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# incompPolyhedra -- returns the pairs of incompatible polyhedra

## Synopsis

• Usage:
Lpairs = incompPolyhedra L
Lpairs = incompPolyhedra(X,PC)
Lpairs = incompPolyhedra(PC,X)
• Inputs:
• Outputs:

## Description

If incompPolyhedra is applied to a list of polyhedra and polyhedral complexes, then it returns the pairs of elements whose intersection is not a face of each. For a Polyhedron P and a PolyhedralComplex PC in the list this means there is at least one generating Polyhedron of PC whose intersection with P is not a face of each. For two polyhedral complexes in the list this means there is at least one generating polyhedron each such that their intersection is not a face of each. If applied to a pair consisting of a polyhedron and a polyhedral complex or two polyhedral complexes, then it returns the pairs of polyhedra that do not share a common face.
 i1 : P1 = convexHull matrix {{1,0,0},{1,1,0}}; i2 : P2 = convexHull matrix {{1,0,0},{0,-1,0}}; i3 : P3 = convexHull matrix {{-1,0,0},{0,1,0}}; i4 : P4 = convexHull matrix {{1,1,0},{0,1,0}}; i5 : P5 = convexHull matrix {{1,2,0},{2,1,0}}; i6 : L = {P1,P2,P3,P4,P5}; i7 : Lpairs = incompPolyhedra L o7 = {({ambient dimension => 2 }, {ambient dimension => 2 dimension of lineality space => 0 dimension of lineality space => dimension of polyhedron => 2 dimension of polyhedron => 2 number of facets => 3 number of facets => 3 number of rays => 0 number of rays => 0 number of vertices => 3 number of vertices => 3 ------------------------------------------------------------------------ }), ({ambient dimension => 2 }, {ambient dimension => 2 0 dimension of lineality space => 0 dimension of lineality space dimension of polyhedron => 2 dimension of polyhedron => 2 number of facets => 3 number of facets => 3 number of rays => 0 number of rays => 0 number of vertices => 3 number of vertices => 3 ------------------------------------------------------------------------ })} => 0 o7 : List i8 : Lpairs == {(P1,P4),(P1,P5)} o8 = false

## Ways to use incompPolyhedra :

• "incompPolyhedra(List)"
• "incompPolyhedra(PolyhedralComplex,PolyhedralComplex)"
• "incompPolyhedra(PolyhedralComplex,Polyhedron)"
• "incompPolyhedra(Polyhedron,PolyhedralComplex)"

## For the programmer

The object incompPolyhedra is .