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Reference documentation for older polymake versions: release 3.4, release 3.3, release 3.2
BigObject PolyhedralComplex<Scalar>
from application fan
A polyhedral complex. The derivation from PolyhedralFan
works like the derivation of Polytope
from Cone
.
 Type Parameters:
Scalar
: numeric data type used for the coordinates, must be an ordered field. Default isRational
. derived from:
 Example:
The following defines a subdivision of a square in the plane into two triangles.
> $c=new PolyhedralComplex(VERTICES=>[[1,0,0],[1,1,0],[1,0,1],[1,1,1]],MAXIMAL_POLYTOPES=>[[0,1,2],[1,2,3]]);
Properties
Input property
These properties are for input only. They allow redundant information.

INPUT_POLYTOPES
Maybe redundant list of not necessarily maximal polytopes. Indices refer to
POINTS
. Each polytope must list all vertices ofPOINTS
it contains. The polytopes are allowed to contain lineality. An empty complex does not have any polytopes. Input section only. Ask forMAXIMAL_POLYTOPES
if you want to know the maximal polytopes (indexed byVERTICES
). Alias for propertyINPUT_CONES
. Type:
 Example:
We can define a polyhderal complex consisting of two distinct triangles with the following (note that additionally stating one side of one of these triangles does not affect our resulting complex):
> $c = new PolyhedralComplex(POINTS=>[[1,1,0],[1,1,1],[1,0,1],[1,1,0],[1,1,1],[1,0,1]],INPUT_POLYTOPES=>[[0,1,2],[3,4,5],[0,1]]);
print $c→MAXIMAL_POLYTOPES;

POINTS
Points in homogeneous coordinates from which the polytopes are formed. May be redundant. All vectors in the input must be nonzero. You also need to provide
INPUT_POLYTOPES
to define a complex completely. Input section only. Ask forVERTICES
if you want a list of nonredundant points. Alias for propertyINPUT_RAYS
. Type:
Matrix<Scalar,NonSymmetric>
 Example:
To obtain a complex consisting of two triangles we can do this (note that, contrary to a polytope, this complex is not convex):
> $c = new PolyhedralComplex(POINTS=>[[1,0,0],[1,1,0],[1,3,1],[1,0,1]],INPUT_POLYTOPES=>[[0,1,3],[1,2,3]]);
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.

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:
 Example:
The compactification of the positive orthant in three dimensions has the same Hasse diagram as the three dimensional cube.
> $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} {})

MAXIMAL_POLYTOPES
Non redundant list of maximal polytopes. Indices refer to
VERTICES
. An empty complex does not have any polytopes. Alias for propertyMAXIMAL_CONES
. Type:
 Example:
After creating a complex via the
INPUT_POLYTOPES
property, we can display all maximal polytopes rising from that definition:> $c = new PolyhedralComplex(POINTS=>[[1,0,0],[1,1,0],[1,1,1],[1,0,1]],INPUT_POLYTOPES=>[[0,1,2],[0,3],[2]]); > print $c>MAXIMAL_POLYTOPES; {0 1 2} {0 3}

MAXIMAL_POLYTOPES_COMBINATORIAL_DIMS
The combinatorial dimensions of the maximal polytopes. The ith entry refers to the ith entry of
MAXIMAL_POLYTOPES
. Alias for propertyMAXIMAL_CONES_COMBINATORIAL_DIMS
. Type:
 Example:
When connecting two vertices of a triangle to a vertex distinct from that triangle we receive a polyhedral complex with maximal polytopes of dimensions 2, 1 and 1, respectively:
> $c = new PolyhedralComplex(POINTS=>[[1,0,0],[1,1,0],[1,1,1],[1,0,1]],INPUT_POLYTOPES=>[[0,1,2],[0,3],[2,3]]); > print $c>MAXIMAL_POLYTOPES; {0 1 2} {0 3} {2 3}
> print $c>MAXIMAL_POLYTOPES_COMBINATORIAL_DIMS; 2 1 1

MAXIMAL_POLYTOPES_INCIDENCES
Array of incidence matrices of all maximal polytopes. Alias for property
MAXIMAL_CONES_INCIDENCES
. Type:
 Example:
Here we construct a polyhedral complex made of two triangles which share a side; this fact can afterwards be read from this property:
> $c = new PolyhedralComplex(POINTS=>[[1,0,0],[1,1,0],[1,1,1],[1,0,1]],INPUT_POLYTOPES=>[[0,1,2],[1,2,3]]); > print $c>MAXIMAL_POLYTOPES; {0 1 2} {1 2 3}
> print $c>MAXIMAL_POLYTOPES_INCIDENCES; <{1 2} {0 2} {0 1} > <{1 2} {2 3} {1 3} >

MAXIMAL_POLYTOPES_THRU_VERTICES
Alias for property
MAXIMAL_CONES_THRU_RAYS
. Type:

