Editing Polyhedron (section)

=== Lattice polyhedron ===
Convex polyhedra in which all vertices have integer coordinates are called [[convex lattice polytope|lattice polyhedra]] or [[integral polyhedron|integral polyhedra]]. The [[Ehrhart polynomial]] of lattice a polyhedron counts how many points with [[integer]] coordinates lie within a scaled copy of the polyhedron, as a function of the scale factor. The study of these polynomials lies at the intersection of [[combinatorics]] and [[commutative algebra]].<ref name=stanley-97>{{citation
 | last = Stanley | first = Richard P. | author-link = Richard P. Stanley 
 | year = 1997
 | title = Enumerative Combinatorics, Volume I
 | edition = 1
 | publisher = Cambridge University Press
 | pages = 235–239
 | isbn = 978-0-521-66351-9
}}</ref> An example is [[Reeve tetrahedron]].<ref name=k>{{citation
 | last = Kołodziejczyk | first = Krzysztof
 | doi = 10.1007/BF00150027
 | issue = 3
 | journal = Geometriae Dedicata
 | mr = 1397808
 | pages = 271–278
 | title = An "odd" formula for the volume of three-dimensional lattice polyhedra
 | volume = 61
 | year = 1996| s2cid = 121162659
 }}</ref>

There is a far-reaching equivalence between lattice polyhedra and certain [[Algebraic variety|algebraic varieties]] called [[Toric variety|toric varieties]].<ref>{{citation |last=Cox |first=David A. |title=Toric varieties |date=2011 |publisher=American Mathematical Society |others=John B. Little, Henry K. Schenck |isbn=978-0-8218-4819-7 |location=Providence, R.I. |oclc=698027255}}</ref> This was used by Stanley to prove the [[Dehn–Sommerville equations]] for [[simplicial polytope]]s.<ref name=stanley-96>{{citation |last=Stanley |first=Richard P. |title=Combinatorics and commutative algebra |date=1996 |publisher=Birkhäuser |isbn=0-8176-3836-9 |edition=2nd |location=Boston |oclc=33080168}}</ref>