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


apeiro- +‎ -gon


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


apeirogon (plural apeirogons)

  1. (mathematics, geometry) A polygon having an infinite number of sides and vertices.
    • 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 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 978-0-88385-511-9, 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; Egon Schulte, Abstract Regular Polytopes, Cambridge: Cambridge University Press, ISBN 978-0-521-81496-6, 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.
    • 2013, Brent Davis; Moshe Renert, The Math Teachers Know: Profound Understanding of Emergent Mathematics, New York, N.Y.; Abingdon, Oxon.: Routledge, ISBN 9780415858441, page 102:
      They [the students] also came upon new and unusual mathematical figures: the digon, a two-sided polygon on a spherical space, and the apeirogon, an open polygon with infinitely many sides  []. All these discoveries brought up even more questions. Is a circle a polygon? What makes an octagon an octagon – its eight vertices, its eight sides, or both? Can a polygon cross itself? Does a polygon need to be closed?
    • 2014, Daniel Pellicer; 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 978-1-4939-0781-6, 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.


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