-- -*- M2-comint -*- {* hash: -577021660 *}
i1 : V = matrix {{0,2,-2,0},{-1,1,1,1}}
o1 = | 0 2 -2 0 |
| -1 1 1 1 |
2 4
o1 : Matrix ZZ <--- ZZ
i2 : P = convexHull V
o2 = {ambient dimension => 2 }
dimension of lineality space => 0
dimension of polyhedron => 2
number of facets => 3
number of rays => 0
number of vertices => 3
o2 : Polyhedron
i3 : vertices P
o3 = | 0 -2 2 |
| -1 1 1 |
2 3
o3 : Matrix QQ <--- QQ
i4 : (HS,v) = halfspaces P
o4 = (| -1 -1 |, | 1 |)
| 1 -1 | | 1 |
| 0 1 | | 1 |
o4 : Sequence
i5 : hyperplanes P
o5 = (0, 0)
o5 : Sequence
i6 : rays P
o6 = 0
2
o6 : Matrix QQ <--- 0
i7 : linSpace P
o7 = 0
2
o7 : Matrix QQ <--- 0
i8 : R = matrix {{1},{0},{0}}
o8 = | 1 |
| 0 |
| 0 |
3 1
o8 : Matrix ZZ <--- ZZ
i9 : V1 = V || matrix {{1,1,1,1}}
o9 = | 0 2 -2 0 |
| -1 1 1 1 |
| 1 1 1 1 |
3 4
o9 : Matrix ZZ <--- ZZ
i10 : P1 = convexHull(V1,R)
o10 = {ambient dimension => 3 }
dimension of lineality space => 0
dimension of polyhedron => 2
number of facets => 3
number of rays => 1
number of vertices => 2
o10 : Polyhedron
i11 : vertices P1
o11 = | 0 -2 |
| -1 1 |
| 1 1 |
3 2
o11 : Matrix QQ <--- QQ
i12 : rays P1
o12 = | 1 |
| 0 |
| 0 |
3 1
o12 : Matrix QQ <--- QQ
i13 : hyperplanes P1
o13 = (| 0 0 -1 |, | -1 |)
o13 : Sequence
i14 : HS = transpose (V || matrix {{-1,2,0,1}})
o14 = | 0 -1 -1 |
| 2 1 2 |
| -2 1 0 |
| 0 1 1 |
4 3
o14 : Matrix ZZ <--- ZZ
i15 : v = matrix {{1},{1},{1},{1}}
o15 = | 1 |
| 1 |
| 1 |
| 1 |
4 1
o15 : Matrix ZZ <--- ZZ
i16 : HP = matrix {{1,1,1}}
o16 = | 1 1 1 |
1 3
o16 : Matrix ZZ <--- ZZ
i17 : w = matrix {{3}}
o17 = | 3 |
1 1
o17 : Matrix ZZ <--- ZZ
i18 : P2 = intersection(HS,v,HP,w)
o18 = {ambient dimension => 3 }
dimension of lineality space => 0
dimension of polyhedron => 2
number of facets => 3
number of rays => 0
number of vertices => 3
o18 : Polyhedron
i19 : vertices P2
o19 = | 4 4 2 |
| 9 5 5 |
| -10 -6 -4 |
3 3
o19 : Matrix QQ <--- QQ
i20 : P3 = intersection(HS,v)
o20 = {ambient dimension => 3 }
dimension of lineality space => 1
dimension of polyhedron => 3
number of facets => 3
number of rays => 0
number of vertices => 3
o20 : Polyhedron
i21 : vertices P3
o21 = | 0 0 0 |
| 1 1 -3 |
| 0 -2 2 |
3 3
o21 : Matrix QQ <--- QQ
i22 : linSpace P3
o22 = | 1 |
| 2 |
| -2 |
3 1
o22 : Matrix QQ <--- QQ
i23 : P4 = hypercube(3,2)
o23 = {ambient dimension => 3 }
dimension of lineality space => 0
dimension of polyhedron => 3
number of facets => 6
number of rays => 0
number of vertices => 8
o23 : Polyhedron
i24 : vertices P4
o24 = | -2 2 -2 2 -2 2 -2 2 |
| -2 -2 2 2 -2 -2 2 2 |
| -2 -2 -2 -2 2 2 2 2 |
3 8
o24 : Matrix QQ <--- QQ
i25 : P5 = crossPolytope(3,3)
o25 = {ambient dimension => 3 }
dimension of lineality space => 0
dimension of polyhedron => 3
number of facets => 8
number of rays => 0
number of vertices => 6
o25 : Polyhedron
i26 : vertices P5
o26 = | -3 3 0 0 0 0 |
| 0 0 -3 3 0 0 |
| 0 0 0 0 -3 3 |
3 6
o26 : Matrix QQ <--- QQ
