Enneagon

The enneagon sometimes referred to as a nonagon, is a polygon with 9 sides. A regular enneagon has equal sides and equal angles.

Enneagon
(OFF file)
Rank	2
Type	Regular
Space	Spherical
Notation
Bowers style acronym	En
Coxeter diagram	x9o ()
Schläfli symbol	{9}
Elements
Edges	9
Vertices	9
Vertex figure	Dyad, length $2\cos(\pi/9)$
Measures (edge length 1)
Circumradius	$\frac{1}{2\sin\frac\pi9} \approx 1.46190$
Inradius	$\frac{1}{2\tan\frac\pi9} \approx 1.37374$
Area	$\frac{9}{4\tan\frac\pi9} \approx 6.18182$
Angle	140°
Central density	1
Number of external pieces	9
Level of complexity	1
Related polytopes
Army	En
Dual	Enneagon
Conjugates	Enneagram, great enneagram
Abstract & topological properties
Flag count	18
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	I2(9), order 18
Convex	Yes
Net count	1
Nature	Tame

The combining prefix in BSAs is e-, as in edip.

Like regular heptagons, regular enneagons are rarely found in higher polytopes that are objects of study, as they do not occur any non-prismatic uniform polyhedra or Johnson solids. A notable exception is the pairwise augmented cupolas, which are acrohedra. Enneagons also appear in some near-miss Johnson solids, such as the sesquitruncated octahedron.

Naming

The name enneagon is derived from the Ancient Greek ἐννέα (9) and γωνία (angle), referring to the number of vertices.

Other names include:

• En, Bowers style acronym, short for "enneagon".

Vertex coordinates

Coordinates for an enneagon of edge length $2\sin(\pi/9)$, centered at the origin, are:

• $(1, 0)$,
• $(\cos(2\pi/9), \pm\sin(2\pi/9))$,
• $(\cos(4\pi/9), \pm\sin(4\pi/9))$,
• $(–1/2, \pm\sqrt{3}/2)$,
• $(\cos(8\pi/9), \pm\sin(8\pi/9))$.

Variations

Besides the regular enneagon, other enneagons with triangular, mirror, or no symmetry exist. A few higher polytopes, such as certain swirlchora, have trigon-symmetric enneagons as faces.

Stellations

• 1st stellation: Enneagram
• 2nd stellation: Fissal enneagram (compound of three triangles)
• 3rd stellation: Great enneagram

External links

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

• Wikipedia Contributors. "Nonagon".