cpo

$m = new Matrix([
[0,1,0,0,0],
[0,0,1,0,0],
[0,0,0,1,0],
[0,0,0,0,1],
[1,-1,0,0,0],
[1,0,-1,0,0],
[1,0,0,-1,0],
[1,0,0,0,-1],
[-1,1,0,0,1],
[2,-1,0,0,-1],
[-1,0,0,1,1],
[2,0,0,-1,-1],
[3,1,1,1,1]
]);
$p = new Polytope(INEQUALITIES=>$m);
print "lattice polytope? ", $p->LATTICE, "\n";
print "volume: ", $p->VOLUME, "\n";
print "h*-vecotr: ", $p->H_STAR_VECTOR, "\n";
print("if lattice, normal? ", $p->NORMAL, "\n") if ($p->LATTICE);
print "vertices \n", $p->VERTICES, "\n";
print "facets \n", $p->FACETS, "\n";

example 1
1523 5431

$m1 = new Matrix([
[0,1,0,0,0],
[0,0,1,0,0],
[0,0,0,1,0],
[0,0,0,0,1],
[1,-1,0,0,0],
[1,0,-1,0,0],
[1,0,0,-1,0],
[1,0,0,0,-1],
[1,-1,-1,0,0],
[-1,0,0,1,1],
[2,0,-1,-1,-1],
[-2,1,1,1,1]
]);

$p = new Polytope(INEQUALITIES=>$m1);
print "lattice polytope? ", $p->LATTICE, "\n";
print "volume: ", $p->VOLUME, "\n";
print "h*-vecotr: ", $p->H_STAR_VECTOR, "\n";
print("if lattice, normal? ", $p->NORMAL, "\n") if ($p->LATTICE);
print "vertices \n", $p->VERTICES, "\n";
print "facets \n", $p->FACETS, "\n";
$tc = $p->TRIANGULATION;
print $tc ->FACETS, "\n";

dual

$md1 = new Matrix([
[0,1,0,0,0,0],
[0,0,1,0,0,0],
[0,0,0,1,0,0],
[0,0,0,0,1,0],
[0,0,0,0,0,1],
[1,-1,0,0,0,0],
[1,0,-1,0,0,0],
[1,0,0,-1,0,0],
[1,0,0,0,-1,0],
[1,0,0,0,0,-1],
[1,-1,-1,-1,0,-1],
[1,-1,0,-1,-1,-1]
]);

$p = new Polytope(INEQUALITIES=>$md1);
print "lattice polytope? ", $p->LATTICE, "\n";
print "volume: ", $p->VOLUME, "\n";
print "h*-vecotr: ", $p->H_STAR_VECTOR, "\n";
print("if lattice, normal? ", $p->NORMAL, "\n") if ($p->LATTICE);
print "vertices \n", $p->VERTICES, "\n";
print "facets \n", $p->FACETS, "\n";
$tc = $p->TRIANGULATION;
print $tc ->FACETS, "\n";


example 2

(1,2,3,6),(4,3,1),(6,5,4,3)

$m2 = new Matrix([
[0,1,0,0,0,0],
[0,0,1,0,0,0],
[0,0,0,1,0,0],
[0,0,0,0,1,0],
[0,0,0,0,0,1],
[1,-1,0,0,0,0],
[1,0,-1,0,0,0],
[1,0,0,-1,0,0],
[1,0,0,0,-1,0],
[1,0,0,0,0,-1],
[1,-1,-1,0,0,0],
[-1,1,1,1,0,0],
[-2,0,0,1,1,1],
[3,-1,-1,-1,-1,-1]
]);
$p = new Polytope(INEQUALITIES=>$m2);
print "lattice polytope? ", $p->LATTICE, "\n";
print "volume: ", $p->VOLUME, "\n";
print "h*-vecotr: ", $p->H_STAR_VECTOR, "\n";
print("if lattice, normal? ", $p->NORMAL, "\n") if ($p->LATTICE);
print "vertices \n", $p->VERTICES, "\n";
print "facets \n", $p->FACETS, "\n";

