====== BigObject PolyhedralComplex<Scalar> ======
//from application [[..:fan|fan]]//\\
\\
 A polyhedral complex.  The derivation from ''[[..:fan:PolyhedralFan |PolyhedralFan]]'' works like the derivation of ''[[..:polytope:Polytope |Polytope]]'' from ''[[..:polytope:Cone |Cone]]''.

  ? Type Parameters:
  :: ''Scalar'': numeric data type used for the coordinates, must be an ordered field.  Default is ''[[..:common#Rational |Rational]]''.
  ? derived from:
  : ''[[..:fan:PolyhedralFan |PolyhedralFan]]''
  ? Example:
  :: The following defines a subdivision of a square in the plane into two triangles. 
  :: <code perl> > $c=new PolyhedralComplex(VERTICES=>[[1,0,0],[1,1,0],[1,0,1],[1,1,1]],MAXIMAL_POLYTOPES=>[[0,1,2],[1,2,3]]);
</code>
===== Properties =====

==== Input property ====
 These properties are for input only. They allow redundant information.
----
{{anchor:input_polytopes:}}
  ?  **''INPUT_POLYTOPES''**
  :: Alias for property ''[[..:fan:PolyhedralFan#INPUT_CONES |INPUT_CONES]]''.
    ? Type:
    :''[[..:common#IncidenceMatrix |IncidenceMatrix]]<[[..:common#NonSymmetric |NonSymmetric]]>''


----
{{anchor:points:}}
  ?  **''POINTS''**
  :: Alias for property ''[[..:fan:PolyhedralFan#INPUT_RAYS |INPUT_RAYS]]''.
    ? Type:
    :''[[..:common#Matrix |Matrix]]<Scalar,[[..:common#NonSymmetric |NonSymmetric]]>''


----

==== Combinatorics ====
 These properties capture combinatorial information of the object.  Combinatorial properties only depend on combinatorial data of the object like, e.g., the face lattice.
----
{{anchor:compactification:}}
  ?  **''COMPACTIFICATION''**
  :: The Hasse diagram of the compactification of the polyhedral complex. For a simplicial polyhedral complex, this is the cubical compactification (or cubical complex, see [Omid Amini: "The combinatorial Chow ring of products of graphs"]). For tropical varieties, this is the tropical compactification, as in  [Brian Osserman and Joseph Rabinoff: "Lifting nonproper tropical intersections"]. The vertices of the compactification correspond to the faces of the original complex that have the same dimension as their recession cone. We call the face corresponding to a vertex the 'realisation' of the vertex. The decoration has four entries: 1. The face in the vertices of the compactification 2. The rank of the face 3. The realisation of the face. This is the union of the realisations of    the new vertices. 4. The sedentarity of the face. This is the intersection of the    sedentarities of the vertices.
    ? Type:
    :''[[..:graph:Lattice |Lattice]]<[[..:fan#SedentarityDecoration |SedentarityDecoration]],[[..:graph#Nonsequential |Nonsequential]]>''
    ? Example:
    :: The compactification of the positive orthant in three dimensions has the same Hasse diagram as the three dimensional cube.
    :: <code perl> > $pc1 = new PolyhedralComplex(POINTS=>[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]], INPUT_POLYTOPES=>[[0,1,2,3]]);
 > print $pc1->COMPACTIFICATION->DECORATION;
 ({} 0 {} {})
 ({0} 1 {0 1 2 3} {1 2 3})
 ({1} 1 {0 2 3} {2 3})
 ({2} 1 {0 1 3} {1 3})
 ({3} 1 {0 1 2} {1 2})
 ({4} 1 {0 3} {3})
 ({5} 1 {0 2} {2})
 ({6} 1 {0 1} {1})
 ({7} 1 {0} {})
 ({0 1} 2 {0 1 2 3} {2 3})
 ({0 2} 2 {0 1 2 3} {1 3})
 ({0 3} 2 {0 1 2 3} {1 2})
 ({1 4} 2 {0 2 3} {3})
 ({1 5} 2 {0 2 3} {2})
 ({2 4} 2 {0 1 3} {3})
 ({2 6} 2 {0 1 3} {1})
 ({3 5} 2 {0 1 2} {2})
 ({3 6} 2 {0 1 2} {1})
 ({4 7} 2 {0 3} {})
 ({5 7} 2 {0 2} {})
 ({6 7} 2 {0 1} {})
 ({0 1 2 4} 3 {0 1 2 3} {3})
 ({0 1 3 5} 3 {0 1 2 3} {2})
 ({0 2 3 6} 3 {0 1 2 3} {1})
 ({1 4 5 7} 3 {0 2 3} {})
 ({2 4 6 7} 3 {0 1 3} {})
 ({3 5 6 7} 3 {0 1 2} {})
 ({0 1 2 3 4 5 6 7} 4 {0 1 2 3} {})
 ({-1} 5 {-1} {})
</code>


