Hendecagon

Regular hendecagon
A regular hendecagon
Type Regular polygon
Edges and vertices 11
Schläfli symbol {11}
Coxeter diagram
Symmetry group Dihedral (D11), order 2×11
Internal angle (degrees) $\approx 147.273$°
Dual polygon self
Properties convex, cyclic, equilateral, isogonal, isotoxal

In geometry, a hendecagon (also undecagon[1]) is an 11-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".[2])

Regular hendecagon

A regular hendecagon has internal angles of 147.27 degrees.[3] The area of a regular hendecagon with side length a is given by[4]

$A = \frac{11}{4}a^2 \cot \frac{\pi}{11} \simeq 9.36564\,a^2.$

A regular hendecagon is not constructible with compass and straightedge.[5] 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, 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.[6] The animation below shows another approximation.

Use in coinage

The Canadian dollar coin, the loonie, is similar to, but not exactly, a regular hendecagonal prism,[7] as are the Indian 2-rupee coin[8] and several other lesser-used coins of other nations.[9] The cross-section of a loonie is actually a Reuleaux hendecagon.

Related shapes

The hendecagon shares the same set of 11 vertices with four regular hendecagrams, {11/2}, {11/3}, {11/4}, {11/5}.

The regular hendecagon is the Petrie polygon for 10-dimensional uniform polytopes of the simplex family, projected in a skew orthogonal projection.[10][11]

References

1. ^ Undecagon Definition – Math Open Reference
2. ^ Hendecagon – from Wolfram MathWorld
3. ^ McClain, Kay (1998), Glencoe mathematics: applications and connections, Glencoe/McGraw-Hill, p. 357, ISBN 9780028330549.
4. ^ Loomis, Elias (1886), Elements of Plane and Spherical Trigonometry: With Their Applications to Mensuration, Surveying, and Navigation, Harper, p. 72.
5. ^ 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.
6. ^ Heath, Sir Thomas Little (1921), A History of Greek Mathematics: From Aristarchus to Diophantus, The Clarendon Press, p. 329.
7. ^ Mossinghoff, Michael J. (2006), "A \$1 problem", American Mathematical Monthly 113 (5): 385–402, JSTOR 27641947
8. ^ Cuhaj, George S.; Michael, Thomas (2012), 2013 Standard Catalog of World Coins 2001 to Date, Krause Publications, p. 402, ISBN 9781440229657.
9. ^ Cuhaj, George S.; Michael, Thomas (2011), Unusual World Coins (6th ed.), Krause Publications, pp. 23, 222, 233, 526, ISBN 9781440217128.
10. ^ Coxeter, H. S. M. Petrie Polygons. Regular Polytopes, 3rd ed. New York: Dover, 1973. (sec 2.6 Petrie Polygons pp. 24–25)
11. ^ Humphreys, James E. (1992), Reflection Groups and Coxeter Groups, Cambridge University Press, pp. 80 (Section 3.16, Coxeter Elements, table 2, Coxeter number for An is n+1), ISBN 978-0-521-43613-7