i27 : P6 = stdSimplex 2
o27 = {ambient dimension => 3 }
dimension of lineality space => 0
dimension of polyhedron => 2
number of facets => 3
number of rays => 0
number of vertices => 3
o27 : Polyhedron
i28 : vertices P6
o28 = | 1 0 0 |
| 0 1 0 |
| 0 0 1 |
3 3
o28 : Matrix QQ <--- QQ
i29 : P7 = convexHull(P4,P5)
o29 = {ambient dimension => 3 }
dimension of lineality space => 0
dimension of polyhedron => 3
number of facets => 24
number of rays => 0
number of vertices => 14
o29 : Polyhedron
i30 : vertices P7
o30 = | -3 3 0 0 0 -2 2 -2 2 -2 2 -2 2 0 |
| 0 0 -3 3 0 -2 -2 2 2 -2 -2 2 2 0 |
| 0 0 0 0 -3 -2 -2 -2 -2 2 2 2 2 3 |
3 14
o30 : Matrix QQ <--- QQ
i31 : P8 = intersection(P4,P5)
o31 = {ambient dimension => 3 }
dimension of lineality space => 0
dimension of polyhedron => 3
number of facets => 14
number of rays => 0
number of vertices => 24
o31 : Polyhedron
i32 : vertices P8
o32 = | -1 1 -2 2 -2 2 -1 1 -1 1 0 0 -2 2 0 0 -2 2 0 0 -1 1 0 0 |
| -2 -2 -1 -1 1 1 2 2 0 0 -1 1 0 0 -2 2 0 0 -2 2 0 0 -1 1 |
| 0 0 0 0 0 0 0 0 -2 -2 -2 -2 -1 -1 -1 -1 1 1 1 1 2 2 2 2 |
3 24
o32 : Matrix QQ <--- QQ
i33 : P9 = convexHull {(V1,R),P2,P6}
o33 = {ambient dimension => 3 }
dimension of lineality space => 0
dimension of polyhedron => 3
number of facets => 8
number of rays => 1
number of vertices => 5
o33 : Polyhedron
i34 : vertices P9
o34 = | 4 4 2 0 -2 |
| 9 5 5 -1 1 |
| -10 -6 -4 1 1 |
3 5
o34 : Matrix QQ <--- QQ
i35 : Q = convexHull (-V)
o35 = {ambient dimension => 2 }
dimension of lineality space => 0
dimension of polyhedron => 2
number of facets => 3
number of rays => 0
number of vertices => 3
o35 : Polyhedron
i36 : P10 = P + Q
o36 = {ambient dimension => 2 }
dimension of lineality space => 0
dimension of polyhedron => 2
number of facets => 6
number of rays => 0
number of vertices => 6
o36 : Polyhedron
i37 : vertices P10
o37 = | -4 4 -2 2 -2 2 |
| 0 0 -2 -2 2 2 |
2 6
o37 : Matrix QQ <--- QQ
i38 : (C,L,M) = minkSummandCone P10
o38 = ({ambient dimension => 6 }, HashTable{0 =>
dimension of lineality space => 0
dimension of the cone => 4
number of facets => 6
number of rays => 5
1 =>
2 =>
3 =>
4 =>
-----------------------------------------------------------------------
{ambient dimension => 2 }}, | 1 0 |)
dimension of lineality space => 0 | 0 1 |
dimension of polyhedron => 1 | 1 0 |
number of facets => 2 | 1 0 |
number of rays => 0 | 0 1 |
number of vertices => 2
{ambient dimension => 2 }
dimension of lineality space => 0
dimension of polyhedron => 2
number of facets => 3
number of rays => 0
number of vertices => 3
{ambient dimension => 2 }
dimension of lineality space => 0
dimension of polyhedron => 1
number of facets => 2
number of rays => 0
number of vertices => 2
{ambient dimension => 2 }
dimension of lineality space => 0
dimension of polyhedron => 1
number of facets => 2
number of rays => 0
number of vertices => 2
{ambient dimension => 2 }
