user_guide:tutorials:latest:apps_polytope

Differences

This shows you the differences between two versions of the page.

Link to this comparison view

user_guide:tutorials:latest:apps_polytope [2020/01/22 09:02] (current)
Line 1: Line 1:
 +====== Tutorial on Polytopes ======
 +
 +A //​polytope//​ is the convex hull of finitely many points in some Euclidean space. Equivalently,​ a polytope is the bounded intersection of finitely many affine halfspaces. ''​polymake''​ can deal with polytopes in both representations and provides numerous tools for analysis.
 +
 +This tutorial first shows basic ways of defining a polytope from scratch. For larger input (e.g. from a file generated by some other program) have a look at our HowTo on [[data|loading data]] in ''​polymake''​.
 +
 +The second part demonstrates some of the tool ''​polymake''​ provides for handling polytopes by examining a small example. For a complete list of properties of polytopes and functions that ''​polymake''​ provides, see the [[reldocs>​latest/​polytope.html|polytope documentation]].
 +
 +===== Constructing a polytope from scratch =====
 +
 +==== V-Description ====
 +
 +To define a polytope as the convex hull of finitely many points, you can pass a matrix of coordinates to the constructor. Since ''​polymake''​ uses [[coordinates|homogeneous coordinates]],​ you need to set the additional coordinate x<​sub>​0</​sub>​ to 1.
 +
 +<code perl>
 +> $p = new Polytope(POINTS=>​[[1,​-1,​-1],​[1,​1,​-1],​[1,​-1,​1],​[1,​1,​1],​[1,​0,​0]]);​
 +</​code>​
 +The ''​POINTS''​ can be any set of coordinates,​ they are not required to be irredundant nor vertices of their convex hull. To compute the actual vertices of our polytope, we do this:
 +
 +<code perl>
 +> print $p->​VERTICES;​
 +1 -1 -1
 +1 1 -1
 +1 -1 1
 +1 1 1
 +
 +</​code>​
 +You can also add a lineality space via the input property ''​INPUT_LINEALITY''​.
 +
 +<code perl>
 +> $p2 = new Polytope(POINTS=>​[[1,​-1,​-1],​[1,​1,​-1],​[1,​-1,​1],​[1,​1,​1],​[1,​0,​0]],​INPUT_LINEALITY=>​[[0,​1,​0]]);​
 +</​code>​
 +To take a look at what that thing looks like, you can use the ''​VISUAL''​ method:
 +
 +<code perl>
 +> $p2->​VISUAL;​
 +</​code>​
 +See [[visual_tutorial#​application%20polytope|here]] for details on visualizing polytopes.
 +
 +If you are sure that all the points really are //extreme points// (vertices) and your description of the lineality space is complete, you can define the polytope via the properties ''​VERTICES''​ and ''​LINEALITY_SPACE''​ instead of ''​POINTS''​ and ''​INPUT_LINEALITY''​. This way, you can avoid unnecessary redundancy checks.
 +
 +The input properties ''​POINTS''​ / ''​INPUT_LINEALITY''​ may not be mixed with the properties ''​VERTICES''​ / ''​LINEALITY_SPACE''​. Furthermore,​ the ''​LINEALITY_SPACE''​ **must be specified** as soon as the property ''​VERTICES''​ is used:
 +
 +<code perl>
 +> $p3 = new Polytope<​Rational>​(VERTICES=>​[[1,​-1,​-1],​[1,​1,​-1],​[1,​-1,​1],​[1,​1,​1]],​ LINEALITY_SPACE=>​[]);​
 +</​code>​
 +==== H-Description ====
 +
 +It is also possible to define a polytope as an intersection of finitely many halfspaces, i.e., a matrix of inequalities.
 +
 +An inequality a<​sub>​0</​sub>​ + a<​sub>​1</​sub>​ x<​sub>​1</​sub>​ + ... + a<​sub>​d</​sub>​ x<​sub>​d</​sub>​ >= 0 is encoded as a row vector (a<​sub>​0</​sub>,​a<​sub>​1</​sub>,​...,​a<​sub>​d</​sub>​),​ see also [[coordinates|Coordinates for Polyhedra]]. Here is an example:
 +
 +<code perl>
 +> $p4 = new Polytope(INEQUALITIES=>​[[1,​1,​0],​[1,​0,​1],​[1,​-1,​0],​[1,​0,​-1],​[17,​1,​1]]);​
