English edit

 
A magnified portion of a regular apeirogon is a partition of a line with equal edge lengths

Etymology edit

apeiro- +‎ -gon

Pronunciation edit

  • IPA(key): /əˈpiːɹɵɡɑn/, /əˈpeɪ̯ɹɵɡɑn/
    • (file)
  • Hyphenation: apei‧ro‧gon

Noun edit

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][1], 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.

Usage notes edit

  • 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.
  • The Schläfli symbol of an apeirogon is  . (For comparison, the symbol for an  -sided regular polygon is  .)
  • 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   into segments of length  .

Hyponyms edit

  • zerogon (a specific non-regular case)

Related terms edit

Translations edit

See also edit

Further reading edit