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apeiro- +‎ -hedron


  • IPA(key): /əˌpiːɹɵˈhiːdɹən/, /əˌpeɪ̯ɹɵˈhiːdɹən/


apeirohedron (plural apeirohedrons or apeirohedra)

  1. (mathematics, geometry) A polyhedron with an infinite number of faces.
    • 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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