apeirogon

apeiro- +‎ -gon

• IPA^(key): /əˈpiːɹɵɡɑn/, /əˈpeɪ̯ɹɵɡɑn/

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• Hyphenation: apei‧ro‧gon

apeirogon (plural apeirogons)

1. (mathematics, geometry) A type of generalised polygon with a countably infinite number of sides and vertices;
   (in the regular case) the limit case of an n-sided regular polygon as n increases to infinity and the edge length is fixed; typically imagined as a straight line partitioned into equal segments by an infinite number of equally-spaced points.
   • 1984, Coxeter-Festschrift [Mitteilungen aus dem Mathem[atisches] Seminar Giessen]‎, Giessen: Gießen Mathematischen Institut, Justus Liebig-Universität Gießen, page 247:

     Hence the regular polygon ABCD ... can either be a convex n-gon, a star n-gon, a horocylic^{[sic – meaning horocyclic]} apeirogon or a hypercyclic apeirogon.

   • 1994, Steven Schwartzman, The Words of Mathematics: An Etymological Dictionary of Mathematical Terms Used in English, Washington, D.C.: Mathematical Association of America, →ISBN, page 27:

     In geometry, an apeirogon is a limiting case of a regular polygon. The number of sides in an apeirogon is becoming infinite, so the apeirogon as a whole approaches a circle. A magnified view of a small piece of the apeirogon looks like a straight line.

   • 2002, Peter McMullen with Egon Schulte, Abstract Regular Polytopes, Cambridge: Cambridge University Press, →ISBN, page 217:

     [A]n apeirogon (infinite regular polygon) is a linear one {∞}, a planar (skew) one (zigzag apeirogon), which is the blend {∞} # { } with a segment, or helix, which is a blend of {∞} with a bounded regular polygon.

   • 2014, Daniel Pellicer with Egon Schulte, “Polygonal Complexes and Graphs for Crystallographic Groups”, in Robert Connelly, Asia Ivić Weiss, Walter Whiteley, editors, Rigidity and Symmetry, New York, N.Y.: Springer, →ISBN, page 331:

     There are exactly 12 regular apeirohedra that in some sense are reducible and have components that are regular figures of dimensions 1 and 2. These apeirohedra are blends of a planar regular apeirohedron, and a line segment { } or linear apeirogon {∞}. This explains why there are 12 = 6·2 blended (or non-pure) apeirohedra. For example, the blend of the standard square tessellation {4,4} and the infinite apeirogon {∞}, denoted {4,4}#{∞}, is an apeirohedron whose faces are helical apeirogons (over squares), rising above the squares of {4,4}, such that 4 meet at each vertex; the orthogonal projections of {4,4}#{∞} onto their component subspaces recover the original components, the square tessellation and the linear apeirogon.

• Some authors use the term only for the regular apeirogon.
• A regular apeirogon can be described as a partition (or tessellation) of the Euclidean line into infinitely many equal-length segments.
  • For alternative definitions, see Apeirogon § Definitions on Wikipedia.Wikipedia
• The Schläfli symbol of an apeirogon is ${\displaystyle \{\infty \}}$. (For comparison, the symbol for an ${\displaystyle n}$-sided regular polygon is ${\displaystyle \{n\}}$.)
• The limit case of an n-sided regular polygon as n increases to infinity and the perimeter length is fixed (meaning the edge lengths decrease to zero) is a circle, which in this context is sometimes called a zerogon.
• In analogy to the Euclidean case, the regular pseudogon is a partition of the hyperbolic line ${\displaystyle H^{1}}$ into segments of length ${\displaystyle 2\lambda }$.

• zerogon (a specific non-regular case)

• apeirohedron

• infinigon
• pseudogon

• apeirogon on Wikipedia.Wikipedia