Differences

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--- tutorial:face_lattice_tutorial [2012/05/22 11:57] – [Dealing with Large Polytopes] joswig
+++ user_guide:tutorials:face_lattice_tutorial [2019/01/25 09:38] – ↷ Page moved from user_guide:face_lattice_tutorial to user_guide:tutorials:face_lattice_tutorial oroehrig
@@ Line 25: / Line 25: @@
 Very often just a part of the face lattice is interesting.  The following command lists just the 1-dimensional faces.
 <code>
-polytope > for (my $k=$HD->DIMS->[1]; $k<$HD->DIMS->[2]; ++$k) { print $HD->FACES->[$k] }
+polytope > print map { $p->HASSE_DIAGRAM->FACES->[$_] } @{$p->HASSE_DIAGRAM->nodes_of_dim(1)};
-{1 2}{2 3}{3 4}{0 4}{0 1}
+{2 3}{3 4}{0 4}{0 1}{1 2}
 </code>
-The above works as described since DIMS return an array with the first nodes of each dimension.
-Face lattices of polytopes can be huge.  So it may be an advantage to compute only a part.  The following computes the 2-skeleton of an 8-dimensional cube.
+Face lattices of polytopes can be huge.  So it may be an advantage to compute only a part.  The following computes the 2-skeleton of an 8-dimensional cube. We need to specify 3 as a limit since this function uses the rank which is one more than the dimension.
 <code>
 polytope > $c=cube(8);
-polytope > $HD_partial = hasse_diagram($c->VERTICES_IN_FACETS,2);
+polytope > $HD_partial = lower_hasse_diagram($c->VERTICES_IN_FACETS,3);
-polytope > print $HD_partial->DIMS;
+polytope > print $HD_partial->INVERSE_RANK_MAP;
-257 1281 3073
+{(0 (0 0)) (1 (1 256)) (2 (257 1280)) (3 (1281 3072)) (4 (3073 3073))}
 </code>
-Instead of listing all those thousands of faces here we give only the vector of starting nodes.  This can be treated just like above.  The (partial) f-vector is obtained by taking successive differences.
+Instead of listing all those thousands of faces here we give only the pairs indicating the start-node and end-node for each rank.
 <code>
-polytope > $partial_f = $HD_partial->DIMS;
+polytope > print map { $HD_partial->nodes_of_rank($_)->size," " } (1..3);
-polytope > for (my $i=1; $i<$partial_f->size(); ++$i) { print $partial_f->[$i]-$partial_f->[$i-1], " " }
 1024 1792
 </code>
@@ Line 65: / Line 63: @@
 The subsequent second stage looks as above; but the difference is that VERTICES_IN_FACETS is known already.
 <code>
-polytope > $HD_partial = hasse_diagram($p->VERTICES_IN_FACETS,2);
+polytope > $HD_partial = lower_hasse_diagram($p->VERTICES_IN_FACETS,3);
-polytope > print $HD_partial->DIMS;
+polytope > print map { $HD_partial->nodes_of_rank($_)->size," " } (1..3);
-101 2082 12534
+2004 10638
 </code>
-The executive summary: While polymake is designed to to all kinds of things automatically, you might have to guide it a little if you are computing with large or special input.
+The executive summary: While polymake is designed to do all kinds of things automatically, you might have to guide it a little if you are computing with large or special input.
 One more caveat:  A //d//-polytope with //n// vertices has at most //O(n^(d/2))// facets.  This is the consequence of the Upper-Bound-Theorem.
   * McMullen, Peter:  The maximum numbers of faces of a convex polytope. Mathematika 17 (1970) 179-184.
 This number is actually attained by neighborly polytopes; for example, by the cyclic polytopes.