$md2 = new Matrix([
[0,1,0,0,0,0,0],
[0,0,1,0,0,0,0],
[0,0,0,1,0,0,0],
[0,0,0,0,1,0,0],
[0,0,0,0,0,1,0],
[0,0,0,0,0,0,1],
[1,-1,0,0,0,0,0],
[1,0,-1,0,0,0,0],
[1,0,0,-1,0,0,0],
[1,0,0,0,-1,0,0],
[1,0,0,0,0,-1,0],
[1,0,0,0,0,0,-1],
[1,-1,-1,-1,0,0,-1],
[1,-1,0,-1,-1,0,-1],
[1,0,0,-1,-1,-1,-1]
]);

$p = new Polytope(INEQUALITIES=>$md2);
print "lattice polytope? ", $p->LATTICE, "\n";
print "volume: ", $p->VOLUME, "\n";
print "h*-vecotr: ", $p->H_STAR_VECTOR, "\n";
print("if lattice, normal? ", $p->NORMAL, "\n") if ($p->LATTICE);
print "vertices \n", $p->VERTICES, "\n";
print "facets \n", $p->FACETS, "\n";
$tc = $p->TRIANGULATION;
print $tc ->FACETS, "\n";

example

(1,2,3),(3,4,5),(1,2,4,5)

$md = new Matrix([
[0,1,0,0,0,0],
[0,0,1,0,0,0],
[0,0,0,1,0,0],
[0,0,0,0,1,0],
[0,0,0,0,0,1],
[1,-1,0,0,0,0],
[1,0,-1,0,0,0],
[1,0,0,-1,0,0],
[1,0,0,0,-1,0],
[1,0,0,0,0,-1],
[1,-1,-1,-1,0,0],
[1,0,0,-1,-1,-1],
[1,-1,-1,0,-1,-1]
]);
$p = new Polytope(INEQUALITIES=>$md);
print $p -> VOLUME ,"\n";
print $p->LATTICE, "\n";
print $p ->H_STAR_VECTOR, "\n";
print $p->NORMAL, "\n";
print $p->VERTICES, "\n";
print $p->FACETS, "\n";

example 3
(1,2,4),(1,3,4)

$m3 = new Matrix([
[0,1,0,0,0],
[0,0,1,0,0],
[0,0,0,1,0],
[0,0,0,0,1],
[1,-1,0,0,0],
[1,0,-1,0,0],
[1,0,0,-1,0],
[1,0,0,0,-1],
[1,0,-1,-1,0],
[1,-1,-1,0,0]
]);

$p = new Polytope(INEQUALITIES=>$m3);
print $p -> VOLUME ,"\n";
print $p->LATTICE, "\n";
print $p ->H_STAR_VECTOR, "\n";
print $p->NORMAL, "\n";
print $p->VERTICES, "\n";
print $p->FACETS, "\n";

dual

$md3 = new Matrix([
[0,1,0,0,0],
[0,0,1,0,0],
[0,0,0,1,0],
[0,0,0,0,1],
[1,-1,0,0,0],
[1,0,-1,0,0],
[1,0,0,-1,0],
[1,0,0,0,-1],
[1,0,-1,-1,-1],
[1,-1,-1,0,-1]
]);

$p = new Polytope(INEQUALITIES=>$md3);
print $p -> VOLUME ,"\n";
print $p->LATTICE, "\n";
print $p ->H_STAR_VECTOR, "\n";
print $p->NORMAL, "\n";
print $p->VERTICES, "\n";
print $p->FACETS, "\n";

example
124,134,321,432

$md = new Matrix([
[0,1,0,0],
[0,0,1,0],
[0,0,0,1],
[1,-1,0,0],
[1,0,-1,0],
[1,0,0,-1],
[-1,1,1,0],
[-1,0,1,1],
[2,-1,-1,-1]
]);
$p = new Polytope(INEQUALITIES=>$md);
print $p -> VOLUME ,"\n";
print $p->LATTICE, "\n";
print $p ->H_STAR_VECTOR, "\n";
print $p->NORMAL, "\n";
print $p->VERTICES, "\n";
print $p->FACETS, "\n";
print $p->SIMPLE, "\n";

