{{Short description|Shape with eleven sides}} {{Use dmy dates|date=May 2019}} {{Regular polygon db|Regular polygon stat table|p11}} In geometry, a '''hendecagon''' (also '''undecagon'''<ref>{{citation|title=Construction of the regular undecagon by a sextic curve|department=Discussions|journal=American Mathematical Monthly|volume=29|issue=10|year=1922|first=Cyrus B.|last=Haldeman|jstor=2299029|doi=10.2307/2299029}}.</ref><ref name="loomis"/> or '''endecagon'''<ref>{{citation|title=Errors of speech and of spelling|last=Brewer|first=Ebenezer Cobham|year=1877|location=London|publisher=W. Tegg and co.|url=https://archive.org/details/errorsspeechand00brewgoog|page=iv}}.</ref>) 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".<ref>[http://mathworld.wolfram.com/Hendecagon.html Hendecagon – from Wolfram MathWorld<!--Bot generated title-->]</ref>)

==Regular hendecagon==

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

A regular hendecagon has internal angles of 147.<span style="text-decoration: overline">27</span> degrees (=147 <math>\tfrac{3}{11}</math> degrees).<ref>{{citation|title=Glencoe mathematics: applications and connections|first=Kay|last=McClain|publisher=Glencoe/McGraw-Hill|year=1998|isbn=9780028330549|page=[https://archive.org/details/glencoemathemati0000unse/page/357 357]|url=https://archive.org/details/glencoemathemati0000unse/page/357}}.</ref> The area of a regular hendecagon with side length ''a'' is given by<ref name="loomis">{{citation|title=Elements of Plane and Spherical Trigonometry: With Their Applications to Mensuration, Surveying, and Navigation|first=Elias|last=Loomis|publisher=Harper|year=1859|page=65|url=https://archive.org/details/elementsplanean00loomgoog}}.</ref> :<math>A = \frac{11}{4}a^2 \cot \frac{\pi}{11} \simeq 9.36564\,a^2.</math>

As 11 is not a Fermat prime, the regular hendecagon is not constructible with compass and straightedge.<ref>As Gauss proved, a polygon with a prime number ''p'' of sides can be constructed if and only if ''p''&nbsp;&minus;&nbsp;1 is a power of two, which is not true for 11. See {{citation|title=Mathematical Thought From Ancient to Modern Times|volume=2|first=Morris|last=Kline|author-link=Morris Kline|publisher=Oxford University Press|year=1990|isbn=9780199840427|pages=753–754|url=https://books.google.com/books?id=VOcUBvbUXlEC&pg=PA753}}.</ref> Because 11 is not a Pierpont prime, construction of a regular hendecagon is still impossible even 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 a unit circle as being 14/25 units long.<ref>{{citation|title=A History of Greek Mathematics, Vol. II: From Aristarchus to Diophantus|first=Sir Thomas Little|last=Heath|author-link=Thomas Little Heath|publisher=The Clarendon Press|year=1921|page=329|url=https://archive.org/stream/historyofgreekma00heat#page/328/mode/2up}}.</ref>

The hendecagon can be constructed exactly via neusis construction<ref>{{cite journal | last1=Benjamin | first1=Elliot | last2=Snyder | first2=C. | title=On the construction of the regular hendecagon by marked ruler and compass | journal=Mathematical Proceedings of the Cambridge Philosophical Society | volume=156 | issue=3 | date=May 2014 | pages=409-424 | doi=10.1017/S0305004113000753}}</ref> and also via two-fold origami.<ref>{{Cite journal|last=Lucero|first=J. C.|date=2018|title=Construction of a regular hendecagon by two-fold origami|url=https://cms.math.ca/wp-content/uploads/crux-pdfs/CRUXv44n5.pdf|journal=Crux Mathematicorum|volume=44|pages=207–213|access-date=20 June 2018|archive-date=18 July 2024|archive-url=https://web.archive.org/web/20240718132303/https://www2.cms.math.ca/crux/v44/n5/CRUXv44n5.pdf}}</ref>

