A Longer Introduction to Polyhedral Computations in Sage

Author: Jean-Philippe Labbé <labbe@math.fu-berlin.de>

This tutorial aims to showcase some of the possibilities of Sage concerning polyhedral geometry and combinatorics.

The classic literature on the topic includes:

  • Convex Polytopes, Branko Grünbaum, [Gru1967]
  • An Introduction to Convex Polytopes, Arne Brøndsted, [Bro1983]
  • Lectures on Polytopes, Günter M. Ziegler, [Zie2007]

For the nomenclature, Sage tries to follow:

  • Handbook of Discrete and Computational Geometry, Chapter 15, [Goo2004]

Preparation of this document was supported in part by the OpenDreamKit project and written during the SageDays 84 in Olot (Spain).

Lecture 0: Basic definitions and constructions

A real \((k\times d)\)-matrix \(A\) and a real vector \(b\) in \(\mathbb{R}^d\) define a (convex) polyhedron \(P\) as the set of solutions of the system of linear inequalities:

\[A\cdot x + b \geq 0.\]

Each row of \(A\) defines a closed half-space of \(\mathbb{R}^d\). Hence a polyhedron is the intersection of finitely many closed half-spaces in \(\mathbb{R}^d\). The matrix \(A\) may contain equal rows, which may lead to a set of equalities satisfied by the polyhedron. If there are no redundant rows in the above definition, this definition is referred to as the \(\mathbf{H}\) -representation of a polyhedron.

A maximal affine subspace \(L\) contained in a polyhedron is a lineality space of \(P\). Fixing a point \(o\) of the lineality space \(L\) to act as the origin, one can write every point \(p\) inside a polyhedron as a combination

\[p = \ell +\sum_{i=1}^{n}\lambda_iv_i+\sum_{i=1}^{m}\mu_ir_i,\]

where \(\ell\in L\) (using \(o\) as the origin), \(\sum_{i=1}^n\lambda_i=1\), \(\lambda_i\geq0\), \(\mu_i\geq0\), and \(r_i\neq0\) for all \(0\leq i\leq m\) and the set of \(r_i\) ‘s are positively independent (the origin is not in their positive span). For a given point \(p\) there may be many equivalent ways to write the above using different sets \(\{v_i\}_{i=1}^{n}\) and \(\{r_i\}_{i=1}^{m}\). Hence we require the sets to be inclusion minimal sets such that we can get the above property equality for any point \(p\in P\).

The vectors \(v_i\) are called the vertices of \(P\) and the vectors \(r_i\) are called the rays of \(P\). This way to represent a polyhedron is referred to as the \(\mathbf{V}\) -representation of a polyhedron. The first sum represents the convex hull of the vertices \(v_i\) ‘s and the second sum represents a pointed polyhedral cone generated by finitely many rays.

When the lineality space and the rays are reduced to a point (i.e. no rays and no lines) the object is often referred to as a polytope.

Note

As mentioned in the documentation of the constructor when typing Polyhedron?:

You may either define it with vertex/ray/line or inequalities/equations data, but not both. Redundant data will automatically be removed (unless “minimize=False”), and the complementary representation will be computed.

Here is the documentation for the constructor function of Polyhedra.

\(V\)-representation

First, let’s define a polyhedron object as the convex hull of a set of points and some rays.

sage: P1 = Polyhedron(vertices = [[1, 0], [0, 1]], rays = [[1, 1]])
sage: P1
A 2-dimensional polyhedron in ZZ^2 defined as the convex hull of 2 vertices and 1 ray

The string representation already gives a lot of information:

  • the dimension of the polyhedron (the smallest affine space containing it)
  • the dimension of the space in which it is defined
  • the base ring (\(\mathbb{Z}^2\)) over which the polyhedron lives (this specifies the parent class, see Parents for Polyhedra)
  • the number of vertices
  • the number of rays

Of course, you want to know what this object looks like:

sage: P1.plot()
Graphics object consisting of 5 graphics primitives

We can also add a lineality space.