$m = new Matrix([
[0,1,0,0,0],
[0,0,1,0,0],
[0,0,0,1,0],
[0,0,0,0,1],
[1,-1,0,0,0],
[1,0,-1,0,0],
[1,0,0,-1,0],
[1,0,0,0,-1],
[1,-1,-1,-1,-1]
]);
$p = new Polytope(INEQUALITIES=>$m);
print $p -> VOLUME ,"\n";
print $p->LATTICE, "\n";
print $p ->H_STAR_VECTOR, "\n";
print $p->NORMAL, "\n";
print $p->VERTICES, "\n";
print $p->FACETS, "\n";
print $p->SIMPLE, "\n";

example 4

(1,2,3,5),(6,5,4,2),(1,2,6,5),(2,3,5,4)

$md4 = new Matrix([
[0,1,0,0,0,0,0],
[0,0,1,0,0,0,0],
[0,0,0,1,0,0,0],
[0,0,0,0,1,0,0],
[0,0,0,0,0,1,0],
[0,0,0,0,0,0,1],
[1,-1,0,0,0,0,0],
[1,0,-1,0,0,0,0],
[1,0,0,-1,0,0,0],
[1,0,0,0,-1,0,0],
[1,0,0,0,0,-1,0],
[1,0,0,0,0,0,-1],
[1,-1,-1,-1,0,-1,0],
[1,0,-1,0,-1,-1,-1,],
[1,-1,-1,0,0,-1,-1],
[1,0,-1,-1,-1,-1,0]
]);

$p = new Polytope(INEQUALITIES=>$md4);
print $p -> VOLUME ,"\n";
print $p->LATTICE, "\n";
print $p ->H_STAR_VECTOR, "\n";
print $p->NORMAL, "\n";
print $p->VERTICES, "\n";
print $p->FACETS, "\n";
print $p->SIMPLE, "\n";

1 0 1 0 0 0 0                                                                                                                
1 0 0 0 0 0 1                                                                                                                
1 0 0 1 0 0 0                                                                                                                
1 0 0 0 0 1 0                                                                                                                
1 0 0 1 0 0 1                                                                                                                
1 1 0 0 1 0 0                                                                                                                
1 1 0 0 0 0 0                                                                                                                
1 0 0 0 0 0 0                                                                                                                
1 0 0 0 1 0 0


example
(1,2,3) (3,4,5) (1,2,4,5)
evil triple (2,3,4)

$m = new Matrix([
[0,1,0,0,0,0],
[0,0,1,0,0,0],
[0,0,0,1,0,0],
[0,0,0,0,1,0],
[0,0,0,0,0,1],
[1,-1,0,0,0,0],
[1,0,-1,0,0,0],
[1,0,0,-1,0,0],
[1,0,0,0,-1,0],
[1,0,0,0,0,-1],
[1,-1,-1,0,0,0],
[1,0,0,-1,-1,0],
[1,-1,0,0,-1,-1]
]);
$p = new Polytope(INEQUALITIES=>$m);
print $p->LATTICE;
print $p ->VOLUME;
print $p ->H_STAR_VECTOR;
print $p->NORMAL;
print $p->VERTICES;
print $p->FACETS;
print $p->SIMPLE;

$md = new Matrix([
[0,1,0,0,0,0],
[0,0,1,0,0,0],
[0,0,0,1,0,0],
[0,0,0,0,1,0],
[0,0,0,0,0,1],
[1,-1,0,0,0,0],
[1,0,-1,0,0,0],
[1,0,0,-1,0,0],
[1,0,0,0,-1,0],
[1,0,0,0,0,-1],
[1,-1,-1,-1,0,0],
[1,0,0,-1,-1,-1],
[1,-1,-1,0,-1,-1]
]);
$p = new Polytope(INEQUALITIES=>$md);
print $p->LATTICE;
print $p ->VOLUME;
print $p ->H_STAR_VECTOR;
print $p->NORMAL;
print $p->VERTICES;
print $p->FACETS;
print $p->SIMPLE;

example, not a lattice polytope but volume counts circular extensions
126, 234, 531, 345, 615, 642

