org/wikipedia/commons/thumb/5/50/Equilateral_pentagonal_dissection_of_regular_octadecagon.svg/220px-Equilateral_pentagonal_dissection_of_regular_octadecagon.svg.png)
Coxeterstates that everyzonogon(a 2m-gon whose opposite sides are parallel and of equal length) can be dissected intom(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 theregular octadecagon,m=9, and it can be divided into 36: 4 sets of 9 rhombs. This decomposition is based on aPetrie polygonprojection of a9-cube,with 36 of 4608 faces. The listOEIS:A006245enumerates the number of solutions as 112018190, including up to 18-fold rotations and chiral forms in reflection.
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Uses
[edit]
A regular triangle, nonagon, and octadecagon can completely surround a point in the plane, one of 17 different combinations of regular polygons with this property.[7]However, this pattern cannot be extended to anArchimedean tilingof 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 atruncated hexagonal tiling,and the second thetruncated trihexagonal tiling.
Related figures
[edit]Anoctadecagramis an 18-sided star polygon, represented by symbol {18/n}. There are two regularstar 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 twoenneagons,{18/3} is reduced to 3{6} or threehexagons,{18/4} and {18/8} are reduced to 2{9/2} and 2{9/4} or twoenneagrams,{18/6} is reduced to 6{3} or 6 equilateral triangles, and finally {18/9} is reduced to 9{2} as ninedigons.
Compounds and star polygons | |||||||||
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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]
Petrie polygons
[edit]A regular skew octadecagon is thePetrie polygonfor a number of higher-dimensional polytopes, shown in these skeworthogonal projectionsfromCoxeter planes:
Octadecagonal petrie polygons | |||||||
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A17 | B9 | D10 | E7 | ||||
![]() 17-simplex |
![]() 9-orthoplex |
![]() 9-cube |
![]() 711 |
![]() 171 |
![]() 321 |
![]() 231 |
![]() 132 |
References
[edit]- ^Kinsey, L. Christine;Moore, Teresa E. (2002),Symmetry, Shape, and Surfaces: An Introduction to Mathematics Through Geometry,Springer, p. 86,ISBN9781930190092.
- ^Adams, Henry (1907),Cassell's Engineer's Handbook: Comprising Facts and Formulæ, Principles and Practice, in All Branches of Engineering,D. McKay, p. 528.
- ^Conway, John B. (2010),Mathematical Connections: A Capstone Course,American Mathematical Society, p. 31,ISBN9780821849798.
- ^John H. Conway, Heidi Burgiel,Chaim Goodman-Strauss,(2008) The Symmetries of Things,ISBN978-1-56881-220-5(Chapter 20, Generalized Schaefli symbols, Types of symmetry of a polygon pp. 275-278)
- ^Hirschhorn & Hunt 1985.
- ^Coxeter,Mathematical recreations and Essays, Thirteenth edition, p.141
- ^Dallas, Elmslie William (1855),The Elements of Plane Practical Geometry, Etc,John W. Parker & Son, p. 134.
- ^The Lighter Side of Mathematics: Proceedings of the Eugène Strens Memorial Conference on Recreational Mathematics and its History, (1994),Metamorphoses of polygons,Branko Grünbaum
- Hirschhorn, M. D.; Hunt, D. C. (1985),"Equilateral convex pentagons which tile the plane"(PDF),Journal of Combinatorial Theory, Series A,39(1): 1–18,doi:10.1016/0097-3165(85)90078-0,ISSN1096-0899,MR0787713,retrieved2020-10-30
- octadecagon