# PolytopeFace Class

This class handles all computations relating to faces of lattice polytopes.

info

Generally, objects of this class should not be constructed directly by the user. Instead, they should be created by the faces function of the Polytope class.

## Constructor​

### cytools.polytopeface.PolytopeFace​

Description: Constructs a PolytopeFace object describing a face of a lattice polytope. This is handled by the hidden __init__ function.

Arguments:

• ambient_poly (Polytope): The ambient polytope.
• vertices (array_like): The list of vertices.
• saturated_ineqs (frozenset): A frozenset containing the indices of the inequalities that this face saturates.
• dim (int, optional): The dimension of the face. If it is not given then it is computed.
Example

Since objects of this class should not be directly created by the end user, we demostrate how to construct these objects using the faces function of the Polytope class.

from cytools import Polytopep = Polytope([[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1],[-1,-1,-1,-1]])faces_3 = p.faces(3) # Find the 3-dimensional facesprint(faces_3[0]) # Print the first 3-face# A 3-dimensional face of a 4-dimensional polytope in ZZ^4

## Functions​

### ambient_dimension​

Description: Returns the dimension of the ambient lattice.

Arguments: None.

Returns: The dimension of the ambient lattice.

Aliases: ambient_dim.

### ambient_poly​

Description: Returns the ambient polytope.

Arguments: None.

Returns: The ambient polytope.

### as_polytope​

Description: Returns the face as a Polytope object.

Arguments: None.

Returns: The Polytope corresponding to the face.

Example

We construct a face object and then convert it into a Polytope object.

p = Polytope([[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1],[-1,-1,-1,-1]])f = p.faces(3)[0] # Pick one of the 3-facesf_poly = f.as_polytope()print(f_poly)# A 3-dimensional lattice polytope in ZZ^4

### boundary_points​

Description: Returns the boundary lattice points of the face.

Arguments:

• as_indices: Return the points as indices of the full list of points of the polytope.

Returns: The list of boundary lattice points of the face.

Aliases: boundary_pts.

Example

We construct a face object and find its boundary lattice points.

p = Polytope([[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1],[-1,-1,-1,-1]])f = p.faces(3)[0] # Pick one of the 3-facesf.boundary_points()# array([[-1, -1, -1, -1],#        [ 0,  0,  0,  1],#        [ 0,  0,  1,  0],#        [ 0,  1,  0,  0]])

### clear_cache​

Description: Clears the cached results of any previous computation.

Arguments: None.

Returns: Nothing.

Example

We construct a face object and find its lattice points, then we clear the cache and compute the points again.

p = Polytope([[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1],[-1,-1,-1,-1]])f = p.faces(3)[0] # Pick one of the 3-facespts = f.points() # Find the lattice pointsf.clear_cache() # Clears the results of any previos computationpts = f.points() # Find the lattice points again

### dimension​

Description: Returns the dimension of the face.

Arguments: None.

Returns: (int) The dimension of the face.

Aliases: dim.

### dual_face​

Description: Returns the dual face of the dual polytope.

note

This duality is only implemented for reflexive polytopes. An exception is raised if the polytope is not reflexive.

Arguments: None.

Returns: The dual face.

Aliases: dual.

Example

We construct a face object from a polytope, then find the dual face in the dual polytope.

p = Polytope([[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1],[-1,-1,-1,-1]])f = p.faces(2)[0] # Pick one of the 2-facesf_dual = f.dual_face()print(f_dual)# A 1-dimensional face of a 4-dimensional polytope in ZZ^4

### faces​

Description: Computes the faces of the face.

Arguments:

• d: Optional parameter that specifies the dimension of the desired faces.

Returns: A tuple of PolytopeFace objects of dimension d, if specified. Otherwise, a tuple of tuples of PolytopeFace objects organized in ascending dimension.

Example

We construct a face from a polytope and find its vertices.

p = Polytope([[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1],[-1,-1,-1,-1]])f = p.faces(3)[0] # Pick one of the 3-facesprint(f.faces(2)[0]) # Print one of its 2-faces# A 2-dimensional face of a 4-dimensional polytope in ZZ^4

### interior_points​

Description: Returns the interior lattice points of the face.