$m = new Matrix([
[0,1,0,0,0,0,0],
[0,0,1,0,0,0,0],
[0,0,0,1,0,0,0],
[0,0,0,0,1,0,0],
[0,0,0,0,0,1,0],
[0,0,0,0,0,0,1],
[1,-1,0,0,0,0,0],
[1,0,-1,0,0,0,0],
[1,0,0,-1,0,0,0],
[1,0,0,0,-1,0,0],
[1,0,0,0,0,-1,0],
[1,0,0,0,0,0,-1],
[1,-1,-1,0,0,0,-1],
[1,0,-1,-1,-1,0,0],
[1,-1,0,-1,0,-1,0],
[1,0,0,-1,-1,-1,0],
[1,-1,0,0,0,-1,-1],
[1,0,-1,0,-1,0,-1]
]);
$p = new Polytope(INEQUALITIES=>$m);
print "lattice polytope? ", $p->LATTICE, "\n";
print "volume: ", $p->VOLUME, "\n";
print "h*-vecotr: ", $p->H_STAR_VECTOR, "\n";
print("if lattice, normal? ", $p->NORMAL, "\n") if ($p->LATTICE);
print "vertices \n", $p->VERTICES, "\n";
print "facets \n", $p->FACETS, "\n";
$tc = $p->TRIANGULATION;
print $tc ->FACETS, "\n";

example
126, 234, 531, 456, 615, 642

$m = new Matrix([
[0,1,0,0,0,0],
[0,0,1,0,0,0],
[0,0,0,1,0,0],
[0,0,0,0,1,0],
[0,0,0,0,0,1],
[1,-1,0,0,0,0],
[1,0,-1,0,0,0],
[1,0,0,-1,0,0],
[1,0,0,0,-1,0],
[1,0,0,0,0,-1],
[1,0,0,-1,-1,0],
[-1,1,1,1,1,0],
[1,0,-1,-1,0,0],
[-1,0,1,1,1,1],
[3,-1,-1,-1,-1,-1]
]);
$p = new Polytope(INEQUALITIES=>$m);
print $p->LATTICE;
print $p ->VOLUME;
print $p ->H_STAR_VECTOR;
print $p->NORMAL;
print $p->VERTICES;
print $p->FACETS;
print $p->SIMPLE;

example, volume of chain polytope does not count circular extensions

1235, 642, 651, 543

$m = new Matrix([
[0,1,0,0,0,0,0],
[0,0,1,0,0,0,0],
[0,0,0,1,0,0,0],
[0,0,0,0,1,0,0],
[0,0,0,0,0,1,0],
[0,0,0,0,0,0,1],
[1,-1,0,0,0,0,0],
[1,0,-1,0,0,0,0],
[1,0,0,-1,0,0,0],
[1,0,0,0,-1,0,0],
[1,0,0,0,0,-1,0],
[1,0,0,0,0,0,-1],
[1,-1,-1,-1,0,-1,0],
[1,0,-1,0,-1,0,-1],
[1,-1,0,0,0,-1,-1],
[1,0,0,-1,-1,-1,0]
]);
$p = new Polytope(INEQUALITIES=>$m);
print "lattice polytope? ", $p->LATTICE, "\n";
print "volume: ", $p->VOLUME, "\n";
print "h*-vecotr: ", $p->H_STAR_VECTOR, "\n";
print("if lattice, normal? ", $p->NORMAL, "\n") if ($p->LATTICE);
print "vertices \n", $p->VERTICES, "\n";
print "facets \n", $p->FACETS, "\n";

generated by algorithm

[0,1,0,0,0,0,0],
[0,0,1,0,0,0,0],
[0,0,0,1,0,0,0],
[0,0,0,0,1,0,0],
[0,0,0,0,0,1,0],
[0,0,0,0,0,0,1],
[1,-1,0,0,0,0,0],
[1,0,-1,0,0,0,0],
[1,0,0,-1,0,0,0],
[1,0,0,0,-1,0,0],
[1,0,0,0,0,-1,0],
[1,0,0,0,0,0,-1],
[1,-1,-1,-1,0,-1,0],
[1,0,-1,0,-1,0,-1],
[1,-1,0,0,0,-1,-1],
[1,0,0,-1,-1,-1,0]

