user_guide:tutorials:latest:apps_polytope

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 — 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​ to 1. + + + > $p = new Polytope(POINTS=>​[[1,​-1,​-1],​[1,​1,​-1],​[1,​-1,​1],​[1,​1,​1],​[1,​0,​0]]);​ + ​ + The ''​POINTS''​ can be any set of coordinates,​ they are not required to be irredundant nor vertices of their convex hull. To compute the actual vertices of our polytope, we do this: + + + > print$p->​VERTICES;​ + 1 -1 -1 + 1 1 -1 + 1 -1 1 + 1 1 1 + + ​ + You can also add a lineality space via the input property ''​INPUT_LINEALITY''​. + + + > $p2 = new Polytope(POINTS=>​[[1,​-1,​-1],​[1,​1,​-1],​[1,​-1,​1],​[1,​1,​1],​[1,​0,​0]],​INPUT_LINEALITY=>​[[0,​1,​0]]);​ + ​ + To take a look at what that thing looks like, you can use the ''​VISUAL''​ method: + + + >$p2->​VISUAL;​ + ​ + 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: + + + > $p3 = new Polytope<​Rational>​(VERTICES=>​[[1,​-1,​-1],​[1,​1,​-1],​[1,​-1,​1],​[1,​1,​1]],​ LINEALITY_SPACE=>​[]);​ + ​ + ==== H-Description ==== + + It is also possible to define a polytope as an intersection of finitely many halfspaces, i.e., a matrix of inequalities. + + An inequality a<​sub>​0​ + a<​sub>​1​ x<​sub>​1​ + ... + a<​sub>​d​ x<​sub>​d​ >= 0 is encoded as a row vector (a<​sub>​0,​a<​sub>​1,​...,​a<​sub>​d​),​ see also [[coordinates|Coordinates for Polyhedra]]. Here is an example: + + + >$p4 = new Polytope(INEQUALITIES=>​[[1,​1,​0],​[1,​0,​1],​[1,​-1,​0],​[1,​0,​-1],​[17,​1,​1]]);​ + ​ + To display the inequalities in a nice way, use the ''​print_constraints''​ method. + + + > print_constraints($p4->​INEQUALITIES);​ + 0: x1 >= -1 + 1: x2 >= -1 + 2: -x1 >= -1 + 3: -x2 >= -1 + 4: x1 + x2 >= -17 + + ​ + The last inequality means 17+x<​sub>​1​+x<​sub>​2​ <​html><​html>​≥<​html>​ 0, hence it does not represent a facet of the polytope. If you want to take a look at the acutal facets, do this: + + + > print$p4->​FACETS;​ + 1 1 0 + 1 0 1 + 1 -1 0 + 1 0 -1 + + ​ + If your polytope lies in an affine subspace then you can specify its equations via the input property ''​EQUATIONS''​. + + + > $p5 = new Polytope(INEQUALITIES=>​[[1,​1,​0,​0],​[1,​0,​1,​0],​[1,​-1,​0,​0],​[1,​0,​-1,​0]],​EQUATIONS=>​[[0,​0,​0,​1],​[0,​0,​0,​2]]);​ + ​ + Again, if you are sure that all your inequalities are facets, you can use the properties ''​FACETS''​ and ''​AFFINE_HULL''​ instead. Note that this pair of properties is dual to the pair ''​VERTICES''​ / ''​LINEALITY_SPACE''​ described above. + + ===== Convex Hulls ===== + + Of course, ''​polymake''​ can convert the V-description of a polytope to its H-description and vice versa. Depending on the individual configuration polymake chooses one of the several convex hull computing algorithms that have a ''​polymake''​ interface. Available algorithms are double description ([[http://​www.ifor.math.ethz.ch/​~fukuda/​cdd_home/​cdd.html|cdd]] of [[http://​bugseng.com/​products/​ppl|ppl]]),​ reverse search ([[http://​cgm.cs.mcgill.ca/​~avis/​C/​lrs.html|lrs]]),​ and beneath beyond (internal). It is also possible to specify explicitly which method to use by using the ''​prefer''​ command: + + + > prefer "​lrs"; ​ # use lrs until revoked by another '​prefer'​ or '​reset_preference "​lrs"'​ + >$p = new Polytope(POINTS=>​[[1,​1],​[1,​0]]);​ + > print $p->​FACETS;​ + polymake: used package lrs + Implementation of the reverse search algorithm of Avis and Fukuda. + Copyright by David Avis. + http://​cgm.cs.mcgill.ca/​~avis/​lrs.html + ​ + 1 -1 + 0 1 + ​ + ===== A Neighborly Cubical Polytope ===== + + ''​polymake''​ provides a variety of standard polytope constructions and transformations. This example construction introduces some of them. Check out the [[https://​polymake.org/​release_docs/​latest/​polytope.html|documentation]] for a comprehensive list. + + The goal is to construct a 4-dimensional cubical polytope which has the same graph as the 5-dimensional cube. It is an example of a //​neighborly cubical// polytope as constructed in + + * Joswig & Ziegler: Neighborly cubical polytopes. Discrete Comput. Geom. 24 (2000), no. 2-3, 325--344, [[http://​www.springerlink.com/​content/​m73pqv6kr80rw4b1/​|DOI 10.1007/​s004540010039]] + + This is the entire construction in a few lines of ''​polymake''​ code: + + + >$c1 = cube(2); + > $c2 = cube(2,2); + >$p1x2 = product($c1,​$c2);​ + > $p2x1 = product($c2,​$c1);​ + >$nc = conv($p1x2,​$p2x1);​ + ​ + Let us examine more closely what this is about. First we constructed a square ''​$c1''​ via calling the function ''​cube''​. The only parameter ''​2''​ is the dimension of the cube to be constructed. It is not obvious how the coordinates are chosen; so let us check. + + + > print$c1->​VERTICES;​ + 1 -1 -1 + 1 1 -1 + 1 -1 1 + 1 1 1 + + ​ + 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: + + + > print $c1->​VOLUME;​ + 4 + + ​ + Here the volume is the Euclidean volume of the ambient space. Hence the volume of a polytope which is not full-dimensional is always zero. + + The second polytope ''​$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 ''​$p1x2''​ as the cartesian product of the two squares. Clearly, this is a four-dimensional polytope which is combinatorially (even affinely) equivalent to a cube, but not congruent. This is easy to verify: + + + > print isomorphic($p1x2,​cube(4));​ + 1 + > print congruent($p1x2,​cube(4));​ + 0 + + ​ + Both return values are boolean, represented by the numbers ''​1''​ and ''​0'',​ respectively. This questions are decided via a reduction to a graph isomorphism problem which in turn is solved via ''​polymake'''​s interface to ''​nauty''​. + + The polytope ''​$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. + + + > print $p1x2->​VERTICES->​[0];​ + 1 -1 -1 -2 -2 + > print$p2x1->​VERTICES->​[0];​ + 1 -2 -2 -1 -1 + + ​ + 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. + + + > print $nc->​SIMPLE,​ " ",$nc->​SIMPLICIAL;​ + 0 0 + + ​ + 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. + + + > print $nc->​F_VECTOR;​ + 32 80 72 24 + + ​ + This is a first hint that our initial claim is indeed valid. The polytope constructed has 32 vertices and 80 = 32*5/2 edges, as many as the 5-dimensional cube: + + + > print cube(5)->​F_VECTOR;​ + 32 80 80 40 10 + + ​ + What is left is to check whether the vertex-edge graphs of the two polytopes actually are the same, and if all proper faces are combinatorially equivalent to cubes. + + + > print isomorphic($nc->​GRAPH->​ADJACENCY,​cube(5)->​GRAPH->​ADJACENCY);​ + 1 + > print \$nc->​CUBICAL;​ + 1 + + ​ + See the [[apps_graph|tutorial on graphs]] for more on that subject. +
• user_guide/tutorials/latest/apps_polytope.txt