sage: P2 = Polyhedron(vertices = [[1/2, 0, 0], [0, 1/2, 0]],
....:                 rays = [[1, 1, 0]],
....:                 lines = [[0, 0, 1]])
sage: P2
A 3-dimensional polyhedron in QQ^3 defined as the convex hull of 2 vertices, 1 ray, 1 line
sage: P2.plot()
Graphics3d Object

Notice that the base ring changes because of the value \(\frac{1}{2}\). Indeed, Sage finds an appropriate ring to define the object.

sage: P1.parent()
Polyhedra in ZZ^2
sage: P2.parent()
Polyhedra in QQ^3

The chosen ring depends on the input format.

sage: P3 = Polyhedron(vertices = [[0.5, 0], [0, 0.5]])
sage: P3
A 1-dimensional polyhedron in RDF^2 defined as the convex hull of 2 vertices
sage: P3.parent()
Polyhedra in RDF^2

Warning

The base ring RDF should be used with care. As it is not an exact ring, certain computations may break or silently produce wrong results, for example when dealing with non-simplicial polyhedra.

The following example demonstrates the limitations of RDF.

sage: D = polytopes.dodecahedron()
sage: D
A 3-dimensional polyhedron in (Number Field in sqrt5 with defining polynomial x^2 - 5)^3 defined as the convex hull of 20 vertices
sage: D_RDF = Polyhedron(vertices = [n(v.vector(),digits=6) for v in D.vertices()], base_ring=RDF)
Traceback (most recent call last):
...
ValueError: *Error: Numerical inconsistency is found.  Use the GMP exact arithmetic.

If the input of the polyhedron consists of python float, it automatically converts the data to RDF:

sage: Polyhedron(vertices=[[float(1.1)]])
A 0-dimensional polyhedron in RDF^1 defined as the convex hull of 1 vertex

It is also possible to define polyhedron over algebraic numbers.

sage: sqrt_2 = AA(2)^(1/2)
sage: cbrt_2 = AA(2)^(1/3)
sage: timeit('Polyhedron(vertices = [[sqrt_2, 0], [0, cbrt_2]])')  # random
5 loops, best of 3: 43.2 ms per loop
sage: P4 = Polyhedron(vertices = [[sqrt_2, 0], [0, cbrt_2]]); P4
A 1-dimensional polyhedron in AA^2 defined as the convex hull of 2 vertices

There is another way to create a polyhedron over algebraic numbers:

sage: K.<a> = NumberField(x^2 - 2, embedding=AA(2)**(1/2))
sage: L.<b> = NumberField(x^3 - 2, embedding=AA(2)**(1/3))
sage: timeit('Polyhedron(vertices = [[a, 0], [0, b]])')  # random
5 loops, best of 3: 39.9 ms per loop
sage: P5 = Polyhedron(vertices = [[a, 0], [0, b]]); P5
A 1-dimensional polyhedron in AA^2 defined as the convex hull of 2 vertices

If the base ring is known it may be a good option to use the proper sage.rings.number_field.number_field.number_field.composite_fields():

sage: J = K.composite_fields(L)[0]
sage: timeit('Polyhedron(vertices = [[J(a), 0], [0, J(b)]])')  # random
25 loops, best of 3: 9.8 ms per loop
sage: P5_comp = Polyhedron(vertices = [[J(a), 0], [0, J(b)]]); P5_comp
A 1-dimensional polyhedron in (Number Field in ab with defining polynomial x^6 - 6*x^4 - 4*x^3 + 12*x^2 - 24*x - 4)^2 defined as the convex hull of 2 vertices

Polyhedral computations with the Symbolic Ring is not implemented. It is not possible to define a polyhedron over it:

sage: sqrt_2s = sqrt(2)
sage: cbrt_2s = 2^(1/3)
sage: Polyhedron(vertices = [[sqrt_2s, 0], [0, cbrt_2s]])
Traceback (most recent call last):
...
ValueError: the only allowed inexact ring is 'RDF' with backend 'cdd'

Similarly, it is not possible to create polyhedron objects over RR (no matter how many bits of precision).

sage: F45 = RealField(45)
sage: F100 = RealField(100)
sage: f = 1.1
sage: Polyhedron(vertices=[[F45(f)]])
Traceback (most recent call last):
...
ValueError: the only allowed inexact ring is 'RDF' with backend 'cdd'
sage: Polyhedron(vertices=[[F100(f)]])
Traceback (most recent call last):
...
ValueError: the only allowed inexact ring is 'RDF' with backend 'cdd'

