# Hendecagon

In geometry, a **hendecagon** (also **undecagon**[1][2] or **endecagon**[3]) or 11-gon is an eleven-sided polygon. (The name *hendecagon*, from Greek *hendeka* "eleven" and *–gon* "corner", is often preferred to the hybrid *undecagon*, whose first part is formed from Latin *undecim* "eleven".[4])

Regular hendecagon | |
---|---|

A regular hendecagon | |

Type | Regular polygon |

Edges and vertices | 11 |

Schläfli symbol | {11} |

Coxeter diagram | |

Symmetry group | Dihedral (D_{11}), order 2×11 |

Internal angle (degrees) | ≈147.273° |

Dual polygon | Self |

Properties | Convex, cyclic, equilateral, isogonal, isotoxal |

## Regular hendecagon

A *regular hendecagon* is represented by Schläfli symbol {11}.

A regular hendecagon has internal angles of 147.27 degrees (=147 degrees).[5] The area of a regular hendecagon with side length *a* is given by[2]

As 11 is not a Fermat prime, the regular hendecagon is not constructible with compass and straightedge.[6] Because 11 is not a Pierpont prime, construction of a regular hendecagon is still impossible even with the usage of an angle trisector. It can, however, be constructed via neusis construction[7] and also via two-fold origami.[8]

Close approximations to the regular hendecagon can be constructed, however. For instance, the ancient Greek mathematicians approximated the side length of a hendecagon inscribed in a unit circle as being 14/25 units long.[9]

## Approximate construction

The following construction description is given by T. Drummond from 1800:[10]

- "
*Draw the radius*"**A B**, bisect it in**C**—with an opening of the compasses equal to half the radius, upon**A**and**C**as centres describe the arcs**C D I**and**A D**—with the distance**I D**upon**I**describe the arc**D O**and draw the line**C O**, which will be the extent of one side of an endecagon sufficiently exact for practice.

On a unit circle:

- Constructed hendecagon side length
- Theoretical hendecagon side length
- Absolute error – if AB is 10 m then this error is approximately 2.3 mm.

## Symmetry

The *regular hendecagon* has Dih_{11} symmetry, order 22. Since 11 is a prime number there is one subgroup with dihedral symmetry: Dih_{1}, and 2 cyclic group symmetries: Z_{11}, and Z_{1}.

These 4 symmetries can be seen in 4 distinct symmetries on the hendecagon. John Conway labels these by a letter and group order.[11] Full symmetry of the regular form is **r22** 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 **g11** subgroup has no degrees of freedom but can seen as directed edges.

## Use in coinage

The Canadian dollar coin, the loonie, is similar to, but not exactly, a regular hendecagonal prism,[12] as are the Indian 2-rupee coin[13] and several other lesser-used coins of other nations.[14] The cross-section of a loonie is actually a Reuleaux hendecagon. The United States Susan B. Anthony dollar has a hendecagonal outline along the inside of its edges.[15]

## Related figures

The hendecagon shares the same set of 11 vertices with four regular hendecagrams:

{11/2} |
{11/3} |
{11/4} |
{11/5} |

## See also

- 10-simplex - can be seen as a complete graph in a regular hendecagonal orthogonal projection

## References

- Haldeman, Cyrus B. (1922), "Construction of the regular undecagon by a sextic curve", Discussions,
*American Mathematical Monthly*,**29**(10), doi:10.2307/2299029, JSTOR 2299029. - Loomis, 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, ISBN 9780028330549. - As Gauss proved, a polygon with a prime number
*p*of sides can be constructed if and only if*p*− 1 is a power of two, not true for 11. See Kline, Morris (1990),*Mathematical Thought From Ancient to Modern Times*,**2**, Oxford University Press, pp. 753–754, ISBN 9780199840427. - 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. - Heath, Sir Thomas Little (1921),
*A History of Greek Mathematics, Vol. II: From Aristarchus to Diophantus*, The Clarendon Press, p. 329. - T. Drummond, (1800) The Young Ladies and Gentlemen's AUXILIARY, in Taking Heights and Distances ..., Construction description pp. 15–16 Fig. 40: scroll from page 69 ... to page 76 Part I. Second Edition, retrieved on 26 March 2016
- John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, (2008) The Symmetries of Things, ISBN 978-1-56881-220-5 (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, JSTOR 27641947 - Cuhaj, George S.; Michael, Thomas (2012),
*2013 Standard Catalog of World Coins 2001 to Date*, Krause Publications, p. 402, ISBN 9781440229657. - Cuhaj, George S.; Michael, Thomas (2011),
*Unusual World Coins*(6th ed.), Krause Publications, pp. 23, 222, 233, 526, ISBN 9781440217128. - U.S. House of Representatives, 1978, p. 7.