 +</​code>​
 +To display the inequalities in a nice way, use the ''​print_constraints''​ method.
 +
 +<code perl>
 +> print_constraints($p4->​INEQUALITIES);​
 +0: x1 >= -1
 +1: x2 >= -1
 +2: -x1 >= -1
 +3: -x2 >= -1
 +4: x1 + x2 >= -17
 +
 +</​code>​
 +The last inequality means 17+x<​sub>​1</​sub>​+x<​sub>​2</​sub>​ <​html><​html></​html>​≥<​html></​html></​html>​ 0, hence it does not represent a facet of the polytope. If you want to take a look at the acutal facets, do this:
 +
 +<code perl>
 +> print $p4->​FACETS;​
 +1 1 0
 +1 0 1
 +1 -1 0
 +1 0 -1
 +
 +</​code>​
 +If your polytope lies in an affine subspace then you can specify its equations via the input property ''​EQUATIONS''​.
 +
 +<code perl>
 +> $p5 = new Polytope(INEQUALITIES=>​[[1,​1,​0,​0],​[1,​0,​1,​0],​[1,​-1,​0,​0],​[1,​0,​-1,​0]],​EQUATIONS=>​[[0,​0,​0,​1],​[0,​0,​0,​2]]);​
 +</​code>​
 +Again, if you are sure that all your inequalities are facets, you can use the properties ''​FACETS''​ and ''​AFFINE_HULL''​ instead. Note that this pair of properties is dual to the pair ''​VERTICES''​ / ''​LINEALITY_SPACE''​ described above.
 +
 +===== Convex Hulls =====
 +
 +Of course, ''​polymake''​ can convert the V-description of a polytope to its H-description and vice versa. Depending on the individual configuration polymake chooses one of the several convex hull computing algorithms that have a ''​polymake''​ interface. Available algorithms are double description ([[http://​www.ifor.math.ethz.ch/​~fukuda/​cdd_home/​cdd.html|cdd]] of [[http://​bugseng.com/​products/​ppl|ppl]]),​ reverse search ([[http://​cgm.cs.mcgill.ca/​~avis/​C/​lrs.html|lrs]]),​ and beneath beyond (internal). It is also possible to specify explicitly which method to use by using the ''​prefer''​ command:
 +
 +<code perl>
 +> prefer "​lrs"; ​                             # use lrs until revoked by another '​prefer'​ or '​reset_preference "​lrs"'​
 +> $p = new Polytope(POINTS=>​[[1,​1],​[1,​0]]);​
 +> print $p->​FACETS;​
 +polymake: used package lrs
 +  Implementation of the reverse search algorithm of Avis and Fukuda.
 +  Copyright by David Avis.
 +  http://​cgm.cs.mcgill.ca/​~avis/​lrs.html
 +    ​
 +1 -1
 +0 1
 +</​code>​
 +===== A Neighborly Cubical Polytope =====
 +
 +''​polymake''​ provides a variety of standard polytope constructions and transformations. This example construction introduces some of them. Check out the [[https://​polymake.org/​release_docs/​latest/​polytope.html|documentation]] for a comprehensive list.
 +
 +The goal is to construct a 4-dimensional cubical polytope which has the same graph as the 5-dimensional cube. It is an example of a //​neighborly cubical// polytope as constructed in
 +
 +  * Joswig & Ziegler: Neighborly cubical polytopes. Discrete Comput. Geom. 24 (2000), no. 2-3, 325--344, [[http://​www.springerlink.com/​content/​m73pqv6kr80rw4b1/​|DOI 10.1007/​s004540010039]]
 +
 +This is the entire construction in a few lines of ''​polymake''​ code:
 +
 +<code perl>
 +> $c1 = cube(2);
 +> $c2 = cube(2,2);
 +> $p1x2 = product($c1,​$c2);​
 +> $p2x1 = product($c2,​$c1);​
 +> $nc = conv($p1x2,​$p2x1);​
 +</​code>​
 +Let us examine more closely what this is about. First we constructed a square ''​$c1''​ via calling the function ''​cube''​. The only parameter ''​2''​ is the dimension of the cube to be constructed. It is not obvious how the coordinates are chosen; so let us check.
 +
 +<code perl>
 +> print $c1->​VERTICES;​
 +1 -1 -1
 +1 1 -1
 +1 -1 1