There is one exception, when the number of bits of precision is 53, then the base ring is converted to RDF:

sage: F53 = RealField(53)
sage: Polyhedron(vertices=[[F53(f)]])
A 0-dimensional polyhedron in RDF^1 defined as the convex hull of 1 vertex
sage: type(Polyhedron(vertices=[[F53(f)]]))
<class 'sage.geometry.polyhedron.parent.Polyhedra_RDF_cdd_with_category.element_class'>

This behavior can be seen as wrong, but it allows the following to be acceptable by Sage:

sage: Polyhedron([(1.0, 2.3), (3.5, 2.0)])
A 1-dimensional polyhedron in RDF^2 defined as the convex hull of 2 vertices

without having specified the base ring RDF by the user.

\(H\)-representation

If a polyhedron object was constructed via a \(V\)-representation, Sage can provide the \(H\)-representation of the object.

sage: for h in P1.Hrepresentation():
....:     print(h)
An inequality (1, 1) x - 1 >= 0
An inequality (1, -1) x + 1 >= 0
An inequality (-1, 1) x + 1 >= 0

Each line gives a row of the matrix \(A\) and an entry of the vector \(b\). The variable \(x\) is a vector in the ambient space where P1 is defined. The \(H\)-representation may contain equations:

sage: P3.Hrepresentation()
(An inequality (-2.0, 0.0) x + 1.0 >= 0,
 An inequality (1.0, 0.0) x + 0.0 >= 0,
 An equation (1.0, 1.0) x - 0.5 == 0)

The construction of a polyhedron object via its \(H\)-representation, requires a precise format. Each inequality \((a_{i1}, \dots, a_{id})\cdot x + b_i \geq 0\) must be written as [b_i,a_i1, ..., a_id].

sage: P3_H = Polyhedron(ieqs = [[1.0, -2, 0], [0, 1, 0]], eqns = [[-0.5, 1, 1]])
sage: P3 == P3_H
True
sage: P3_H.Vrepresentation()
(A vertex at (0.0, 0.5), A vertex at (0.5, 0.0))

It is worth using the parameter eqns to shorten the construction of the object. In the following example, the first four rows are the negative of the second group of four rows.

sage: H = [[0, 0, 0, 0, 0, 0, 0, 0, 1],
....:  [0, 0, 0, 0, 0, 0, 1, 0, 0],
....:  [-2, 1, 1, 1, 1, 1, 0, 0, 0],
....:  [0, 0, 0, 0, 0, 0, 0, 1, 0],
....:  [0, 0, 0, 0, 0, 0, 0, 0, -1],
....:  [0, 0, 0, 0, 0, 0, -1, 0, 0],
....:  [2, -1, -1, -1, -1, -1, 0, 0, 0],
....:  [0, 0, 0, 0, 0, 0, 0, -1, 0],
....:  [2, -1, -1, -1, -1, 0, 0, 0, 0],
....:  [0, 0, 0, 0, 1, 0, 0, 0, 0],
....:  [0, 0, 0, 1, 0, 0, 0, 0, 0],
....:  [0, 0, 1, 0, 0, 0, 0, 0, 0],
....:  [-1, 1, 1, 1, 1, 0, 0, 0, 0],
....:  [1, 0, 0, -1, 0, 0, 0, 0, 0],
....:  [0, 1, 0, 0, 0, 0, 0, 0, 0],
....:  [1, 0, 0, 0, -1, 0, 0, 0, 0],
....:  [1, 0, -1, 0, 0, 0, 0, 0, 0],
....:  [1, -1, 0, 0, 0, 0, 0, 0, 0]]
sage: timeit('Polyhedron(ieqs = H)')  # random
125 loops, best of 3: 5.99 ms per loop
sage: timeit('Polyhedron(ieqs = H[8:], eqns = H[:4])')  # random
125 loops, best of 3: 4.78 ms per loop
sage: Polyhedron(ieqs = H) == Polyhedron(ieqs = H[8:], eqns = H[:4])
True

Of course, this is a toy example, but it is generally worth to preprocess the data before defining the polyhedron if possible.

