Editing Hendecagram

{{short description|11-pointed star polygon}}
{{Infobox polygon
| name        = Hendecagram
| image       = HendecagramTypes.png
| caption     = The four regular hendecagrams
| edges       = 11
| schläfli    = {11/2}, {11/3}<br />{11/4}, {11/5}
| coxeter     ={{CDD|node_1|11|rat|d2|node}}, {{CDD|node_1|11|rat|d3|node}}<br />{{CDD|node_1|11|rat|d4|node}}, {{CDD|node_1|11|rat|d5|node}}
| symmetry    = [[Dihedral symmetry|Dih<sub>11</sub>]], order 22
| area        =
| angle       = ≈114.545° {11/2}<br />≈81.8182° {11/3}<br />≈49.0909° {11/4}<br />≈16.3636° {11/5}
| properties  =}}
In [[geometry]], a '''hendecagram''' (also '''endecagram''' or '''endekagram''') is a [[star polygon]] that has eleven [[Vertex (geometry)|vertices]].

The name ''hendecagram'' combines a Greek [[numeral prefix]], ''[[wikt:hendeca-|hendeca-]]'', with the [[Greek language|Greek]] suffix ''[[wikt:-gram|-gram]]''. The ''hendeca-'' prefix derives from Greek ἕνδεκα (ἕν + δέκα, one + ten) meaning "[[11 (number)|eleven]]". The ''-gram'' suffix derives from γραμμῆς (''grammēs'') meaning a line.<ref>{{citation |first1=Henry George |last1=Liddell |first2=Robert |last2=Scott |title=A Greek-English Lexicon: γραμμή |location=Oxford |publisher=[[Clarendon Press]] |year=1940 |url=https://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.04.0057%3Aentry%3Dgrammh%2F}}</ref>

==Regular hendecagrams==
There are four '''regular hendecagrams''',<ref>{{citation|title=Geometry: an investigative approach|first1=Phares G.|last1=O'Daffer|first2=Stanley R.|last2=Clemens|publisher=[[Addison-Wesley]]|year=1976|isbn=9780201054200|at=[https://archive.org/details/geometry00odaf/page/ Exercise&nbsp;7, p.&nbsp;62]|url-access=registration|url=https://archive.org/details/geometry00odaf/page/}}.</ref> which can be described by the notation
{11/2}, {11/3}, {11/4}, and {11/5}; in this notation, the number after the slash indicates the number of steps between pairs of points that are connected by edges. These same four forms can also be considered as [[Stellation#Stellating polygons|stellations]] of a regular [[hendecagon]].<ref>{{citation|title=Elementary Geometry|volume=43|series=Student mathematical library|publisher=[[American Mathematical Society]]|first1=Ilka|last1=Agricola|author1-link= Ilka Agricola |first2=Thomas|last2=Friedrich|year=2008|isbn=9780821890677|page=96|url=https://books.google.com/books?id=Ts20OwbWfPkC&pg=PA96}}.</ref>

Since 11 is prime, all hendecagrams are star polygons and not compound figures.

==Construction==
As with all odd regular polygons and star polygons whose orders are not products of distinct [[Fermat prime]]s, the regular hendecagrams cannot be constructed with compass and straightedge.<ref>{{citation|title=Abstract Algebra: Applications to Galois Theory, Algebraic Geometry, and Cryptography|volume=11|series=Sigma series in pure mathematics|first1=Celine|last1=Carstensen|first2=Benjamin|last2=Fine|first3=Gerhard|last3=Rosenberger|publisher=[[Walter de Gruyter]]|year=2011|isbn=9783110250084|page=88|url=https://books.google.com/books?id=Xo6iSxRfXz0C&pg=PA88|quote=On the other hand a regular 11-gon is not constructible.}}</ref>
However, {{harvtxt|Hilton|Pedersen|1986}} describe folding patterns for making the hendecagrams {11/3}, {11/4}, and {11/5} out of strips of paper.<ref>{{citation|last1=Hilton |first1=Peter|author1-link=Peter Hilton |last2=Pedersen |first2=Jean |author2-link=Jean Pedersen|year=1986 |title=Symmetry in mathematics |journal=Computers & Mathematics with Applications |volume=12 |issue=1–2 |doi=10.1016/0898-1221(86)90157-4 |mr=838152 |pages=315–328|doi-access= }}</ref>

==Applications==
[[File:Flickr - The U.S. Army - The Golden Knights land at Statue of Liberty in New York City (2).jpg|thumb|[[Liberty Island#Fort Wood|Fort Wood]]'s star-shaped walls became the base of the Statue of Liberty.]]
[[Prism (geometry)|Prism]]s over the hendecagrams {11/3} and {11/4} may be used to approximate the shape of [[DNA]] molecules.<ref>{{citation |title=DNA enclosing forms from scaled growth forms of snow crystals |first=Aloysio |last=Janner |journal=Crystal Engineering |volume=4 |issue=2–3 |date=June 2001 |pages=119–129 |doi=10.1016/S1463-0184(01)00005-3|bibcode=2001CrEng...4..119J }}</ref>

[[File:Momine Fragment.jpg|thumb|An 11-pointed star from the [[Momine Khatun Mausoleum]]]]
[[Liberty Island#Fort Wood|Fort Wood]], now the base of the [[Statue of Liberty]] in [[New York City]], is a [[star fort]] in the form of an irregular 11-point star.<ref>{{citation|title=The Hudson River Guidebook|first=Arthur G.|last=Adams|publisher=[[Fordham Univ Press]]|year=1996|isbn=9780823216796|page=66|url=https://books.google.com/books?id=KIkHuzvZTGYC&pg=PA66}}.</ref>

The [[Topkapı Scroll]] contains images of an 11-pointed star [[Girih]] form used in [[Islamic art]]. The star in this scroll is not one of the regular forms of the hendecagram, but instead uses lines that connect the vertices of a [[hendecagon]] to nearly-opposite midpoints of the hendecagon's edges.<ref>{{citation|url=http://archive.bridgesmathart.org/2009/bridges2009-147.pdf|title=Bridges 2009: Mathematics, Music, Art, Architecture, Culture|first=B. Lynn|last=Bodner|contribution=The eleven–pointed star polygon design of the ''Topkapı Scroll''|pages=147–154|year=2009}}.</ref> 11-pointed star Girih patterns are also used on the exterior of the [[Momine Khatun Mausoleum]]; Eric Broug writes that its pattern "can be considered a high point in Islamic geometric design".<ref>{{citation|title=Islamic Geometric Design|first=Eric|last=Broug|page=182|year=2013|publisher=[[Thames & Hudson]]}}</ref>

An 11-point star-shaped cross-section was used in the [[Space Shuttle Solid Rocket Booster]], for the core of the forward section of the rocket (the hollow space within which the fuel burns). This design provided more surface area and greater thrust in the earlier part of a launch, and a slower burn rate and reduced thrust after the points of the star were burned away, at approximately the same time as the rocket passed the [[sound barrier]].<ref>{{citation|title=Encyclopedia of Space and Astronomy|first=Joseph A.|last=Angelo|publisher=[[Infobase Publishing]]|year=2009|isbn=9781438110189|page=511|url=https://books.google.com/books?id=VUWno1sOwnUC&pg=PA511}}.</ref>

==See also==
*[[Hendecagrammic prism]]

==References==
{{Star polygons}}
{{reflist}}

==External links==
*{{Mathworld |urlname=Polygram |title=Polygram|mode=cs2}}

{{Polygons}}

[[Category:Star polygons|11]]
[[Category:11 (number)]]