----
{{anchor:maximal_polytopes:}}
  ?  **''MAXIMAL_POLYTOPES''**
  :: Alias for property ''[[..:fan:PolyhedralFan#MAXIMAL_CONES |MAXIMAL_CONES]]''.
    ? Type:
    :''[[..:common#IncidenceMatrix |IncidenceMatrix]]<[[..:common#NonSymmetric |NonSymmetric]]>''


----
{{anchor:maximal_polytopes_combinatorial_dims:}}
  ?  **''MAXIMAL_POLYTOPES_COMBINATORIAL_DIMS''**
  :: Alias for property ''[[..:fan:PolyhedralFan#MAXIMAL_CONES_COMBINATORIAL_DIMS |MAXIMAL_CONES_COMBINATORIAL_DIMS]]''.
    ? Type:
    :''[[..:common#Array |Array]]<[[..:common#Int |Int]]>''


----
{{anchor:maximal_polytopes_incidences:}}
  ?  **''MAXIMAL_POLYTOPES_INCIDENCES''**
  :: Array of incidence matrices of all [[..:fan:PolyhedralComplex#MAXIMAL_POLYTOPES |maximal polytopes]]. Alias for property ''[[..:fan:PolyhedralFan#MAXIMAL_CONES_INCIDENCES |MAXIMAL_CONES_INCIDENCES]]''.
    ? Type:
    :''[[..:common#Array |Array]]<[[..:common#IncidenceMatrix |IncidenceMatrix]]<[[..:common#NonSymmetric |NonSymmetric]]%%>>%%''


----
{{anchor:maximal_polytopes_thru_vertices:}}
  ?  **''MAXIMAL_POLYTOPES_THRU_VERTICES''**
  :: Alias for property ''[[..:fan:PolyhedralFan#MAXIMAL_CONES_THRU_RAYS |MAXIMAL_CONES_THRU_RAYS]]''.
    ? Type:
    :''[[..:common#IncidenceMatrix |IncidenceMatrix]]<[[..:common#NonSymmetric |NonSymmetric]]>''


----
{{anchor:n_maximal_polytopes:}}
  ?  **''N_MAXIMAL_POLYTOPES''**
  :: Number of ''[[..:fan:PolyhedralComplex#MAXIMAL_POLYTOPES |MAXIMAL_POLYTOPES]]''. Alias for property ''[[..:fan:PolyhedralFan#N_MAXIMAL_CONES |N_MAXIMAL_CONES]]''.
    ? Type:
    :''[[..:common#Int |Int]]''


----
{{anchor:n_polytopes:}}
  ?  **''N_POLYTOPES''**
  :: Alias for property ''[[..:fan:PolyhedralFan#N_CONES |N_CONES]]''.
    ? Type:
    :''[[..:common#Int |Int]]''


----
{{anchor:polytopes:}}
  ?  **''POLYTOPES''**
  :: Alias for property ''[[..:fan:PolyhedralFan#CONES |CONES]]''.
    ? Type:
    :''[[..:common#Array |Array]]<[[..:common#IncidenceMatrix |IncidenceMatrix]]<[[..:common#NonSymmetric |NonSymmetric]]%%>>%%''


