Heptagon

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Heptagon
Rank2
TypeRegular
Notation
Bowers style acronymHeg
Coxeter diagramx7o ()
Schläfli symbolTemplate:Schlafi
Elements
Edges7
Vertices7
Vertex figureDyad, length 2cos(π/7)
Measures (edge length 1)
Circumradius
Inradius
Area
Angle
Central density1
Number of external pieces7
Level of complexity1
Related polytopes
ArmyHeg
DualHeptagon
ConjugatesHeptagram, great heptagram
Abstract & topological properties
Flag count14
Euler characteristic0
OrientableYes
Properties
SymmetryI2(7), order 14
Flag orbits1
ConvexYes
NatureTame

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 2sin(π/7), centered at the origin, are:

  • ,
  • ,
  • ,
  • .

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]

Variations[edit | edit source]

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

Tiling the plane[edit | edit source]

While convex heptagons cannot tile the plane[2], certain non-convex heptagons can.

Stellations[edit | edit source]

External links[edit | edit source]

References[edit | edit source]

  1. Online Encyclopedia of Integer Sequences. "A003401".
  2. 2.0 2.1 Semidoc (March 9, 2018). "No tiling by convex heptagons".
  3. Hull (2009:1)
  4. Hull (2009:5)
  5. Lang (2015)

Bibliography[edit | edit source]

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