apeirogon

English

WOTD – 20 December 2015

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

Etymology

apeiro- +‎ -gon

Pronunciation

IPA^(key): /əˈpiːɹɵɡɑn/, /əˈpeɪ̯ɹɵɡɑn/
- Audio (RP) (file)
Hyphenation: apei‧ro‧gon

Noun

apeirogon (plural apeirogons)

(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; 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; 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

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 ${\\displaystyle 2\\lambda }$ .