----

==== Geometry ====
 These properties capture geometric information of the object.  Geometric properties depend on geometric information of the object, like, e.g., vertices or facets.
----
{{anchor:affine_hull:}}
  ?  **''AFFINE_HULL''**
  :: Alias for property ''[[..:fan:PolyhedralFan#LINEAR_SPAN_NORMALS |LINEAR_SPAN_NORMALS]]''.
    ? Type:
    :''[[..:common#Matrix |Matrix]]<Scalar,[[..:common#NonSymmetric |NonSymmetric]]>''


----
{{anchor:bounded:}}
  ?  **''BOUNDED''**
  :: True if each object in ''[[..:fan:PolyhedralComplex#MAXIMAL_POLYTOPES |MAXIMAL_POLYTOPES]]'' is [[..:polytope:Polytope#BOUNDED |bounded]].
    ? Type:
    :''[[..:common#Bool |Bool]]''


----
{{anchor:far_vertices:}}
  ?  **''FAR_VERTICES''**
  :: Indices of vertices that are rays.
    ? Type:
    :''[[..:common#Set |Set]]<[[..:common#Int |Int]]>''


----
{{anchor:maximal_polytopes_affine_hull_normals:}}
  ?  **''MAXIMAL_POLYTOPES_AFFINE_HULL_NORMALS''**
  :: Alias for property ''[[..:fan:PolyhedralFan#MAXIMAL_CONES_LINEAR_SPAN_NORMALS |MAXIMAL_CONES_LINEAR_SPAN_NORMALS]]''.
    ? Type:
    :''[[..:common#IncidenceMatrix |IncidenceMatrix]]<[[..:common#NonSymmetric |NonSymmetric]]>''


----
{{anchor:maximal_polytopes_facets:}}
  ?  **''MAXIMAL_POLYTOPES_FACETS''**
  :: Alias for property ''[[..:fan:PolyhedralFan#MAXIMAL_CONES_FACETS |MAXIMAL_CONES_FACETS]]''.
    ? Type:
    :''[[..:common#SparseMatrix |SparseMatrix]]<[[..:common#Int |Int]],[[..:common#NonSymmetric |NonSymmetric]]>''


----
{{anchor:n_points:}}
  ?  **''N_POINTS''**
  :: Number of ''[[..:fan:PolyhedralComplex#POINTS |POINTS]]''. Alias for property ''[[..:fan:PolyhedralFan#N_INPUT_RAYS |N_INPUT_RAYS]]''.
    ? Type:
    :''[[..:common#Int |Int]]''


----
{{anchor:n_vertices:}}
  ?  **''N_VERTICES''**
  :: Number of ''[[..:fan:PolyhedralComplex#VERTICES |VERTICES]]''. Alias for property ''[[..:fan:PolyhedralFan#N_RAYS |N_RAYS]]''.
    ? Type:
    :''[[..:common#Int |Int]]''


----
{{anchor:vertices:}}
  ?  **''VERTICES''**
  :: Alias for property ''[[..:fan:PolyhedralFan#RAYS |RAYS]]''.
    ? Type:
    :''[[..:common#Matrix |Matrix]]<Scalar,[[..:common#NonSymmetric |NonSymmetric]]>''