vertices                                                                                                                     
1 0 0 1 0 0 0                                                                                                                
1 0 1 0 0 0 0                                                                                                                
1 0 0 0 0 1 0                                                                                                                
1 0 0 0 0 0 1                                                                                                                
1 0 0 1 0 0 1                                                                                                                
1 1 0 0 1 0 0                                                                                                                
1 1 0 0 0 0 0                                                                                                                
1 1/2 1/2 0 0 0 1/2                                                                                                          
1 0 1/2 0 0 1/2 1/2                                                                                                          
1 0 0 0 0 0 0                                                                                                                
1 0 0 0 1 0 0                                                                                                                
1 0 0 0 1/2 1/2 1/2                                                                                                          
1 0 1/3 0 1/3 2/3 1/3                                                                                                        
1 0 1/2 1/2 1/2 0 0                                                                                                          
1 0 1/2 0 1/2 1/2 0

$md = new Matrix([
[0,1,0,0,0,0],
[0,0,1,0,0,0],
[0,0,0,1,0,0],
[0,0,0,0,1,0],
[0,0,0,0,0,1],
[1,-1,0,0,0,0],
[1,0,-1,0,0,0],
[1,0,0,-1,0,0],
[1,0,0,0,-1,0],
[1,0,0,0,0,-1],
[1,-1,-1,0,0,0],
[-1,1,1,1,1,0],
[2,-1,-1,-1,-1,0],
[-2,1,1,1,1,1],
[3,-1,-1,-1,-1,-1],
[-1,0,1,1,1,1],
[-1,0,0,1,1,0]
]);
$p = new Polytope(INEQUALITIES=>$md);
print "lattice polytope? ", $p->LATTICE, "\n";
print "volume: ", $p->VOLUME, "\n";
print "h*-vecotr: ", $p->H_STAR_VECTOR, "\n";
print("if lattice, normal? ", $p->NORMAL, "\n") if ($p->LATTICE);
print "vertices \n", $p->VERTICES, "\n";
print "facets \n", $p->FACETS, "\n";

example flow polytope

$m = new Matrix<Int>([
[0,1,0,0,0,0],
[0,0,1,0,0,0],
[0,0,0,1,0,0],
[0,0,0,0,1,0],
[0,0,0,0,0,1],
[1,-1,0,0,0,0],
[1,0,-1,0,0,0],
[1,0,0,-1,0,0],
[1,0,0,0,-1,0],
[1,0,0,0,0,-1],
[1,-1,-1,-1,0,0],
[1,-1,0,-1,-1,0],
]);
$p = new Polytope(INEQUALITIES=>$m);
print "lattice polytope? ", $p->LATTICE, "\n";
print "volume: ", $p->VOLUME, "\n";
print "h*-vecotr: ", $p->H_STAR_VECTOR, "\n";
print("if lattice, normal? ", $p->NORMAL, "\n") if ($p->LATTICE);
print "vertices \n", $p->VERTICES, "\n";
print "facets \n", $p->FACETS, "\n";

$m = new Matrix<Int>([
[0,1,0,0,0,0],
[0,0,1,0,0,0],
[0,0,0,1,0,0],
[0,0,0,0,1,0],
[0,0,0,0,0,1],
[1,-1,0,0,0,0],
[1,0,-1,0,0,0],
[1,0,0,-1,0,0],
[1,0,0,0,-1,0],
[1,0,0,0,0,-1],
[1,-1,-1,-1,-1,0],
[1,-1,-1,0,-1,-1]
]);
$p = new Polytope(INEQUALITIES=>$m);
print "lattice polytope? ", $p->LATTICE, "\n";
print "volume: ", $p->VOLUME, "\n";
print "h*-vecotr: ", $p->H_STAR_VECTOR, "\n";
print("if lattice, normal? ", $p->NORMAL, "\n") if ($p->LATTICE);
print "vertices \n", $p->VERTICES, "\n";
print "facets \n", $p->FACETS, "\n";

Example 123, 531, 234, 345

$m = new Matrix<Int>([
[0,1,0,0,0,0],
[0,0,1,0,0,0],
[0,0,0,1,0,0],
[0,0,0,0,1,0],
[0,0,0,0,0,1],
[1,-1,0,0,0,0],
[1,0,-1,0,0,0],
[1,0,0,-1,0,0],
[1,0,0,0,-1,0],
[1,0,0,0,0,-1],
[1,-1,-1,-1,0,0],
[1,0,-1,-1,-1,0],
[1,-1,0,-1,0,-1],
[1,0,0,-1,-1,-1]
]);

