apeirogon
Sign in to save260px|thumb|A partition of the Euclidean line into infinitely many equal-length segments can be understood as a regular apeirogon. In geometry, an apeirogon () or infinite polygon is a polygon with an infinite number of sides. Apeirogons are the rank 2 case of infinite polytopes. In some literature, the term "apeirogon" may refer only to the regular apeirogon, with an infinite dihedral group of symmetries.
Key facts
- Polygon.name
- Apeirogon (regular)
- Polygon.image
- regular apeirogon.svg
- Polygon.edges
- ∞
- Polygon.schläfli
- {∞}
- Polygon.angle
- 180°
- Polygon.dual
- Self-dual
via Wikipedia infobox
Article
13 sectionsContents
- Definitions
- Geometric apeirogon
- Hyperbolic pseudogon
- Abstract apeirogon
- Symmetries
- Moduli space
- Classification of Euclidean apeirogons
- Generalizations
- Higher rank
- See also
- Notes
- References
- External links
{{Infobox Polygon | name = Apeirogon (regular) | image = regular apeirogon.svg | edges = ∞ | schläfli = {∞} | coxeter = | angle = 180° | dual = Self-dual }} 260px|thumb|A partition of the Euclidean line into infinitely many equal-length segments can be understood as a regular apeirogon. In geometry, an apeirogon () or infinite polygon is a polygon with an infinite number of sides. Apeirogons are the rank 2 case of infinite polytopes. In some literature, the term "apeirogon" may refer only to the regular apeirogon, with an infinite dihedral group of symmetries.
==Definitions== ===Geometric apeirogon=== Given a point A0 in a Euclidean space and a translation S, define the point Ai to be the point obtained from i applications of the translation S to A0, so Ai = Si(A0). The set of vertices Ai with i any integer, together with edges connecting adjacent vertices, is a sequence of equal-length segments of a line, and is called the regular apeirogon as defined by H. S. M. Coxeter.