 +1 1 1
 +
 +</​code>​
 +The four vertices are listed line by line in homogeneous coordinates,​ where the homogenizing coordinate is the leading one. As shown the vertices correspond to the four choices of ''​+/​-1''​ in two positions. So the area of this square equals four, which is verified as follows:
 +
 +<code perl>
 +> print $c1->​VOLUME;​
 +4
 +
 +</​code>​
 +Here the volume is the Euclidean volume of the ambient space. Hence the volume of a polytope which is not full-dimensional is always zero.
 +
 +The second polytope ''​$c2''​ constructed is also a square. However, the optional second parameter says that ''​+/​-2''​-coordinates are to be used rather than ''​+/​-1''​ as in the default case. The optional parameter is also allowed to be ''​0''​. In this case a cube with ''​0/​1''​-coordinates is returned. You can access the documentation of functions by typing their name in the ''​polymake''​ shell and then hitting F1.
 +
 +The third command constructs the polytope ''​$p1x2''​ as the cartesian product of the two squares. Clearly, this is a four-dimensional polytope which is combinatorially (even affinely) equivalent to a cube, but not congruent. This is easy to verify:
 +
 +<code perl>
 +> print isomorphic($p1x2,​cube(4));​
 +1
 +> print congruent($p1x2,​cube(4));​
 +0
 +
 +</​code>​
 +Both return values are boolean, represented by the numbers ''​1''​ and ''​0'',​ respectively. This questions are decided via a reduction to a graph isomorphism problem which in turn is solved via ''​polymake'''​s interface to ''​nauty''​.
 +
 +The polytope ''​$p2x1''​ does not differ that much from the previous. In fact, the construction is twice the same, except for the ordering of the factors in the call of the function ''​product''​. Let us compare the first vertices of the two products. One can see how the coordinates are induced by the ordering of the factors.
 +
 +<code perl>
 +> print $p1x2->​VERTICES->​[0];​
 +1 -1 -1 -2 -2
 +> print $p2x1->​VERTICES->​[0];​
 +1 -2 -2 -1 -1
 +
 +</​code>​
 +In fact, one of these two products is obtained from the other by exchanging coordinate directions. Thats is to say, they are congruent but distinct as subsets of Euclidean 4-space. This is why taking their joint convex hull yields something interesting. Let us explore what kind of polytope we got.
 +
 +<code perl>
 +> print $nc->​SIMPLE,​ " ", $nc->​SIMPLICIAL;​
 +0 0
 +
 +</​code>​
 +This says the polytope is neither simple nor simplicial. A good idea then is to look at the f-vector. Beware, however, this usually requires to build the entire face lattice of the polytope, which is extremely costly. Therefore this is computationally infeasible for most high-dimensional polytopes.
 +
 +<code perl>
 +> print $nc->​F_VECTOR;​
 +32 80 72 24
 +
 +</​code>​
 +This is a first hint that our initial claim is indeed valid. The polytope constructed has 32 vertices and 80 = 32*5/2 edges, as many as the 5-dimensional cube:
 +
 +<code perl>
 +> print cube(5)->​F_VECTOR;​
 +32 80 80 40 10
 +
 +</​code>​
 +What is left is to check whether the vertex-edge graphs of the two polytopes actually are the same, and if all proper faces are combinatorially equivalent to cubes.
 +
 +<code perl>
 +> print isomorphic($nc->​GRAPH->​ADJACENCY,​cube(5)->​GRAPH->​ADJACENCY);​
 +1
 +> print $nc->​CUBICAL;​
 +1
 +
 +</​code>​
 +See the [[apps_graph|tutorial on graphs]] for more on that subject.
 +
  
  • user_guide/tutorials/latest/apps_polytope.txt
  • Last modified: 2020/01/22 09:02
  • (external edit)