dimension of lineality space => 0
dimension of polyhedron => 2
number of facets => 3
number of rays => 0
number of vertices => 3
o38 : Sequence
i39 : apply(values L, vertices)
o39 = {| 0 4 |, | 0 4 2 |, | 0 2 |, | 0 2 |, | 0 4 2 |}
| 0 0 | | 0 0 -2 | | 0 2 | | 0 -2 | | 0 0 2 |
o39 : List
i40 : P11 = P * Q
o40 = {ambient dimension => 4 }
dimension of lineality space => 0
dimension of polyhedron => 4
number of facets => 6
number of rays => 0
number of vertices => 9
o40 : Polyhedron
i41 : vertices P11
o41 = | 0 -2 2 0 -2 2 0 -2 2 |
| -1 1 1 -1 1 1 -1 1 1 |
| -2 -2 -2 2 2 2 0 0 0 |
| -1 -1 -1 -1 -1 -1 1 1 1 |
4 9
o41 : Matrix QQ <--- QQ
i42 : ambDim P11
o42 = 4
i43 : fVector P11
o43 = {9, 18, 15, 6, 1}
o43 : List
i44 : L = faces(1,P11)
o44 = {{ambient dimension => 4 }, {ambient dimension => 4
dimension of lineality space => 0 dimension of lineality space =>
dimension of polyhedron => 3 dimension of polyhedron => 3
number of facets => 5 number of facets => 5
number of rays => 0 number of rays => 0
number of vertices => 6 number of vertices => 6
-----------------------------------------------------------------------
}, {ambient dimension => 4 }, {ambient dimension => 4
0 dimension of lineality space => 0 dimension of lineality space
dimension of polyhedron => 3 dimension of polyhedron => 3
number of facets => 5 number of facets => 5
number of rays => 0 number of rays => 0
number of vertices => 6 number of vertices => 6
-----------------------------------------------------------------------
}, {ambient dimension => 4 },
=> 0 dimension of lineality space => 0
dimension of polyhedron => 3
number of facets => 5
number of rays => 0
number of vertices => 6
-----------------------------------------------------------------------
{ambient dimension => 4 }}
dimension of lineality space => 0
dimension of polyhedron => 3
number of facets => 5
number of rays => 0
number of vertices => 6
o44 : List
i45 : apply(L,vertices)
o45 = {| 0 -2 0 -2 0 -2 |, | 0 2 0 2 0 2 |, | -2 2 -2 2 -2 2 |, |
| -1 1 -1 1 -1 1 | | -1 1 -1 1 -1 1 | | 1 1 1 1 1 1 | |
| -2 -2 2 2 0 0 | | -2 -2 2 2 0 0 | | -2 -2 2 2 0 0 | |
| -1 -1 -1 -1 1 1 | | -1 -1 -1 -1 1 1 | | -1 -1 -1 -1 1 1 | |
-----------------------------------------------------------------------
0 -2 2 0 -2 2 |, | 0 -2 2 0 -2 2 |, | 0 -2 2 0 -2 2 |}
-1 1 1 -1 1 1 | | -1 1 1 -1 1 1 | | -1 1 1 -1 1 1 |
-2 -2 -2 2 2 2 | | -2 -2 -2 0 0 0 | | 2 2 2 0 0 0 |
-1 -1 -1 -1 -1 -1 | | -1 -1 -1 1 1 1 | | -1 -1 -1 1 1 1 |
o45 : List
i46 : L = latticePoints P11
o46 = {| 0 |, | 0 |, | 0 |, | 0 |, | 0 |, | 0 |, | 0 |, | 0 |, | 0
| -1 | | -1 | | -1 | | -1 | | -1 | | -1 | | -1 | | -1 | | -1
| -2 | | -1 | | 0 | | 1 | | 2 | | -1 | | 0 | | 1 | | 0
| -1 | | -1 | | -1 | | -1 | | -1 | | 0 | | 0 | | 0 | | 1
-----------------------------------------------------------------------
|, | -1 |, | -1 |, | -1 |, | -1 |, | -1 |, | -1 |, | -1 |, | -1 |, | -1