==Approximate construction== {{multiple image | align = right | image1 = 01-Endecagon-Drummond.gif | width1 = 400 | alt1 = | caption1 = Hendecagon inscribed in a circle, a continuation of the basic construction according to T. Drummond as animation.<br/>Corresponds to the copper engraving by Anton Ernst Burkhard of Birckenstein. | image2 = Fotothek df tg 0004812 Geometrie ^ Architektur ^ Festungsbau ^ Vermessung.jpg | width2 = 187 | alt2 = | caption2 = Hendecagon, copper engraving by 1698 by Anton Ernst Burkhard of Birckenstein | footer = }} The following construction description is given by T. Drummond from 1800:<ref>T. Drummond, (1800) [https://books.google.com/books?id=gR5kAAAAcAAJ&dq=Endecagon&pg=PA15 The Young Ladies and Gentlemen's AUXILIARY, in Taking Heights and Distances ..., Construction description pp. 15–16] [https://books.google.com/books?id=gR5kAAAAcAAJ&pg=PA69 Fig. 40: scroll from page 69 ... to page 76] Part I. Second Edition, retrieved on 26 March 2016</ref>

{{blockquote|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 a hendecagon sufficiently exact for practice.}}

On a unit circle: * Constructed hendecagon side length <math>b=0.563692\ldots</math> * Theoretical hendecagon side length <math>a=2\sin(\frac{\pi}{11})=0.563465\ldots</math> * Absolute error <math>\delta=b-a=2.27\ldots\cdot10^{-4}</math> – if {{Overline|AB}} is 10&nbsp;m then this error is approximately 2.3&nbsp;mm.

== Symmetry== thumb|200px|Symmetries of a regular hendecagon. Vertices are colored by their symmetry positions. Blue mirror lines are drawn through vertices and edge. Gyration orders are given in the center.

The ''regular hendecagon'' has Dih<sub>11</sub> symmetry, order 22. Since 11 is a prime number there is one subgroup with dihedral symmetry: Dih<sub>1</sub>, and 2 cyclic group symmetries: Z<sub>11</sub>, and Z<sub>1</sub>.

These 4 symmetries can be seen in 4 distinct symmetries on the hendecagon. John Conway labels these by a letter and group order.<ref>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)</ref> 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 be seen as directed edges.

==Use in coinage== The Canadian dollar coin, the loonie, is similar to, but not exactly, a regular hendecagonal prism,<ref>{{citation|title=A $1 problem|first=Michael J.|last=Mossinghoff|journal=American Mathematical Monthly|volume=113|issue=5|year=2006|pages=385–402|jstor=27641947|url=http://www.maa.org/sites/default/files/images/upload_library/22/Ford/mossinghoff385.pdf|doi=10.2307/27641947}}</ref> as are the Indian 2-rupee coin<ref>{{citation|title=2013 Standard Catalog of World Coins 2001 to Date|first1=George S.|last1=Cuhaj|first2=Thomas|last2=Michael|publisher=Krause Publications|year=2012|isbn=9781440229657|page=402|url=https://books.google.com/books?id=jI5QOJGScHgC&pg=PA402}}.</ref> and several other lesser-used coins of other nations.<ref>{{citation|title=Unusual World Coins|first1=George S.|last1=Cuhaj|first2=Thomas|last2=Michael|edition=6th|publisher=Krause Publications|year=2011|isbn=9781440217128|pages=23, 222, 233, 526}}.</ref> 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.{{sfn|U.S. House of Representatives, 1978|p=7}}

==Related figures== The hendecagon shares the same set of 11 vertices with four regular hendecagrams: {| class=wikitable |- align=center |80px<br>{11/2} |80px<br>{11/3} |80px<br>{11/4} |80px<br>{11/5} |}

== See also== *10-simplex - can be seen as a complete graph in a regular hendecagonal orthogonal projection

==References== {{Reflist}}

===Works cited=== * {{cite book |author=United States House of Representatives |title=Proposed Smaller One-Dollar Coin |year=1978 |publisher=Government Printing Office |location=Washington, D.C. |ref={{sfnRef|U.S. House of Representatives, 1978}}}}

==External links== *[http://www.mathopenref.com/undecagon.html Properties of an Undecagon (hendecagon)] With interactive animation *{{MathWorld |title=Hendecagon |urlname=Hendecagon}} *[https://commons.wikimedia.org/wiki/Category:Regular_hendecagons Regular hendecagons] *[https://commons.wikimedia.org/wiki/File:01-Elfeck-3.svg Regular hendecagon, an approximate construction] * [https://semjonadlaj.com/Excerpts/Hendecagon.pdf Exact construction via quintisection]

{{Polygons}}

Category:Polygons by the number of sides Category:11 (number) Category:Elementary shapes