Hendecagon

Hendecagon
(OFF file)
Rank	2
Type	Regular
Notation
Bowers style acronym	Heng
Coxeter diagram	x11o ()
Schläfli symbol	{11}
Elements
Edges	11
Vertices	11
Vertex figure	Dyad, length 2cos(π/11)
Measures (edge length 1)
Circumradius	${\frac {1}{2\sin \left({\frac {\pi }{11}}\right)}}\approx 1.77473$
Inradius	${\frac {1}{2\tan \left({\frac {\pi }{11}}\right)}}\approx 1.70284$
Area	${\frac {11}{4\tan \left({\frac {\pi }{11}}\right)}}\approx 9.36564$
Angle	${\frac {9\pi }{11}}\approx 147.27273^{\circ }$
Central density	1
Number of external pieces	11
Level of complexity	1
Related polytopes
Army	Heng
Dual	Hendecagon
Conjugates	Small hendecagram, hendecagram, great hendecagram, grand hendecagram
Abstract & topological properties
Flag count	22
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	I2(11), order 22
Flag orbits	1
Convex	Yes
Nature	Tame

The hendecagon is a polygon with 11 sides. A regular hendecagon has equal sides and equal angles.

The combining prefix in BSAs is hen-, as in hentet, or han-, as in handip.

It has four stellations, these being the small hendecagram, the hendecagram, the great hendecagram, and the grand hendecagram.

It is the smallest regular convex polygon that cannot form a planar vertex with only other regular convex polygons.

Higher polytopes containing regular hendecagons that are "interesting" in some sense are rare. An exception is three members of the family of pairwise augmented cupolae, which are 11-4-3 acrohedra (all faces are regular).

Naming

The name hendecagon is derived from the Ancient Greek ἕνδεκα (11) and γωνία (angle), referring to the number of vertices.

Other names include:

• heng, Bowers style acronym, short for "hendecagon"

Vertex coordinates

Vertex coordinates for a hendecagon of edge length $2\sin(\pi /11)$, centered at the origin, are:

• $(1,\,0)$,
• $(\cos(2\pi /11),\,\pm \sin(2\pi /11))$,
• $(\cos(4\pi /11),\,\pm \sin(4\pi /11))$,
• $(\cos(6\pi /11),\,\pm \sin(6\pi /11))$,
• $(\cos(8\pi /11),\,\pm \sin(8\pi /11))$,
• $(\cos(10\pi /11),\,\pm \sin(10\pi /11))$.

External links

• Bowers, Jonathan. "Regular Polygons and Other Two Dimensional Shapes".

• Wikipedia contributors. "Hendecagon".