;### Produce a new polytope with rational coordinates.
$p=new Polytope<Rational>;
$p->POINTS=<<".";
1 0 0 0
1 1 0 0
1 0 1 0
1 0 0 1
1 1 1 0
1 1 0 1
1 0 1 1
1 1 1 1
.
;### Calculate and print some properties of p.
print join (", ",$p->list_properties);
print $p->VERTICES;
print join (", ",$p->list_properties);
print rows_labeled($p->FACETS);
show_credits;
print $p->N_VERTICES, "\t", $p->N_FACETS, "\t", $p->SIMPLE;
@a=@{$p->FACET_SIZES};
print @a;
print "@a";
print join(", ", @a);
;### A for loop in Perl.
for($i=0;$i<$p->N_FACETS;++$i){print $a[$i]." ";}
;### Vizualization.
$p->VISUAL;
jreality($p->VISUAL);
;### Produce a pyramid over p.
$all_zeros=new Vector<Rational>([ (0)x($p->N_VERTICES) ]);
$point=new Vector<Rational>(1,3/4,1/4,1/2);
$height=3; 
$m=( ($p->VERTICES | $all_zeros) / ($point|$height));
$pyr=new Polytope(POINTS=>$m);
$pyr->SCHLEGEL;
$pyr->SCHLEGEL(FACET=>6, EdgeColor=>"blue");
help 'functions/Producing a new polyhedron from others/';
$pyr2=pyramid($p,$height);
print isomorphic($pyr,$pyr2);
;### Other help: F1, apropos, documentation, forum  
;
;### Cones and fans.
print rows_labeled($p->FACETS);
$indices=new Array<Int>(0,1,2); 
$c=new Polytope(INEQUALITIES=>($p->FACETS->minor($indices,All)));
print $c->VERTICES;
$c=new Cone(INPUT_RAYS=>[[1,0,0],[0,1,0],[0,-1,0]]);
print $c->DIM,"\t",$c->AMBIENT_DIM;
print $c->LINEALITY_SPACE;
print $c->FACETS;
application "fan";
help '/';
$f=new PolyhedralFan(INPUT_RAYS=>[[1,0,0],[0,1,0],[0,-1,0]],INPUT_CONES=>[[0,1],[0,2]]);
prefer("jreality");
$f->VISUAL;
print $f->FACET_NORMALS;
print $f->MAXIMAL_CONES_FACETS;
$p2=polytope::transportation((new Vector<Int>('3 6 5')),(new Vector<Int>('7 7')));
print $p2->AMBIENT_DIM;
$t=polytope::projection_full( $p2 );
$t->VISUAL->LATTICE_COLORED;
print $t->INTERIOR_LATTICE_POINTS;
;### ILP tomorrow
save($t,"transportation.poly");
### Graphs with polymake.
application "graph";
$g=new Graph(ADJACENCY=>[[1,2,4],[0,2],[0,1,3,4],[2,4],[0,1,2,3]]);
$g->VISUAL;
graphviz($g->VISUAL);
$g=new props::Graph(5);
$g->edge(0,1);$g->edge(1,2);$g->edge(0,2);
print $g;
$g->delete_node(0);
;### Linear programming with polymake.
application "polytope";
$t=load("transportation.poly");
$objective=new LinearProgram<Rational>(LINEAR_OBJECTIVE=>[0,1,1]);
$lp=new Polytope<Rational>(VERTICES=>$t->VERTICES, LP=>$objective);
print $lp->LP->MAXIMAL_VALUE;
print $lp->LP->MAXIMAL_FACE;
;### The end of BEG.
;### type 'exit;' to exit polymake