| | 0 | | 0 | | 0 | | 0 | | 0 | | 0 | | 0 | | 0 | | 0
| | -2 | | -1 | | 0 | | 1 | | 2 | | -1 | | 0 | | 1 | | 0
| | -1 | | -1 | | -1 | | -1 | | -1 | | 0 | | 0 | | 0 | | 1
-----------------------------------------------------------------------
|, | 0 |, | 0 |, | 0 |, | 0 |, | 0 |, | 0 |, 0, | 0 |, | 0 |, |
| | 0 | | 0 | | 0 | | 0 | | 0 | | 0 | | 0 | | 0 | |
| | -2 | | -1 | | 0 | | 1 | | 2 | | -1 | | 1 | | 0 | |
| | -1 | | -1 | | -1 | | -1 | | -1 | | 0 | | 0 | | 1 | |
-----------------------------------------------------------------------
1 |, | 1 |, | 1 |, | 1 |, | 1 |, | 1 |, | 1 |, | 1 |, | 1 |, | -2
0 | | 0 | | 0 | | 0 | | 0 | | 0 | | 0 | | 0 | | 0 | | 1
-2 | | -1 | | 0 | | 1 | | 2 | | -1 | | 0 | | 1 | | 0 | | -2
-1 | | -1 | | -1 | | -1 | | -1 | | 0 | | 0 | | 0 | | 1 | | -1
-----------------------------------------------------------------------
|, | -2 |, | -2 |, | -2 |, | -2 |, | -1 |, | -1 |, | -1 |, | -1 |, | -1
| | 1 | | 1 | | 1 | | 1 | | 1 | | 1 | | 1 | | 1 | | 1
| | -1 | | 0 | | 1 | | 2 | | -2 | | -1 | | 0 | | 1 | | 2
| | -1 | | -1 | | -1 | | -1 | | -1 | | -1 | | -1 | | -1 | | -1
-----------------------------------------------------------------------
|, | 0 |, | 0 |, | 0 |, | 0 |, | 0 |, | 1 |, | 1 |, | 1 |, | 1
| | 1 | | 1 | | 1 | | 1 | | 1 | | 1 | | 1 | | 1 | | 1
| | -2 | | -1 | | 0 | | 1 | | 2 | | -2 | | -1 | | 0 | | 1
| | -1 | | -1 | | -1 | | -1 | | -1 | | -1 | | -1 | | -1 | | -1
-----------------------------------------------------------------------
|, | 1 |, | 2 |, | 2 |, | 2 |, | 2 |, | 2 |, | -2 |, | -1 |, | 0
| | 1 | | 1 | | 1 | | 1 | | 1 | | 1 | | 1 | | 1 | | 1
| | 2 | | -2 | | -1 | | 0 | | 1 | | 2 | | -1 | | -1 | | -1
| | -1 | | -1 | | -1 | | -1 | | -1 | | -1 | | 0 | | 0 | | 0
-----------------------------------------------------------------------
|, | 1 |, | 2 |, | -2 |, | -1 |, | 0 |, | 1 |, | 2 |, | -2 |, | -1 |,
| | 1 | | 1 | | 1 | | 1 | | 1 | | 1 | | 1 | | 1 | | 1 |
| | -1 | | -1 | | 0 | | 0 | | 0 | | 0 | | 0 | | 1 | | 1 |
| | 0 | | 0 | | 0 | | 0 | | 0 | | 0 | | 0 | | 0 | | 0 |
-----------------------------------------------------------------------
| 0 |, | 1 |, | 2 |, | -2 |, | -1 |, | 0 |, | 1 |, | 2 |}
| 1 | | 1 | | 1 | | 1 | | 1 | | 1 | | 1 | | 1 |
| 1 | | 1 | | 1 | | 0 | | 0 | | 0 | | 0 | | 0 |
| 0 | | 0 | | 0 | | 1 | | 1 | | 1 | | 1 | | 1 |
o46 : List
i47 : #L
o47 = 81
i48 : C = tailCone P1
o48 = {ambient dimension => 3 }
dimension of lineality space => 0
dimension of the cone => 1
number of facets => 1
number of rays => 1
o48 : Cone
i49 : rays C
o49 = | 1 |
| 0 |
| 0 |
3 1
o49 : Matrix QQ <--- QQ
i50 : P12 = polar P11
o50 = {ambient dimension => 4 }
dimension of lineality space => 0
dimension of polyhedron => 4
number of facets => 9
number of rays => 0
number of vertices => 6
o50 : Polyhedron
i51 : vertices P12
o51 = | 0 -1 1 0 0 0 |
| -1 1 1 0 0 0 |
| 0 0 0 -1 1 0 |
| 0 0 0 -1 -1 1 |
4 6
o51 : Matrix QQ <--- QQ
i52 :