----

==== Symmetry ====
 These properties capture information of the object that is concerned with the action of permutation groups.
----
{{anchor:group:}}
  ?  **''GROUP''**
  ::
    ? derived from:
    : ''[[..:fan:PolyhedralFan#GROUP |GROUP]]''
    ? Type:
    :''[[..:group:Group |Group]]''
    ? Properties of GROUP:
    :
      ?  **''COORDINATE_ACTION''**
      ::
        ? Type:
        :''[[..:group:PermutationAction |PermutationAction]]<[[..:common#Int |Int]],[[..:common#Rational |Rational]]>''
        ? Properties of COORDINATE_ACTION:
        :
          ?  **''N_POINTS_GENERATORS''**
          :: Alias for property ''[[..:group:Action#N_INPUT_RAYS_GENERATORS |N_INPUT_RAYS_GENERATORS]]''.
            ? Type:
            :''[[..:common#Int |Int]]''
          ?  **''N_VERTICES_GENERATORS''**
          :: Alias for property ''[[..:group:Action#N_RAYS_GENERATORS |N_RAYS_GENERATORS]]''.
            ? Type:
            :''[[..:common#Int |Int]]''
          ?  **''POINTS_GENERATORS''**
          :: Alias for property ''[[..:group:Action#INPUT_RAYS_GENERATORS |INPUT_RAYS_GENERATORS]]''.
            ? Type:
            :''[[..:common#Matrix |Matrix]]<[[..:common#Rational |Rational]],[[..:common#NonSymmetric |NonSymmetric]]>''
          ?  **''VERTICES_GENERATORS''**
          :: Alias for property ''[[..:group:Action#RAYS_GENERATORS |RAYS_GENERATORS]]''.
            ? Type:
            :''[[..:common#Matrix |Matrix]]<[[..:common#Rational |Rational]],[[..:common#NonSymmetric |NonSymmetric]]>''
      ?  **''INPUT_POLYTOPES_ACTION''**
      :: Alias for property ''[[..:group:Group#INPUT_CONES_ACTION |INPUT_CONES_ACTION]]''.
        ? Type:
        :''[[..:group:PermutationAction |PermutationAction]]<[[..:common#Int |Int]],[[..:common#Rational |Rational]]>''
      ?  **''MATRIX_ACTION_ON_COMPLEX''**
      ::
        ? Type:
        :''[[..:group:MatrixActionOnVectors |MatrixActionOnVectors]]<Scalar>''
        ? Properties of MATRIX_ACTION_ON_COMPLEX:
        :
          ?  **''VERTICES_GENERATORS''**
          :: Alias for property ''[[..:group:Action#RAYS_GENERATORS |RAYS_GENERATORS]]''.
            ? Type:
            :''[[..:common#Matrix |Matrix]]<OrbitGeneratorScalarType,[[..:common#NonSymmetric |NonSymmetric]]>''
      ?  **''MAXIMAL_POLYTOPES_ACTION''**
      ::
        ? derived from:
        : ''[[..:fan:PolyhedralFan#MAXIMAL_CONES_ACTION |MAXIMAL_CONES_ACTION]]''
        ? Type:
        :''[[..:group:PermutationAction |PermutationAction]]<[[..:common#Int |Int]],[[..:common#Rational |Rational]]>''
        ? Properties of MAXIMAL_POLYTOPES_ACTION:
        :
          ?  **''MAXIMAL_POLYTOPES_GENERATORS''**
          :: Alias for property ''[[..:group:Action#MAXIMAL_CONES_GENERATORS |MAXIMAL_CONES_GENERATORS]]''.
            ? Type:
            :''[[..:common#IncidenceMatrix |IncidenceMatrix]]<[[..:common#NonSymmetric |NonSymmetric]]>''
          ?  **''N_POINTS_GENERATORS''**
          :: Alias for property ''[[..:group:Action#N_INPUT_RAYS_GENERATORS |N_INPUT_RAYS_GENERATORS]]''.
            ? Type:
            :''[[..:common#Int |Int]]''
          ?  **''N_VERTICES_GENERATORS''**
          :: Alias for property ''[[..:group:Action#N_RAYS_GENERATORS |N_RAYS_GENERATORS]]''.
            ? Type:
            :''[[..:common#Int |Int]]''
          ?  **''POINTS_GENERATORS''**
          :: Alias for property ''[[..:group:Action#INPUT_RAYS_GENERATORS |INPUT_RAYS_GENERATORS]]''.
            ? Type:
            :''[[..:common#Matrix |Matrix]]<[[..:common#Rational |Rational]],[[..:common#NonSymmetric |NonSymmetric]]>''
          ?  **''VERTICES_GENERATORS''**
          :: Alias for property ''[[..:group:Action#RAYS_GENERATORS |RAYS_GENERATORS]]''.
            ? Type:
            :''[[..:common#Matrix |Matrix]]<[[..:common#Rational |Rational]],[[..:common#NonSymmetric |NonSymmetric]]>''
      ?  **''POINTS_ACTION''**
      :: Alias for property ''[[..:group:Group#INPUT_RAYS_ACTION |INPUT_RAYS_ACTION]]''.
        ? Type:
        :''[[..:group:PermutationAction |PermutationAction]]<[[..:common#Int |Int]],[[..:common#Rational |Rational]]>''
      ?  **''REPRESENTATIVE_VERTICES''**
      :: Alias for property ''[[..:fan:PolyhedralFan#REPRESENTATIVE_RAYS |REPRESENTATIVE_RAYS]]''.
        ? Type:
        :''[[..:common#Matrix |Matrix]]<Scalar,[[..:common#NonSymmetric |NonSymmetric]]>''
      ?  **''VERTICES_ACTION''**
      :: Alias for property ''[[..:group:Group#RAYS_ACTION |RAYS_ACTION]]''.
        ? Type:
        :''[[..:group:PermutationAction |PermutationAction]]<[[..:common#Int |Int]],[[..:common#Rational |Rational]]>''