$p = new Polytope(INEQUALITIES=>$m);
print "lattice polytope? ", $p->LATTICE, "\n";
print "volume: ", $p->VOLUME, "\n";
print "h*-vecotr: ", $p->H_STAR_VECTOR, "\n";
print("if lattice, normal? ", $p->NORMAL, "\n") if ($p->LATTICE);
print "vertices \n", $p->VERTICES, "\n";
print "facets \n", $p->FACETS, "\n";

Example

123 431 654

$m = new Matrix<Int>([
[0,1,0,0,0,0],
[0,0,1,0,0,0],
[0,0,0,1,0,0],
[0,0,0,0,1,0],
[0,0,0,0,0,1],
[1,-1,0,0,0,0],
[1,0,-1,0,0,0],
[1,0,0,-1,0,0],
[1,0,0,0,-1,0],
[1,0,0,0,0,-1],
[1,-1,-1,0,0,0],
[-1,1,1,1,0,0],
[-2,1,1,1,1,1],
[-1,0,0,0,1,1]
]);
$p = new Polytope(INEQUALITIES=>$m);
print "lattice polytope? ", $p->LATTICE, "\n";
print "volume: ", $p->VOLUME, "\n";
print "h*-vecotr: ", $p->H_STAR_VECTOR, "\n";
print("if lattice, normal? ", $p->NORMAL, "\n") if ($p->LATTICE);
print "vertices \n", $p->VERTICES, "\n";
print "facets \n", $p->FACETS, "\n";

$m = new Matrix<Int>([
[0,1,0,0,0,0,0],
[0,0,1,0,0,0,0],
[0,0,0,1,0,0,0],
[0,0,0,0,1,0,0],
[0,0,0,0,0,1,0],
[0,0,0,0,0,0,1],
[1,-1,0,0,0,0,0],
[1,0,-1,0,0,0,0],
[1,0,0,-1,0,0,0],
[1,0,0,0,-1,0,0],
[1,0,0,0,0,-1,0],
[1,0,0,0,0,0,-1],
[1,-1,-1,-1,0,0,0],
[1,-1,0,-1,-1,0,0],
[1,0,0,0,-1,-1,-1]
]);
$p = new Polytope(INEQUALITIES=>$m);
print "lattice polytope? ", $p->LATTICE, "\n";
print "volume: ", $p->VOLUME, "\n";
print "h*-vecotr: ", $p->H_STAR_VECTOR, "\n";
print("if lattice, normal? ", $p->NORMAL, "\n") if ($p->LATTICE);
print "vertices \n", $p->VERTICES, "\n";
print "facets \n", $p->FACETS, "\n";


Example

123 431 654

$m = new Matrix<Int>([
[0,1,0,0,0,0],
[0,0,1,0,0,0],
[0,0,0,1,0,0],
[0,0,0,0,1,0],
[0,0,0,0,0,1],
[1,-1,0,0,0,0],
[1,0,-1,0,0,0],
[1,0,0,-1,0,0],
[1,0,0,0,-1,0],
[1,0,0,0,0,-1],
[1,-1,-1,0,0,0],
[-1,1,1,1,0,0],
[-2,1,1,1,1,1],
[-1,0,0,0,1,1]
]);
$p = new Polytope(INEQUALITIES=>$m);
print "lattice polytope? ", $p->LATTICE, "\n";
print "volume: ", $p->VOLUME, "\n";
print "h*-vecotr: ", $p->H_STAR_VECTOR, "\n";
print("if lattice, normal? ", $p->NORMAL, "\n") if ($p->LATTICE);
print "vertices \n", $p->VERTICES, "\n";
print "facets \n", $p->FACETS, "\n";

