Heptagon

Heptagon
(OFF file)
Rank	2
Type	Regular
Space	Spherical
Notation
Bowers style acronym	Heg
Coxeter diagram	x7o ()
Schläfli symbol	{7}
Elements
Edges	7
Vertices	7
Vertex figure	Dyad, length 2cos(π/7)
Measures (edge length 1)
Circumradius	$\frac{1}{2\sin\frac\pi7} ≈ 1.15238$
Inradius	$\frac{1}{2\tan\frac\pi7} ≈ 1.03826$
Area	$\frac{7}{4\tan\frac\pi7} ≈ 3.63391$
Angle	$\frac{5\pi}{7} ≈ 128.57143^\circ$
Central density	1
Number of pieces	7
Level of complexity	1
Related polytopes
Army	Heg
Dual	Heptagon
Conjugates	Heptagram, great heptagram
Abstract properties
Flag count	14
Euler characteristic	0
Topological properties
Orientable	Yes
Properties
Symmetry	I2(7), order 14
Convex	Yes
Nature	Tame

The heptagon is a polygon with 7 sides. A regular heptagon has equal sides and equal angles.

The combining prefix in BSAs is he-, as in hedip.

It has two stellations, these being the heptagram and the great heptagram.

The regular heptagon has several properties that distinguish it from all smaller regular polygons. It is the simplest polygon not to appear on any non-prismatic uniform polyhedron. This is partially due to its I2(7) symmetry group not being embedded in any higher fundamental Coxeter group. It's also the simplest polygon that cannot be constructed with a straightedge and a compass, as the expressions for its coordinates involve cube roots. No heptagon appears in the Johnson solids, and acrohedra containing regular heptagons are particularly hard to discover (although a few are known, such as the 7-4-3 pairwise augmented cupolae, 7-6-4, and the 7-7-3 small supersemicupola).

Furthermore, in contrast to polygons with fewer sides, there is no single convex heptagon that can tile the plane without overlap. Intuitively, this is because the average angles around each vertex would have to be at least (15/14)×360°, a clear impossibility. This intuition may be formalized with bounds involving the Euler characteristic. Nevertheless, regular heptagons can tile the hyperbolic plane, as in the order-3 heptagonal tiling, for example.

Naming

The name heptagon is derived from the Ancient Greek ἑπτά (7) and γωνία (angle), referring to the number of vertices.

Other names include:

• Heg, Bowers style acronym, short for "heptagon".
• Septagon, based on Latin septum.

Vertex coordinates

Coordinates for a regular heptagon of edge length 2sin(π/7), centered at the origin, are:

• $(1, 0)$,
• $(\cos(2\pi/7), \pm\sin(2\pi/7))$,
• $(\cos(4\pi/7), \pm\sin(4\pi/7))$,
• $(\cos(6\pi/7), \pm\sin(6\pi/7))$.

Construction

The regular heptagon cannot be constructed with a compass and straightedge. It is the smallest regular polygon which cannot be constructed this way. It can however be constructed via origami construction or neusis construction.

Variations

Besides the regular heptagon, other less regular heptagons with mirror or no symmetry exist. Some appear as vertex figures of polyhedra.

Tiling the plane

While convex heptagons cannot tile the plane, certain non-convex heptagons can.

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Stellations

• 1st stellation: Heptagram.
• 2nd stellation: Great heptagram.

External links

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

• Wikipedia Contributors. "Heptagon".

Bibliography

• Hull, Thomas (2009), Folding Regular Heptagons (PDF)
• Lang, Wolfdieter (2015), A255240: Archimedes's Construction of the Regular Heptagon