Hendecagon

Regular hendecagon

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

A regular hendecagon has internal angles of 147.27 degrees (=147 \tfrac{3}{11} degrees). The area of a regular hendecagon with side length a is given by :A = \frac{11}{4}a^2 \cot \frac{\pi}{11} \simeq 9.36564,a^2.

As 11 is not a Fermat prime, the regular hendecagon is not constructible with compass and straightedge. 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.

The hendecagon can be constructed exactly via neusis construction{{cite journal

Approximate construction

Corresponds to the copper engraving by Anton Ernst Burkhard of Birckenstein. The following construction description is given by T. Drummond from 1800:

On a unit circle:

• Constructed hendecagon side length b=0.563692\ldots
• Theoretical hendecagon side length a=2\sin(\frac{\pi}{11})=0.563465\ldots
• Absolute error \delta=b-a=2.27\ldots\cdot10^{-4} – if is 10 m then this error is approximately 2.3 mm.

Symmetry

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. 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, as are the Indian 2-rupee coin and several other lesser-used coins of other nations. 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.

Related figures

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

References

Works cited

References

1. Haldeman, Cyrus B.. (1922). "Construction of the regular undecagon by a sextic curve". [[American Mathematical Monthly]].
2. Brewer, Ebenezer Cobham. (1877). "Errors of speech and of spelling". W. Tegg and co..
3. [http://mathworld.wolfram.com/Hendecagon.html Hendecagon – from Wolfram MathWorld]
4. McClain, Kay. (1998). "Glencoe mathematics: applications and connections". Glencoe/McGraw-Hill.
5. Loomis, Elias. (1859). "Elements of Plane and Spherical Trigonometry: With Their Applications to Mensuration, Surveying, and Navigation". Harper.
6. Kline, Morris. (1990). "Mathematical Thought From Ancient to Modern Times". Oxford University Press.
7. Heath, Sir Thomas Little. (1921). "A History of Greek Mathematics, Vol. II: From Aristarchus to Diophantus". The Clarendon Press.
8. Lucero, J. C.. (2018). "Construction of a regular hendecagon by two-fold origami". Crux Mathematicorum.
9. 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
10. 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)
11. Mossinghoff, Michael J.. (2006). "A $1 problem". [[American Mathematical Monthly]].
12. (2012). "2013 Standard Catalog of World Coins 2001 to Date". Krause Publications.
13. (2011). "Unusual World Coins". Krause Publications.