# apeirohedron

## English[edit]

### Etymology[edit]

### Pronunciation[edit]

- IPA
^{(key)}: /əˌpiːɹɵˈhiːdɹən/, /əˌpeɪ̯ɹɵˈhiːdɹən/

### Noun[edit]

**apeirohedron** (*plural* **apeirohedrons** *or* **apeirohedra**)

- (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, 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.

- There are exactly 12 regular