Arguments:

• as_indices: Return the points as indices of the full list of points of the polytope.

Returns: The list of interior lattice points of the face.

Aliases: interior_pts.

Example

We construct a face object and find its interior lattice points.

p = Polytope([[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1],[-1,-1,-6,-9]])f = p.faces(3)[2] # Pick one of the 3-facesf.interior_points()# array([[ 0,  0, -1, -2],#        [ 0,  0,  0, -1]])

### labels​

Description: Returns the labels of lattice points in the face.

Arguments: None.

Returns: The labels of lattice points in the face.

### labels_bdry​

Description: Returns the labels of boundary lattice points in the face.

Arguments: None.

Returns: The labels of boundary lattice points in the face.

### labels_int​

Description: Returns the labels of interior lattice points in the face.

Arguments: None.

Returns: The labels of interior lattice points in the face.

### labels_vertices​

Description: Returns the labels of vertices in the face.

Arguments: None.

Returns: The labels of vertices in the face.

### points​

Description: Returns the lattice points of the face.

Arguments:

• as_indices: Return the points as indices of the full list of points of the polytope.

Returns: (numpy.ndarray) The list of lattice points of the face.

Aliases: pts.

Example

We construct a face object and find its lattice points.

p = Polytope([[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1],[-1,-1,-1,-1]])f = p.faces(3)[0] # Pick one of the 3-facesf.points()# array([[-1, -1, -1, -1],#        [ 0,  0,  0,  1],#        [ 0,  0,  1,  0],#        [ 0,  1,  0,  0]])

### triangulate​

Description: Returns a single regular triangulation of the face.

Just a simple wrapper for the Triangulation constructor.

Also see Polytope.triangulate

Arguments:

• heights: A list of heights specifying the regular triangulation. When not specified, it will return the Delaunay triangulation when using CGAL, a triangulation obtained from random heights near the Delaunay when using QHull, or the placing triangulation when using TOPCOM. Heights can only be specified when using CGAL or QHull as the backend.
• simplices: A list of simplices specifying the triangulation. This is useful when a triangulation was previously computed and it needs to be used again. Note that the order of the points needs to be consistent with the order that the Polytope class uses.
• check_input_simplices: Flag that specifies whether to check if the input simplices define a valid triangulation.
• backend: Specifies the backend used to compute the triangulation. The available options are "qhull", "cgal", and "topcom". CGAL is the default one as it is very fast and robust.
• verbosity: The verbosity level.

Returns: A Triangulation object describing a triangulation of the polytope.

### vertices​

Description: Returns the vertices of the face.

Arguments: None.

Returns: The list of vertices of the face.

Example

We construct a face from a polytope and find its vertices.

p = Polytope([[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1],[-1,-1,-1,-1]])f = p.faces(2)[0] # Pick one of the 2-facesf.vertices()# array([[-1, -1, -1, -1],#        [ 0,  0,  0,  1],#        [ 0,  0,  1,  0]])

## Hidden Functions​

### __init__​

Description: Initializes a PolytopeFace object.

Arguments:

• ambient_poly: The ambient polytope.
• vert_labels: The vertices, specified by labels in ambient_poly.
• saturated_ineqs: Indices of inequalities that this face saturates.
• dim: The dimension of this face. If not given, then it's computed.

Returns: Nothing.

Example

This is the function that is called when creating a new PolytopeFace object. Thus, it is used in the following example.

from cytools import Polytopep = Polytope([[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1],[-1,-1,-1,-1]])faces_3 = p.faces(3) # Find the 3-dimensional facesprint(faces_3[0]) # Print the first 3-face# A 3-dimensional face of a 4-dimensional polytope in ZZ^4

### __repr__​

Description: Returns a string describing the face.

Arguments: None.

Returns: (str) A string describing the face.

Example

This function can be used to convert the face to a string or to print information about the face.

p = Polytope([[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1],[-1,-1,-1,-1]])f = p.faces(3)[0]face_info = str(f) # Converts to stringprint(f) # Prints face info

### _process_points​

Description: Grabs the labels of the lattice points of the face along with the indices of the hyperplane inequalities that they saturate.

Arguments: None.

Returns: Nothing.