polytope.miraheze.org

Heptagon

️Thu Nov 14 2024

Heptagon
(OFF file)
Rank	2
Type	Regular
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	${\frac {1}{2\sin {\frac {\pi }{7}}}}\approx 1.15238$
Inradius	${\frac {1}{2\tan {\frac {\pi }{7}}}}\approx 1.03826$
Area	${\frac {7}{4\tan {\frac {\pi }{7}}}}\approx 3.63391$
Angle	${\frac {5\pi }{7}}\approx 128.57143^{\circ }$
Central density	1
Number of external pieces	7
Level of complexity	1
Related polytopes
Army	Heg
Dual	Heptagon
Conjugates	Heptagram, great heptagram
Abstract & topological properties
Flag count	14
Euler characteristic	0
Orientable	Yes
Symmetry properties
Symmetry	I₂(7), order 14
Flag orbits	1
Other properties
Convex	Yes
Nature	Tame

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

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 finite 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.^[1] 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.^[2] Nevertheless, regular heptagons can tile the hyperbolic plane, as in the order-3 heptagonal tiling, for example.

Naming[edit | edit source]

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.
7-gon

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

Vertex coordinates[edit | edit source]

Coordinates for a regular heptagon of edge length 1, centered at the origin, are:

$\left({\frac {1}{2\sin(\pi /7)}},\,0\right)$ ,
$\left({\frac {\cos(2\pi /7)}{2\sin(\pi /7)}},\,\pm {\frac {\sin(2\pi /7)}{2\sin(\pi /7)}}\right)$ ,
$\left({\frac {\cos(4\pi /7)}{2\sin(\pi /7)}},\,\pm {\frac {\sin(4\pi /7)}{2\sin(\pi /7)}}\right)$ ,
$\left({\frac {\cos(6\pi /7)}{2\sin(\pi /7)}},\,\pm {\frac {\sin(6\pi /7)}{2\sin(\pi /7)}}\right)$ .

Construction[edit | edit source]

The regular heptagon cannot be constructed with a compass and straightedge.^[3] It is the smallest regular polygon which cannot be constructed this way. It can however be constructed via origami construction^[4] or neusis construction.^[5]