Lecture 1: Representation objects

Many objects are related to the \(H\)- and \(V\)-representations. Sage has classes implemented for them.

\(H\)-representation

You can store the \(H\)-representation in a variable and use the inequalities and equalities as objects.

sage: P3_QQ = Polyhedron(vertices = [[0.5, 0], [0, 0.5]], base_ring=QQ)
sage: HRep = P3_QQ.Hrepresentation()
sage: H1 = HRep[0]; H1
An equation (2, 2) x - 1 == 0
sage: H2 = HRep[1]; H2
An inequality (0, -2) x + 1 >= 0
sage: H1.<tab>   # not tested
sage: H1.A()
(2, 2)
sage: H1.b()
-1
sage: H1.is_equation()
True
sage: H1.is_inequality()
False
sage: H1.contains(vector([0,0]))
False
sage: H2.contains(vector([0,0]))
True
sage: H1.is_incident(H2)
True

It is possible to obtain the different objects of the \(H\)-representation as follows.

sage: P3_QQ.equations()
(An equation (2, 2) x - 1 == 0,)
sage: P3_QQ.inequalities()
(An inequality (0, -2) x + 1 >= 0, An inequality (0, 1) x + 0 >= 0)

Note

It is recommended to use equations or equation_generator (and similarly for inequalities) if one wants to iterate over them instead of equations_list.

\(V\)-representation

Similarly, you can access vertices, rays and lines of the polyhedron.

sage: VRep = P2.Vrepresentation(); VRep
(A line in the direction (0, 0, 1),
 A vertex at (0, 1/2, 0),
 A vertex at (1/2, 0, 0),
 A ray in the direction (1, 1, 0))
sage: L = VRep[0]; L
A line in the direction (0, 0, 1)
sage: V = VRep[1]; V
A vertex at (0, 1/2, 0)
sage: R = VRep[3]; R
A ray in the direction (1, 1, 0)
sage: L.is_line()
True
sage: L.is_incident(V)
True
sage: R.is_incident(L)
False
sage: L.vector()
(0, 0, 1)
sage: V.vector()
(0, 1/2, 0)

It is possible to obtain the different objects of the \(V\)-representation as follows.

sage: P2.vertices()
(A vertex at (0, 1/2, 0), A vertex at (1/2, 0, 0))
sage: P2.rays()
(A ray in the direction (1, 1, 0),)
sage: P2.lines()
(A line in the direction (0, 0, 1),)

sage: P2.vertices_matrix()
[  0 1/2]
[1/2   0]
[  0   0]

Note

It is recommended to use vertices or vertex_generator (and similarly for rays and lines) if one wants to iterate over them instead of vertices_list.

Lecture 2: Backends for polyhedral computations

To deal with polyhedron objects, Sage currently has four backends available. These backends offer various functionalities and have their own specific strengths and limitations.

The default backend is ppl. Whenever one needs speed it is good to try out the different backends. The backend field is not specifically designed for dealing with extremal computations but can do computations in exact coordinates.

The cdd backend

In order to use a specific backend, we specify the backend parameter.

sage: P1_cdd = Polyhedron(vertices = [[1, 0], [0, 1]], rays = [[1, 1]], backend='cdd')
sage: P1_cdd
A 2-dimensional polyhedron in QQ^2 defined as the convex hull of 2 vertices and 1 ray

A priori, it seems that nothing changed, but …

sage: P1_cdd.parent()
Polyhedra in QQ^2

The polyhedron P1_cdd is now considered as a rational polyhedron by the backend cdd. We can also check the backend and the parent using type:

sage: type(P1_cdd)
<class 'sage.geometry.polyhedron.parent.Polyhedra_QQ_cdd_with_category.element_class'>
sage: type(P1)
<class 'sage.geometry.polyhedron.parent.Polyhedra_ZZ_ppl_with_category.element_class'>

We see

  • the backend used (ex: backend_cdd)
  • followed by a dot ‘.’
  • the parent (ex: Polyhedra_QQ) followed again by the backend,

and you can safely ignore the rest for the purpose of this tutorial.

