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In higher-dimensional geometry, the faces of a [[polytope]] are features of all dimensions.<ref name="m"/><ref name="g">{{citation|title=Convex Polytopes|title-link=Convex Polytopes|first=Branko|last=Grünbaum|author-link=Branko Grünbaum|page=[https://books.google.com/books?id=ISHO86XJ1CsC&pg=PA17 17]|volume=221|edition=2nd|series=Graduate Texts in Mathematics|publisher=Springer|year=2003}}.</ref><ref name="z">{{citation|first=Günter M.|last=Ziegler|author-link=Günter M. Ziegler|title=Lectures on Polytopes|at=Definition 2.1, p. 51|url=https://books.google.com/books?id=xd25TXSSUcgC&pg=PA51|volume=152|series=Graduate Texts in Mathematics|publisher=Springer|year=1995|isbn=9780387943657}}.</ref> A face of dimension ''k'' is called a ''k''-face. For example, the polygonal faces of an ordinary polyhedron are 2-faces. In [[set theory]], the set of faces of a polytope includes the polytope itself and the empty set, where the empty set is for consistency given a "dimension" of −1. For any ''n''-polytope (''n''-dimensional polytope), −1 ≤ ''k'' ≤ ''n''.
For example, with this meaning, the faces of a [[cube]] comprise the cube itself (3-face), its (square) [[Face (geometry)#Facet or (n-1)-face|facets]] (2-faces), (
*4-face – the 4-dimensional [[4-polytope]] itself
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