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The Combinatorics of Polytopes

Subsections

Points in Polytopes

Given a polygon, one might want to know how many integral points it contains, and one might want to list them. These are computed by different algorithms (although one can of course list the points and then count them). For point counting we use the methods of Barvinok and Pommersheim ([BP99]), as described in [DLHTY04]. More generally, one might want to know the number of points in integral dilations of a polytope: these numbers are the coefficients of a generating function, the Ehrhart series; this is discussed in a later section.

Points(P) : TorPol -> SeqEnum[TorLatElt]
InteriorPoints(P) : TorPol -> SeqEnum[TorLatElt]
BoundaryPoints(P) : TorPol -> SeqEnum[TorLatElt]
The integral toric lattice points, the strictly interior lattice points, or the boundary lattice points of the polytope P.
NumberOfPoints(P) : TorPol -> RngIntElt
The number of integral toric lattice points of the polytope P.

Ehrhart Theory of Polytopes

EhrhartSeries(P) : TorPol -> FldFunRatUElt
The rational generating function of the Ehrhart series for the polytope P.
EhrhartPolynomial(P) : TorPol -> [RngUPolElt]
A sequence representing the Ehrhart (quasi-)polynomial for the polytope P. That is, a sequence of polynomials [p0, ..., pr - 1] of length r, the quasi-period of the Ehrhart polynomial, so that the number of lattice points of the dilation nP is the value of ps(k) where n=kr + s is the Euclidean division of n by r; in other words, s is the least residue of n modulo r. Note that since Magma indexes sequences from 1, we have that pi = EhrhartPolynomial(P)[i+1].
EhrhartCoefficients(P,l) : TorPol,RngIntElt -> [RngIntElt]
The first l + 1 coefficients of the Ehrhart series for the polytope P (starting with 0P up to and including lP).
EhrhartCoefficient(P,k) : TorPol,RngIntElt -> RngIntElt
The number of lattice points in the (non-negative, integral) dilation kP of the polytope P.

Automorphisms of a Polytope

AutomorphismGroup(P) : TorPol -> GrpMat
The subgroup of GL(n, (Z)) (acting on the ambient lattice) which leaves the polyhedron P unchanged.

Example Polyhedra_polytope-automorphism-example (H143E5)

> P:=CrossPolytope(2);      
> P;
2-dimensional polytope P with 4 vertices:
    ( 1,  0),
    ( 0,  1),
    (-1,  0),
    ( 0, -1)
> AutomorphismGroup(P);
MatrixGroup(2, Integer Ring)
Generators:
    [0 1]
    [1 0]

    [ 0  1]
    [-1  0]

Operations on Polytopes

Triangulation(P) : TorPol -> SetEnum
A sequence of polytopes that make a triangulation of the polytope P.
TriangulationOfBoundary(P) : TorPol -> SetEnum
A sequence of polytopes that make a triangulation of the boundary of the polytope P.
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Version: V2.19 of Wed Apr 24 15:09:57 EST 2013