----
{{anchor:input_polytopes_reps:}}
  ?  **''INPUT_POLYTOPES_REPS''**
  :: Alias for property ''[[..:fan:PolyhedralFan#INPUT_CONES_REPS |INPUT_CONES_REPS]]''.
    ? Type:
    :''[[..:common#IncidenceMatrix |IncidenceMatrix]]<[[..:common#NonSymmetric |NonSymmetric]]>''


----
{{anchor:maximal_polytopes_in_orbits:}}
  ?  **''MAXIMAL_POLYTOPES_IN_ORBITS''**
  :: Alias for property ''[[..:fan:PolyhedralFan#MAXIMAL_CONES_IN_ORBITS |MAXIMAL_CONES_IN_ORBITS]]''.
    ? Type:
    :''[[..:common#Array |Array]]<[[..:common#Set |Set]]<[[..:common#Int |Int]]%%>>%%''


----
{{anchor:maximal_polytopes_orbit_sizes:}}
  ?  **''MAXIMAL_POLYTOPES_ORBIT_SIZES''**
  :: Alias for property ''[[..:fan:PolyhedralFan#MAXIMAL_CONES_ORBIT_SIZES |MAXIMAL_CONES_ORBIT_SIZES]]''.
    ? Type:
    :''[[..:common#Array |Array]]<[[..:common#Int |Int]]>''


----
{{anchor:maximal_polytopes_reps:}}
  ?  **''MAXIMAL_POLYTOPES_REPS''**
  :: Alias for property ''[[..:fan:PolyhedralFan#MAXIMAL_CONES_REPS |MAXIMAL_CONES_REPS]]''.
    ? Type:
    :''[[..:common#Array |Array]]<[[..:common#Set |Set]]<[[..:common#Int |Int]]%%>>%%''


----
{{anchor:maximal_polytopes_reps_affine_span_normals:}}
  ?  **''MAXIMAL_POLYTOPES_REPS_AFFINE_SPAN_NORMALS''**
  :: Alias for property ''[[..:fan:PolyhedralFan#MAXIMAL_CONES_REPS_LINEAR_SPAN_NORMALS |MAXIMAL_CONES_REPS_LINEAR_SPAN_NORMALS]]''.
    ? Type:
    :''[[..:common#IncidenceMatrix |IncidenceMatrix]]<[[..:common#NonSymmetric |NonSymmetric]]>''


----
{{anchor:maximal_polytopes_reps_dims:}}
  ?  **''MAXIMAL_POLYTOPES_REPS_DIMS''**
  :: Alias for property ''[[..:fan:PolyhedralFan#MAXIMAL_CONES_REPS_DIMS |MAXIMAL_CONES_REPS_DIMS]]''.
    ? Type:
    :''[[..:common#Array |Array]]<[[..:common#Int |Int]]>''