The cdd backend accepts also entries in RDF:

sage: P3_cdd = Polyhedron(vertices = [[0.5, 0], [0, 0.5]], backend='cdd')
sage: P3_cdd
A 1-dimensional polyhedron in RDF^2 defined as the convex hull of 2 vertices

but not algebraic or symbolic values:

sage: P4_cdd = Polyhedron(vertices = [[sqrt_2, 0], [0, cbrt_2]], backend='cdd')
Traceback (most recent call last):
...
ValueError: No such backend (=cdd) implemented for given basering (=Algebraic Real Field).

sage: P5_cdd = Polyhedron(vertices = [[sqrt_2s, 0], [0, cbrt_2s]], backend='cdd')
Traceback (most recent call last):
...
ValueError: the only allowed inexact ring is 'RDF' with backend 'cdd'

It is possible to get the cdd format of any polyhedron object defined over \(\mathbb{Z}\), \(\mathbb{Q}\), or RDF:

sage: print(P1.cdd_Vrepresentation())
V-representation
begin
 3 3 rational
 0 1 1
 1 0 1
 1 1 0
end
sage: print(P3.cdd_Hrepresentation())
H-representation
linearity 1 1
begin
 3 3 real
 -0.5 1.0 1.0
 1.0 -2.0 0.0
 0.0 1.0 0.0
end

You can also write this data to a file using the method .write_cdd_Hrepresentation(filename) or .write_cdd_Vrepresentation(filename), where filename is a string containing a path to a file to be written.

The ppl backend

The default backend for polyhedron objects is ppl.

sage: type(P1)
<class 'sage.geometry.polyhedron.parent.Polyhedra_ZZ_ppl_with_category.element_class'>
sage: type(P2)
<class 'sage.geometry.polyhedron.parent.Polyhedra_QQ_ppl_with_category.element_class'>
sage: type(P3)  # has entries like 0.5
<class 'sage.geometry.polyhedron.parent.Polyhedra_RDF_cdd_with_category.element_class'>

As you see, it does not accepts values in RDF and the polyhedron constructor used the cdd backend.

The polymake backend

The polymake backend is provided when the experimental package polymake for sage is installed.

sage: p = Polyhedron(vertices=[(0,0),(1,0),(0,1)],             # optional - polymake
....:                rays=[(1,1)], lines=[],
....:                backend='polymake', base_ring=QQ)

An example with quadratic field:

sage: V = polytopes.dodecahedron().vertices_list()
sage: Polyhedron(vertices=V, backend='polymake')               # optional - polymake
A 3-dimensional polyhedron in (Number Field in sqrt5 with defining polynomial x^2 - 5)^3 defined as the convex hull of 20 vertices

The field backend

As it turns out, the rational numbers do not suffice to represent all combinatorial types of polytopes. For example, Perles constructed a \(8\)-dimensional polytope with \(12\) vertices which does not have a realization with rational coordinates, see Example 6.21 p. 172 of [Zie2007]. Furthermore, if one wants a realization to have specific geometric property, such as symmetry, one also sometimes need irrational coordinates.

The backend field provides the necessary tools to deal with such examples.

sage: type(D)
<class 'sage.geometry.polyhedron.parent.Polyhedra_field_with_category.element_class'>

Any time that the coordinates should be in an extension of the rational, the backend field is called.

sage: P4.parent()
Polyhedra in AA^2
sage: P5.parent()
Polyhedra in AA^2
sage: type(P4)
<class 'sage.geometry.polyhedron.parent.Polyhedra_field_with_category.element_class'>
sage: type(P5)
<class 'sage.geometry.polyhedron.parent.Polyhedra_field_with_category.element_class'>

The normaliz backend

The fourth backend is normaliz and is an optional Sage package.

