Samiksha Jaiswal (Editor)

Apeirotope

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An apeirotope or infinite polytope is a polytope which has infinitely many facets. There are two main geometric classes of apeirotope:

Contents

  • honeycombs in n dimensions, which completely fill an n-dimensional space.
  • skew apeirotopes, comprising an n-dimensional manifold in a higher space

    • Honeycombs

    In general, a honeycomb in n dimensions is an infinite example of a polytope in n + 1 dimensions.

    Tilings of the plane and close-packed space-fillings of polyhedra are examples of honeycombs in two and three dimensions respectively.

    A line divided into infinitely many finite segments is an example of an apeirogon.

    Skew apeirogons

    A skew apeirogon in two dimensions forms a zig-zag line in the plane. If the zig-zag is even and symmetrical, then the apeirogon is regular.

    Skew apeirogons can be constructed in any number of dimensions. In three dimensions, a regular skew apeirogon traces out a helical spiral and may be either left- or right-handed.

    Infinite skew polyhedra

    There are three regular skew apeirohedra, which look rather like polyhedral sponges:

  • 6 squares around each vertex, Coxeter symbol {4,6|4}
  • 4 hexagons around each vertex, Coxeter symbol {6,4|4}
  • 6 hexagons around each vertex, Coxeter symbol {6,6|3}

    • There are thirty regular apeirohedra in Euclidean space. These include those listed above, as well as (in the plane) polytopes of type: {∞,3}, {∞,4}, {∞,6} and in 3-dimensional space, blends of these with either an apeirogon or a line segment, and the "pure" 3-dimensional apeirohedra (12 in number)

    References

    Apeirotope Wikipedia
    (Text) CC BY-SA