Polymake Mini-Reference

Created especially for The minicourse "Actually Doing It!: A Hands-On Approach to Computational Polyhedral Geometry".

Starting POLYMAKE

Starting with version 2.9, from the Linux prompt, type polymake and press Enter. You should see

polytope >

and then you're in the polymake system. To exit, type "exit;" (without the quotes, but don't forget the semi-colon).

Construct new polytope by points

To construct a new polytope $p as the convex hull of some points

polytope > $p=new Polytope;

polytope > $p->POINTS=<<".";
polytope (2)> 1 0 0 0
polytope (3)> 1 2 0 0
polytope (4)> 1 8 1 0
polytope (5)> 1 1 1 0
polytope (6)> 1 0 -1 1
polytope (7)> .

NOTE: the "1" in front is a part of polymake's internal representation that we promise to explain later. The polytope defined by inputting the above is the convex hull of the points (0,0,0), (2,0,0), (8,1,0), (1,1,0), and (0,-1,1).

Construct new polytope by equations and inequalities

Here is a quick explanation of the order in which you must enter the coefficients of your normal vector and the constant term. Note, the constant term goes in front:

Inequalities: To write A x₁ + B x₂ + C x₃ + D ≥ 0, you must input this as D A B C.
For example, in the input below the first inequality is 1 x₁ + 0 x₂ + ... + 0 x₁₂ + 0 ≥ 0.
Equalities: To write A x₁ + B x₂ + C x₃ + D = 0, you must input this as D A B C.
For example, in the input below, the first equation is 1 x₁ + 0 x₂ + 0 x₃ + 1 x₄ + 0 x₅ + 0 x₆ + 1 x₇ + 0 x₈ + 0 x₉ + 1 x₁₀ + 0 x₁₁ + 0 x₁₂ + (-4) = 0.

$p=new Polytope;

$p->INEQUALITIES=<<".";
0 1 0 0 0 0 0 0 0 0 0 0 0
0 0 1 0 0 0 0 0 0 0 0 0 0
0 0 0 1 0 0 0 0 0 0 0 0 0
0 0 0 0 1 0 0 0 0 0 0 0 0
0 0 0 0 0 1 0 0 0 0 0 0 0
0 0 0 0 0 0 1 0 0 0 0 0 0
0 0 0 0 0 0 0 1 0 0 0 0 0
0 0 0 0 0 0 0 0 1 0 0 0 0
0 0 0 0 0 0 0 0 0 1 0 0 0
0 0 0 0 0 0 0 0 0 0 1 0 0
0 0 0 0 0 0 0 0 0 0 0 1 0
0 0 0 0 0 0 0 0 0 0 0 0 1
.

$p->EQUATIONS=<<".";
-4 1 0 0 1 0 0 1 0 0 1 0 0
-4 0 1 0 0 1 0 0 1 0 0 1 0
-4 0 0 1 0 0 1 0 0 1 0 0 1
-3 1 1 1 0 0 0 0 0 0 0 0 0
.

Example of a for-loop

for ($i=0; $i<50; $i++) { print rand_sphere(4,10)->N_FACETS; }

Affine hull

Gives a minimal list of equalities satisfied by all points in the polytope.

print $p->AFFINE_HULL;

Ambient Dimension

To see the ambient (description) dimension of polytope $p,

print $p->AMBIENT_DIM;

Birkhoff polytope

To construct the n by n Birkhoff polytope, run birkhoff(n). For example, the 4 by 4 Birkhoff polytope is assigned to the variable $p by

$p = birkhoff(4);

Bounded

Test to see if a polyhedron is bounded (i.e., is a polytope). An output of 1 means the polyhedron is bounded. An output of 0 means it is an unbounded polyhedron.

print $p->BOUNDED;

Cube

Constructs a d-dimensional cube. For example, the 3-cube:

$c=cube(3);

Cross Polytope

Constructs a d-dimensional cross polytope (this is dual to the d-cube). For example,

$p=cross(3);

Cyclic polytopes

For n > d ≥ 2, cyclic(d,n) constructs a d-dimensional cyclic polytope with n vertices. See example:

$p=cyclic(3,6);

Diameter

Graph theoretical diameter of GRAPH. The minimal number which guarantees that between any two vertices x and y, there is a path from x to y using no more than this many edges.

print $p->DIAMETER;

Dimension

print $p->DIM;

Exit

To quit the program polymake, type

exit;

Facets

Print a list of the facets. This is one part of the H-representation. See also FACETS.

print $p->FACETS;

f-vector

print $p->F_VECTOR;

Hypersimplex

For n > d ≥ 2, constructs a hypersimplex:

$p=hypersimplex(d,n);

Intersection

To assign $c as the intersection of two polytopes $a and $b that have been previously defined, run

$c = intersection($a,$b);

Lattice points

To see the number of lattice points, run

print $p->N_LATTICE_POINTS;

To see the coordinates of the lattice points of polytope $p, run

print $p->LATTICE_POINTS;

Minkowski Sum

To build the Minkowski sum of polytopes $a and $b, and store the result as $c, run

$c = minkowski_sum($a,$b);

Number of facets

print $p->N_FACETS;

Number of vertices

print $p->N_VERTICES;

To see the actual vertices, see VERTICES.

Permutahedron

Constructs a d-dimensional permutahedron. For example,

$p=permutahedron(3);

Polarization

To construct the dual of a polytope $r, first center it, then run polarize:

$thedual = polarize(center($r));

Random 0/1 polytope

To construct a random d-dimensional 0/1 polytope (the coordinates of all vertices are 0 or 1) with n vertices, run

$r=rand_sphere(d,n);

NOTE: n > d required!

Random polytope

To construct a random 3-dimensional polytope as the convex hull of 10 random points on a sphere, run

$r=rand_sphere(3,10);

Simplex

Constructs a d-dimensional simplex. For example,

$p=simplex(3);

Simple

Test to see if a polytope is simple (every vertex is contained in dimension-many facets). An output of 1 means the polytope is simple. An output of 0 means it is not simple.

print $p->SIMPLE;

Simplicial

Test to see if a polytope is simplicial (every facet is a simplex of one dimension lower). An output of 1 means the polytope is simplicial. An output of 0 means it is not simplicial.

print $p->SIMPLICIAL;