N_MAXIMAL_POLYTOPES
Number of
MAXIMAL_POLYTOPES
. Alias for propertyN_MAXIMAL_CONES
. Type:
 Example:
The number of maximal polytopes of a planar net of a polytope is the number of facets of that polytope; here we see this for the dodecahedron:
> $c = planar_net(dodecahedron()); > print $c>N_MAXIMAL_POLYTOPES; 12

N_POLYTOPES
Alias for property
N_CONES
. Type:

POLYTOPES
List of all polytopes of the complex of each dimension. Indices refer to
VERTICES
. Alias for propertyCONES
. Type:
 Example:
A complex whose only maximal polytope is a triangle also contains 3 line segments and 3 points:
> $c = new PolyhedralComplex(POINTS=>[[1,0,0],[1,1,0],[1,0,1]],INPUT_POLYTOPES=>[[0,1,2]]); > print $c>POLYTOPES; <{1} {2} {0} > <{1 2} {0 2} {0 1} > <{0 1 2} >
Geometry
These properties capture geometric information of the object. Geometric properties depend on geometric information of the object, like, e.g., vertices or facets.

AFFINE_HULL
# The possible linear span normals of all maximal polytopes. Empty if
PURE
andFULL_DIM
, i.e. each maximal polytope has the same dimension as the ambient space. Alias for propertyLINEAR_SPAN_NORMALS
. Type:
Matrix<Scalar,NonSymmetric>
 Example:
In the plane, when we construct a polyhedral complex with a 2dimensional and two 1dimensional maximal polytopes only the latter two will have a linear span with a normal in this ambient space:
> $c = new PolyhedralComplex(POINTS=>[[1,0,0],[1,1,0],[1,0,1],[1,1,0],[1,1,1]],INPUT_POLYTOPES=>[[0,1,2],[0,3],[0,4]]); > print $c>AFFINE_HULL; 0 0 1 0 1 1

BOUNDED
True if each object in
MAXIMAL_POLYTOPES
is bounded. Type:

FAR_VERTICES
Indices of vertices that are rays.
 Type:
 Example:
We construct a PolyhedralComplex consisting only of one unbounded
Polytope
which is the Minkowski sum of an interval and a cone orthogonal to this line. Such a Minkowski sum always has a far vertex:> $c = new PolyhedralComplex(POINTS=>[[1,0,0],[1,1,0],[0,0,1]],INPUT_POLYTOPES=>[[0,1,2]]); > print rows_numbered($c>VERTICES); 0:1 0 0 1:1 1 0 2:0 0 1
> print $c>FAR_VERTICES; {2}

MAXIMAL_POLYTOPES_AFFINE_HULL_NORMALS
A basis of the normal space for each maximal polytope. This uniquely determines the affine hull of the corresponding maximal polytope. Indices refer to
AFFINE_HULL
. Rows correspond toMAXIMAL_POLYTOPES
. An empty row corresponds to a fulldimensional cone. Alias for propertyMAXIMAL_CONES_LINEAR_SPAN_NORMALS
. Type:
 Example:
In the plane, when we construct a polyhedral complex with a 2dimensional and two 1dimensional maximal polytopes only the latter will have a linear span with a normal in this ambient space:
> $c = new PolyhedralComplex(POINTS=>[[1,0,0],[1,1,0],[1,0,1],[1,1,0],[1,1,1]],INPUT_POLYTOPES=>[[0,1,2],[0,3],[0,4]]); > print $c>MAXIMAL_POLYTOPES; {0 1 2} {0 3} {0 4}
> print rows_numbered($c>AFFINE_HULL); 0:0 0 1 1:0 1 1
> print $c>MAXIMAL_POLYTOPES_AFFINE_HULL_NORMALS; {} {0} {1}

MAXIMAL_POLYTOPES_FACETS
Tells for each maximal polytope what are its facet normals, thus implying the facets. Each row corresponds to a maximal polytope and each column to the row with the same index of
AFFINE_HULL
. A negative number means that the corresponding row ofAFFINE_HULL
has to be negated. Alias for propertyMAXIMAL_CONES_FACETS
. Type:
 Example:
Here we see the facet normals of the maximal polytopes of a complex made of two triangles (note that some facet normal appear to be redundant due to usage of homogeneous coordinates and the derivation from PolyhedralFan):
> $c = new PolyhedralComplex(POINTS=>[[1,0,0],[1,1,0],[1,1,1],[1,0,1]],INPUT_POLYTOPES=>[[0,1,2],[0,2,3]]); > print $c>MAXIMAL_POLYTOPES; {0 1 2} {0 2 3}
> print rows_numbered($c>FACET_NORMALS); 0:1 1 0 1:0 1 1 2:0 0 1 3:1 0 1 4:0 1 0
> print $c>MAXIMAL_POLYTOPES_FACETS; 1 1 1 0 0 0 1 0 1 1

N_POINTS
Number of
POINTS
. Alias for propertyN_INPUT_RAYS
. Type:
 Example:
In the plane, glueing two triangles together along one side gives us a complex with four vertices; nevertheless we can specify these two triangles using six points with redundancies:
> $c = new PolyhedralComplex(POINTS=>[[1,0,0],[1,1,0],[1,0,1],[1,1,0],[2,0,2],[1,1,1]],INPUT_POLYTOPES=>[[0,1,2],[3,4,5]]); > print $c>N_VERTICES; 4
> print $c>N_POINTS; 6

N_VERTICES
 Type:

VERTICES
Vertices from which the polytopes are formed. Nonredundant. Coexists with
LINEALITY_SPACE
. Alias for propertyRAYS
. Type:
Matrix<Scalar,NonSymmetric>
Symmetry
These properties capture information of the object that is concerned with the action of permutation groups.