----
{{anchor:maximal_polytopes_reps_facets:}}
  ?  **''MAXIMAL_POLYTOPES_REPS_FACETS''**
  :: Alias for property ''[[..:fan:PolyhedralFan#MAXIMAL_CONES_REPS_FACETS |MAXIMAL_CONES_REPS_FACETS]]''.
    ? Type:
    :''[[..:common#SparseMatrix |SparseMatrix]]<[[..:common#Int |Int]],[[..:common#NonSymmetric |NonSymmetric]]>''


----
{{anchor:n_maximal_polytopes_orbits:}}
  ?  **''N_MAXIMAL_POLYTOPES_ORBITS''**
  :: Alias for property ''[[..:fan:PolyhedralFan#N_MAXIMAL_CONE_ORBITS |N_MAXIMAL_CONE_ORBITS]]''.
    ? Type:
    :''[[..:common#Int |Int]]''


----
{{anchor:n_vertices_orbits:}}
  ?  **''N_VERTICES_ORBITS''**
  :: Alias for property ''[[..:fan:PolyhedralFan#N_RAY_ORBITS |N_RAY_ORBITS]]''.
    ? Type:
    :''[[..:common#Int |Int]]''


----
{{anchor:points_reps:}}
  ?  **''POINTS_REPS''**
  :: Alias for property ''[[..:fan:PolyhedralFan#INPUT_RAYS_REPS |INPUT_RAYS_REPS]]''.
    ? Type:
    :''[[..:common#Matrix |Matrix]]<Scalar,[[..:common#NonSymmetric |NonSymmetric]]>''


----
{{anchor:polytopes_orbit_sizes:}}
  ?  **''POLYTOPES_ORBIT_SIZES''**
  :: Alias for property ''[[..:fan:PolyhedralFan#CONES_ORBIT_SIZES |CONES_ORBIT_SIZES]]''.
    ? Type:
    :''[[..:common#Array |Array]]<[[..:common#Array |Array]]<[[..:common#Int |Int]]%%>>%%''


----
{{anchor:polytopes_reps:}}
  ?  **''POLYTOPES_REPS''**
  :: Alias for property ''[[..:fan:PolyhedralFan#CONES_REPS |CONES_REPS]]''.
    ? Type:
    :''[[..:common#Array |Array]]<[[..:common#Array |Array]]<[[..:common#Set |Set]]<[[..:common#Int |Int]]%%>>%%>''


----
{{anchor:reps_affine_span_normals:}}
  ?  **''REPS_AFFINE_SPAN_NORMALS''**
  :: Alias for property ''[[..:fan:PolyhedralFan#REPS_LINEAR_SPAN_NORMALS |REPS_LINEAR_SPAN_NORMALS]]''.
    ? Type:
    :''[[..:common#Matrix |Matrix]]<Scalar,[[..:common#NonSymmetric |NonSymmetric]]>''


----
{{anchor:vertices_images:}}
  ?  **''VERTICES_IMAGES''**
  :: Alias for property ''[[..:fan:PolyhedralFan#RAYS_IMAGES |RAYS_IMAGES]]''.
    ? Type:
    :''[[..:common#Array |Array]]<[[..:common#Array |Array]]<[[..:common#Int |Int]]%%>>%%''


----
{{anchor:vertices_in_orbits:}}
  ?  **''VERTICES_IN_ORBITS''**
  :: Alias for property ''[[..:fan:PolyhedralFan#RAYS_IN_ORBITS |RAYS_IN_ORBITS]]''.
    ? Type:
    :''[[..:common#Array |Array]]<[[..:common#Set |Set]]<[[..:common#Int |Int]]%%>>%%''


----
{{anchor:vertices_orbit_sizes:}}
  ?  **''VERTICES_ORBIT_SIZES''**
  :: Alias for property ''[[..:fan:PolyhedralFan#RAYS_ORBIT_SIZES |RAYS_ORBIT_SIZES]]''.
    ? Type:
    :''[[..:common#Array |Array]]<[[..:common#Int |Int]]>''


----
{{anchor:vertices_reps:}}
  ?  **''VERTICES_REPS''**
  :: Alias for property ''[[..:fan:PolyhedralFan#RAYS_REPS |RAYS_REPS]]''.
    ? Type:
    :''[[..:common#Matrix |Matrix]]<Scalar,[[..:common#NonSymmetric |NonSymmetric]]>''