sage: P1_normaliz = Polyhedron(vertices = [[1, 0], [0, 1]], rays = [[1, 1]], backend='normaliz')  # optional - pynormaliz
sage: type(P1_normaliz)                                                                           # optional - pynormaliz
<class 'sage.geometry.polyhedron.parent.Polyhedra_ZZ_normaliz_with_category.element_class'>
sage: P2_normaliz = Polyhedron(vertices = [[1/2, 0, 0], [0, 1/2, 0]],                             # optional - pynormaliz
....:                 rays = [[1, 1, 0]],
....:                 lines = [[0, 0, 1]], backend='normaliz')
sage: type(P2_normaliz)                                                                           # optional - pynormaliz
<class 'sage.geometry.polyhedron.parent.Polyhedra_QQ_normaliz_with_category.element_class'>

This backend does not work with RDF, or algebraic numbers or the Symbolic Ring:

sage: P3_normaliz = Polyhedron(vertices = [[0.5, 0], [0, 0.5]], backend='normaliz')             # optional - pynormaliz
Traceback (most recent call last):
...
ValueError: No such backend (=normaliz) implemented for given basering (=Real Double Field).

sage: P4_normaliz = Polyhedron(vertices = [[sqrt_2, 0], [0, cbrt_2]], backend='normaliz')       # optional - pynormaliz
Traceback (most recent call last):
...
ValueError: No such backend (=normaliz) implemented for given basering (=Algebraic Real Field).

sage: P5_normaliz = Polyhedron(vertices = [[sqrt_2s, 0], [0, cbrt_2s]], backend='normaliz')     # optional - pynormaliz
Traceback (most recent call last):
...
ValueError: the only allowed inexact ring is 'RDF' with backend 'cdd'

The backend normaliz provides other methods such as integral_hull, which also works on unbounded polyhedron.

sage: P6 = Polyhedron(vertices = [[0, 0], [3/2, 0], [3/2, 3/2], [0, 3]], backend='normaliz')  # optional - pynormaliz
sage: IH = P6.integral_hull(); IH                                                             # optional - pynormaliz
A 2-dimensional polyhedron in QQ^2 defined as the convex hull of 4 vertices
sage: P6.plot(color='blue')+IH.plot(color='red')                                              # optional - pynormaliz
Graphics object consisting of 12 graphics primitives
sage: P1_normaliz.integral_hull()                                                             # optional - pynormaliz
A 2-dimensional polyhedron in ZZ^2 defined as the convex hull of 2 vertices and 1 ray

Lecture 3: To every polyhedron, the proper parent class

In order to know all the methods that a polyhedron object has one has to look into its base class:

Don’t be surprised if the classes look empty! The classes mainly contain private methods that implement some comparison methods: to verify equality and inequality of numbers in the base ring and other internal functionalities.

To get a full overview of methods offered to you, Base class for polyhedra is the first place you want to go.

Lecture 4: A library of polytopes

There are a lot of polytopes that are readily available in the library, see Library of commonly used, famous, or interesting polytopes. Have a look at them to see if your polytope is already defined!

sage: A = polytopes.buckyball(); A  # can take long
A 3-dimensional polyhedron in (Number Field in sqrt5 with defining polynomial x^2 - 5)^3 defined as the convex hull of 60 vertices
sage: B = polytopes.cross_polytope(4); B
A 4-dimensional polyhedron in ZZ^4 defined as the convex hull of 8 vertices
sage: C = polytopes.cyclic_polytope(3,10); C
A 3-dimensional polyhedron in QQ^3 defined as the convex hull of 10 vertices
sage: E = polytopes.snub_cube(); E
A 3-dimensional polyhedron in RDF^3 defined as the convex hull of 24 vertices
sage: polytopes.<tab>  # not tested, to view all the possible polytopes

Bibliography

[Bro1983]Brondsted, A., An Introduction to Convex Polytopes, volume 90 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1983. ISBN 978-1-4612-7023-2
[Goo2004]J.E. Goodman and J. O’Rourke, editors, CRC Press LLC, Boca Raton, FL, 2004. ISBN 978-1584883012 (65 chapters, xvii + 1539 pages).
[Gru1967]Grünbaum, B., Convex polytopes, volume 221 of Graduate Texts in Mathematics. Springer-Verlag, New York, 2003. ISBN 978-1-4613-0019-9
[Zie2007](1, 2) Ziegler, G. M., Lectures on polytopes, volume 152 of Graduate Texts in Mathematics. Springer-Verlag, New York, 2007. ISBN 978-0-387-94365-7