Ingeometry,ahendecagon(alsoundecagon[1][2]orendecagon[3]) or 11-gon is an eleven-sidedpolygon.(The namehendecagon,from Greekhendeka"eleven" and–gon"corner", is often preferred to the hybridundecagon,whose first part is formed from Latinundecim"eleven".[4])
| Regular hendecagon | |
|---|---|
 A regular hendecagon | |
| Type | Regular polygon |
| Edgesandvertices | 11 |
| Schläfli symbol | {11} |
| Coxeter–Dynkin diagrams |    |
| Symmetry group | Dihedral(D11), order 2×11 |
| Internal angle(degrees) | ≈147.273° |
| Properties | Convex,cyclic,equilateral,isogonal,isotoxal |
| Dual polygon | Self |
Regular hendecagon
editAregularhendecagonis represented bySchläfli symbol{11}.
A regular hendecagon hasinternal anglesof 147.27degrees(=147degrees).[5]The area of a regular hendecagon with side lengthais given by[2]
As 11 is not aFermat prime,the regular hendecagon is notconstructiblewithcompass and straightedge.[6]Because 11 is not aPierpont prime,construction of a regular hendecagon is still impossibleeven with the usage of an angle trisector.
Close approximations to the regular hendecagon can be constructed. For instance, the ancient Greek mathematicians approximated the side length of a hendecagon inscribed in aunit circleas being 14/25 units long.[7]
The hendecagon can be constructed exactly vianeusis construction[8]and also via two-fold origami.[9]
Approximate construction
edit
Corresponds to the copper engraving by Anton Ernst Burkhard of Birckenstein.
The following construction description is given by T. Drummond from 1800:[10]
- "Draw the radiusA B,bisect it inC—with an opening of the compasses equal to half the radius, uponAandCas centres describe the arcsC D IandA D—with the distanceI DuponIdescribe the arcD Oand draw the lineC O,which will be the extent of one side of a hendecagon sufficiently exact for practice."
On a unit circle:
- Constructed hendecagon side length
- Theoretical hendecagon side length
- Absolute error– ifABis 10 m then this error is approximately 2.3 mm.
Symmetry
edit
Theregular hendecagonhasDih11symmetry,order 22. Since 11 is aprime numberthere is one subgroup with dihedral symmetry: Dih1,and 2cyclic groupsymmetries: Z11,and Z1.
These 4 symmetries can be seen in 4 distinct symmetries on the hendecagon.John Conwaylabels these by a letter and group order.[11]Full symmetry of the regular form isr22and 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 theg11subgroup has no degrees of freedom but can be seen asdirected edges.
Use in coinage
editTheCanadian dollarcoin, theloonie,is similar to, but not exactly, a regularhendecagonal prism,[12]as are the Indian 2-rupeecoin[13]and several other lesser-used coins of other nations.[14]The cross-section of a loonie is actually aReuleaux hendecagon.The United StatesSusan B. Anthony dollarhas a hendecagonal outline along the inside of its edges.[15]
Related figures
editThe hendecagon shares the same set of 11 vertices with four regularhendecagrams:
 {11/2} |
 {11/3} |
 {11/4} |
 {11/5} |
See also
edit- 10-simplex- can be seen as a complete graph in a regular hendecagonal orthogonal projection
References
edit- ^Haldeman, Cyrus B. (1922), "Construction of the regular undecagon by a sextic curve", Discussions,American Mathematical Monthly,29(10),doi:10.2307/2299029,JSTOR2299029.
- ^abLoomis, Elias (1859),Elements of Plane and Spherical Trigonometry: With Their Applications to Mensuration, Surveying, and Navigation,Harper, p. 65.
- ^Brewer, Ebenezer Cobham (1877),Errors of speech and of spelling,London: W. Tegg and co., p. iv.
- ^Hendecagon – from Wolfram MathWorld
- ^McClain, Kay (1998),Glencoe mathematics: applications and connections,Glencoe/McGraw-Hill, p.357,ISBN9780028330549.
- ^AsGaussproved, a polygon with a prime numberpof sides can be constructed if and only ifp− 1 is apower of two,which is not true for 11. SeeKline, Morris(1990),Mathematical Thought From Ancient to Modern Times,vol. 2, Oxford University Press, pp. 753–754,ISBN9780199840427.
- ^Heath, Sir Thomas Little(1921),A History of Greek Mathematics, Vol. II: From Aristarchus to Diophantus,The Clarendon Press, p. 329.
- ^Benjamin, Elliot; Snyder, C. Mathematical Proceedings of the Cambridge Philosophical Society156.3 (May 2014): 409-424.;https://dx.doi.org/10.1017/S0305004113000753
- ^Lucero, J. C. (2018)."Construction of a regular hendecagon by two-fold origami".Crux Mathematicorum.44:207–213. Archived fromthe originalon 20 June 2018.Retrieved20 June2018.
- ^T. Drummond, (1800)The Young Ladies and Gentlemen's AUXILIARY, in Taking Heights and Distances..., Construction description pp. 15–16Fig. 40: scroll from page 69... to page 76Part I. Second Edition, retrieved on 26 March 2016
- ^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)
- ^Mossinghoff, Michael J. (2006),"A $1 problem"(PDF),American Mathematical Monthly,113(5): 385–402,doi:10.2307/27641947,JSTOR27641947
- ^Cuhaj, George S.; Michael, Thomas (2012),2013 Standard Catalog of World Coins 2001 to Date,Krause Publications, p. 402,ISBN9781440229657.
- ^Cuhaj, George S.; Michael, Thomas (2011),Unusual World Coins(6th ed.), Krause Publications, pp. 23, 222, 233, 526,ISBN9781440217128.
- ^U.S. House of Representatives, 1978,p. 7.
Works cited
edit- United States House of Representatives (1978).Proposed Smaller One-Dollar Coin.Washington, D.C.: Government Printing Office.
