Skip to main content

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 Polytope
p = 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 faces
print(faces_3[0]) # Print the first 3-face
# A 3-dimensional face of a 4-dimensional polytope in ZZ^4

Functions

ambient_dim

Description: Returns the dimension of the ambient lattice.

Arguments: None.

Returns: (int) The dimension of the ambient lattice.

Example

We construct a face from a polytope and print its ambient dimension.

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-faces
f.ambient_dim()
# 4

ambient_polytope

Description: Returns the ambient polytope of the face.

Arguments: None.

Returns: (Polytope) The ambient polytope.

Example

We construct a face object from a polytope, then find the ambient polytope and verify that it is the starting 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(3)[0] # Pick one of the 3-faces
ambient_poly = f.ambient_polytope()
ambient_poly is p
# True

as_polytope

Description: Returns the face as a Polytope object.

Arguments: None.

Returns: (Polytope) 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-faces
f_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 (bool): Return the points as indices of the full list of points of the polytope.

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

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-faces
f.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-faces
pts = f.points() # Find the lattice points
f.clear_cache() # Clears the results of any previos computation
pts = f.points() # Find the lattice points again

dim

Description: Returns the dimension of the face.

Arguments: None.

Returns: (int) The dimension of the face.

Example

We construct a face from a polytope and print its dimension.

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-faces
f.dim()
# 3

dual

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: (PolytopeFace) The dual face.

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-faces
f_dual = f.dual()
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 (int, optional): Optional parameter that specifies the dimension of the desired faces.

Returns: (tuple) 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-faces
print(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 (bool): Return the points as indices of the full list of points of the polytope.

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

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-faces
f.interior_points()
# array([[ 0, 0, -1, -2],
# [ 0, 0, 0, -1]])

points

Description: Returns the lattice points of the face.

Arguments:

  • as_indices (bool): 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.

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-faces
f.points()
# array([[-1, -1, -1, -1],
# [ 0, 0, 0, 1],
# [ 0, 0, 1, 0],
# [ 0, 1, 0, 0]])

vertices

Description: Returns the vertices of the face.

Arguments: None.

Returns: (numpy.ndarray) 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-faces
f.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 (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.

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 Polytope
p = 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 faces
print(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 string
print(f) # Prints face info

_points_saturated

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

notes
  • Points are sorted in the same way as for the _points_saturated function of the Polytope class.
  • Typically this function should not be called by the user. Instead, it is called by various other functions in the PolytopeFace class.

Arguments: None.

Returns: (list) A list of tuples. The first component of each tuple is the list of coordinates of the point and the second component is a frozenset of the hyperplane inequalities that it saturates.

Example

We construct a face and compute the lattice points along with the inequalities that they saturate. We print the second point and the inequalities that it saturates.

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]
pts_sat = f._points_saturated()
print(pts_sat[1])
# ((0, 0, 0, 1), frozenset({0, 1, 2, 4}))
p.inequalities()[list(pts_sat[1][1])]
# array([[ 4, -1, -1, -1, 1],
# [-1, 4, -1, -1, 1],
# [-1, -1, 4, -1, 1],
# [-1, -1, -1, 4, 1]])