GROUP
 derived from:
 Type:
 Properties of GROUP:

COORDINATE_ACTION
 Type:
 Properties of COORDINATE_ACTION:

N_POINTS_GENERATORS
Alias for property
N_INPUT_RAYS_GENERATORS
. Type:

N_VERTICES_GENERATORS
Alias for property
N_RAYS_GENERATORS
. Type:

POINTS_GENERATORS
Alias for property
INPUT_RAYS_GENERATORS
. Type:

VERTICES_GENERATORS
Alias for property
RAYS_GENERATORS
. Type:


INPUT_POLYTOPES_ACTION
Alias for property
INPUT_CONES_ACTION
. Type:

MATRIX_ACTION_ON_COMPLEX
 Type:
MatrixActionOnVectors<Scalar>
 Properties of MATRIX_ACTION_ON_COMPLEX:

VERTICES_GENERATORS
Alias for property
RAYS_GENERATORS
. Type:
Matrix<OrbitGeneratorScalarType,NonSymmetric>


MAXIMAL_POLYTOPES_ACTION
 derived from:
 Type:
 Properties of MAXIMAL_POLYTOPES_ACTION:

MAXIMAL_POLYTOPES_GENERATORS
Alias for property
MAXIMAL_CONES_GENERATORS
. Type:

N_POINTS_GENERATORS
Alias for property
N_INPUT_RAYS_GENERATORS
. Type:

N_VERTICES_GENERATORS
Alias for property
N_RAYS_GENERATORS
. Type:

POINTS_GENERATORS
Alias for property
INPUT_RAYS_GENERATORS
. Type:

VERTICES_GENERATORS
Alias for property
RAYS_GENERATORS
. Type:


POINTS_ACTION
Alias for property
INPUT_RAYS_ACTION
. Type:

POLYTOPES_ORBIT_SIZES
Alias for property
CONES_ORBIT_SIZES
. Type:

REPRESENTATIVE_MAXIMAL_POLYTOPES
Alias for property
REPRESENTATIVE_MAXIMAL_CONES
. Type:

REPRESENTATIVE_VERTICES
Alias for property
REPRESENTATIVE_RAYS
. Type:
Matrix<Scalar,NonSymmetric>

VERTICES_ACTION
Alias for property
RAYS_ACTION
. Type:

Visualization
These properties are for visualization.

POINT_LABELS
Unique names assigned to the
POINTS
. Similar toVERTEX_LABELS
forVERTICES
. Alias for propertyINPUT_RAY_LABELS
. Type:

VERTEX_LABELS
Unique names assigned to the
VERTICES
. If specified, they are shown by visualization tools instead of vertex indices. For a polyhedral complex built from scratch, you should create this property by yourself, either manually in a text editor, or with a client program. Alias for propertyRAY_LABELS
. Type:
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.

AMBIENT_DIM()
Returns the dimension of the ambient space.
 Returns:
 Example:
The ambient dimension of a point in the line is 1:
> $c = new PolyhedralComplex(POINTS=>[[1,0]],INPUT_POLYTOPES=>[[0]]); > print $c>AMBIENT_DIM; 1

DIM()
Returns the dimension of the linear space spanned by the complex.
 Returns:
 Example:
The dimension of a point in the line is 0:
> $c = new PolyhedralComplex(POINTS=>[[1,0]],INPUT_POLYTOPES=>[[0]]); > print $c>DIM; 0

polytope(Int i)
Returns the ith facet of the complex as a
Polytope
. Parameters:
Int
i
 Returns:
 Example:
The planar net of the 3dimensional cross polytope consists only of triangles (and the according adjacent lines and vertices); asking for any of its polytopes thus gives us a triangle:
> $c = planar_net(cross(3)); > $p = $c>polytope(5); > print rows_numbered($p>VERTICES); 0:1 0 0 1:1 0.707106781186547 1.22474487139159 2:1 0.707106781186549 1.22474487139159
Visualization
These methods are for visualization.

VISUAL_ORBIT_COLORED_GRAPH()
Visualizes the graph of a symmetric cone: All nodes belonging to one orbit get the same color.
 Options:
 option list
Visual::Graph::decorations
 Returns:
no category

VISUAL()
Visualizes the polyhedral complex.
 Options:
 option list
geometric_options
 Returns:

VISUAL_METRIC_TIGHT_SPAN()
UNDOCUMENTED
 Options:
 option list
Visual::Graph::TightSpanDecorations