Hendecagon

Template:Short description Template:Use dmy dates Template:Regular polygon db In geometry, a hendecagon (also undecagon<ref>Template:Citation.</ref><ref name="loomis"/> or endecagon<ref>Template:Citation.</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>Hendecagon – from Wolfram MathWorld</ref>)

Regular hendecagonEdit

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

A regular hendecagon has internal angles of 147.27 degrees (=147 <math>\tfrac{3}{11}</math> degrees).<ref>Template:Citation.</ref> The area of a regular hendecagon with side length a is given by<ref name="loomis">Template:Citation.</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 − 1 is a power of two, which is not true for 11. See Template:Citation.</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>Template:Citation.</ref>

The hendecagon can be constructed exactly via neusis construction<ref>Template:Cite journal</ref> and also via two-fold origami.<ref>Template:Cite journal</ref>

Approximate constructionEdit

Template:Multiple image The following construction description is given by T. Drummond from 1800:<ref>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</ref>

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.{{#if:|{{#if:|}}
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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 Template:Overline is 10 m then this error is approximately 2.3 mm.

SymmetryEdit

File:Symmetries of hendecagon.png

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 Dih₁₁ symmetry, order 22. Since 11 is a prime number there is one subgroup with dihedral symmetry: Dih₁, and 2 cyclic group symmetries: Z₁₁, and Z₁.

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, Template:Isbn (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 coinageEdit

The Canadian dollar coin, the loonie, is similar to, but not exactly, a regular hendecagonal prism,<ref>Template:Citation</ref> as are the Indian 2-rupee coin<ref>Template:Citation.</ref> and several other lesser-used coins of other nations.<ref>Template:Citation.</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.Template:Sfn

Related figuresEdit

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

File:Regular star polygon 11-2.svg {11/2}	File:Regular star polygon 11-3.svg {11/3}	File:Regular star polygon 11-4.svg {11/4}	File:Regular star polygon 11-5.svg {11/5}