----
{{anchor:vertices_reps_labels:}}
  ?  **''VERTICES_REPS_LABELS''**
  :: Alias for property ''[[..:fan:PolyhedralFan#RAYS_REPS_LABELS |RAYS_REPS_LABELS]]''.
    ? Type:
    :''[[..:common#Array |Array]]<[[..:common#String |String]]>''


----

==== Visualization ====
 These properties are for visualization.
----
{{anchor:point_labels:}}
  ?  **''POINT_LABELS''**
  :: Alias for property ''[[..:fan:PolyhedralFan#INPUT_RAY_LABELS |INPUT_RAY_LABELS]]''.
    ? Type:
    :''[[..:common#Array |Array]]<[[..:common#String |String]]>''


----
{{anchor:vertex_labels:}}
  ?  **''VERTEX_LABELS''**
  :: Alias for property ''[[..:fan:PolyhedralFan#RAY_LABELS |RAY_LABELS]]''.
    ? Type:
    :''[[..:common#Array |Array]]<[[..:common#String |String]]>''


----
===== Methods =====

==== Geometry ====
 These methods capture geometric information of the object.  Geometric properties depend on geometric information of the object, like, e.g., vertices or facets.
----
{{anchor:ambient_dim:}}
  ?  **''AMBIENT_DIM()''**
  :: Returns the dimension of the ambient space.
    ? Returns:
    :''[[..:common#Int |Int]]''


----
{{anchor:dim:}}
  ?  **''DIM()''**
  :: Returns the dimension of the linear space spanned by the complex.
    ? Returns:
    :''[[..:common#Int |Int]]''


----
{{anchor:polytope:}}
  ?  **''polytope([[..:common#Int |Int]] i)''**
  :: Returns the //i//-th facet of the complex as a ''[[..:polytope:Polytope |Polytope]]''.
    ? Parameters:
    :: ''[[..:common#Int |Int]]'' ''i''
    ? Returns:
    :''[[..:polytope:Polytope |Polytope]]''


----

==== Visualization ====
 These methods are for visualization.
----
{{anchor:visual_metric_tight_span:}}
  ?  **''VISUAL_METRIC_TIGHT_SPAN()''**
  :: This is a variation of ''[[..:polytope:Polytope#VISUAL_BOUNDED_GRAPH |VISUAL_BOUNDED_GRAPH]]'' for the special case of a tight span. The vertices are embedded according to the metric, the others are hung in between. This only produces meaningful results for extended tight spans produced from metrics, e.g. through ''[[..:fan#metric_extended_tight_span |metric_extended_tight_span]]''.
    ? Options:
    : 
    :: ''[[..:common#Array |Array]]<[[..:common#String |String]]>'' ''Taxa'': Labels for the taxa of the metric.
    :: ''[[..:common#Int |Int]]'' ''seed'': random seed value for the string embedder
    :: ''[[..:common#String |String]]'' ''norm'': which norm to use when calculating the distances between metric vectors ("max" or "square")
    ? Returns:
    :''[[..:graph:Visual_Graph |Visual::Graph]]''


----
{{anchor:visual_orbit_colored_graph:}}
  ?  **''VISUAL_ORBIT_COLORED_GRAPH()''**
  :: Visualizes the graph of a symmetric cone: All nodes belonging to one orbit get the same color.
    ? Options:
    : option list ''[[..:graph#Visual_Graph_decorations |Visual::Graph::decorations]]''
    ? Returns:
    :''[[..:polytope:Visual_PolytopeGraph |Visual::PolytopeGraph]]''


----

==== no category ====
{{anchor:visual:}}
  ?  **''VISUAL()''**
  :: Visualizes the polyhedral complex.
    ? Options:
    : option list ''[[..:common#geometric_options |geometric_options]]''
    ? Returns:
    :''[[..:fan:Visual_PolyhedralFan |Visual::PolyhedralFan]]''


----