$m = new Matrix<Int>([
[0,1,0,0,0,0,0],
[0,0,1,0,0,0,0],
[0,0,0,1,0,0,0],
[0,0,0,0,1,0,0],
[0,0,0,0,0,1,0],
[0,0,0,0,0,0,1],
[1,-1,0,0,0,0,0],
[1,0,-1,0,0,0,0],
[1,0,0,-1,0,0,0],
[1,0,0,0,-1,0,0],
[1,0,0,0,0,-1,0],
[1,0,0,0,0,0,-1],
[1,-1,-1,-1,0,0,0],
[1,-1,0,-1,-1,0,0],
[1,0,0,0,-1,-1,-1],
[1,-1,0,0,-1,0,-1]
]);
$p = new Polytope(INEQUALITIES=>$m);
print "lattice polytope? ", $p->LATTICE, "\n";
print "volume: ", $p->VOLUME, "\n";
print "h*-vecotr: ", $p->H_STAR_VECTOR, "\n";
print("if lattice, normal? ", $p->NORMAL, "\n") if ($p->LATTICE);
print "vertices \n", $p->VERTICES, "\n";
print "facets \n", $p->FACETS, "\n";

Example
123 245 456

$m = new Matrix<Int>([
[0,1,0,0,0,0,0],
[0,0,1,0,0,0,0],
[0,0,0,1,0,0,0],
[0,0,0,0,1,0,0],
[0,0,0,0,0,1,0],
[0,0,0,0,0,0,1],
[1,-1,-1,0,0,0,0],
[1,-1,-1,-1,0,0,0],
[1,0,-1,0,-1,-1,0],
[1,0,0,0,-1,-1,-1]
]);
$p = new Polytope(INEQUALITIES=>$m);
print "lattice polytope? ", $p->LATTICE, "\n";
print "volume: ", $p->VOLUME, "\n";
print "h*-vecotr: ", $p->H_STAR_VECTOR, "\n";
print("if lattice, normal? ", $p->NORMAL, "\n") if ($p->LATTICE);
print "vertices \n", $p->VERTICES, "\n";
print "facets \n", $p->FACETS, "\n";

0123 2345 3456

$m = new Matrix<Int>([
[0,1,0,0,0,0,0,0],
[0,0,1,0,0,0,0,0],
[0,0,0,1,0,0,0,0],
[0,0,0,0,1,0,0,0],
[0,0,0,0,0,1,0,0],
[0,0,0,0,0,0,1,0],
[0,0,0,0,0,0,0,1],
[1,-1,0,0,0,0,0,0],
[1,0,-1,0,0,0,0,0],
[1,0,0,-1,0,0,0,0],
[1,0,0,0,-1,0,0,0],
[1,0,0,0,0,-1,0,0],
[1,0,0,0,0,0,-1,0],
[1,0,0,0,0,0,0,-1],
[1,-1,-1,-1,-1,0,0,0],
[1,0,0,-1,-1,-1,-1,0],
[1,0,0,0,-1,-1,-1,-1]
]);
$p = new Polytope(INEQUALITIES=>$m);
print "lattice polytope? ", $p->LATTICE, "\n";
print "volume: ", $p->VOLUME, "\n";
print "h*-vecotr: ", $p->H_STAR_VECTOR, "\n";
print("if lattice, normal? ", $p->NORMAL, "\n") if ($p->LATTICE);
print "vertices \n", $p->VERTICES, "\n";
print "facets \n", $p->FACETS, "\n";

Example
126, 234, 456, 642

$m = new Matrix<Int>([
[0,1,0,0,0,0,0],
[0,0,1,0,0,0,0],
[0,0,0,1,0,0,0],
[0,0,0,0,1,0,0],
[0,0,0,0,0,1,0],
[0,0,0,0,0,0,1],
[1,-1,0,0,0,0,0],
[1,0,-1,0,0,0,0],
[1,0,0,-1,0,0,0],
[1,0,0,0,-1,0,0],
[1,0,0,0,0,-1,0],
[1,0,0,0,0,0,-1],
[1,-1,-1,0,0,0,-1],
[1,0,-1,-1,-1,0,0],
[1,0,0,0,-1,-1,-1],
[1,0,-1,0,-1,0,-1]
]);
$p = new Polytope(INEQUALITIES=>$m);
print "lattice polytope? ", $p->LATTICE, "\n";
print "volume: ", $p->VOLUME, "\n";
print "h*-vecotr: ", $p->H_STAR_VECTOR, "\n";
print("if lattice, normal? ", $p->NORMAL, "\n") if ($p->LATTICE);
print "vertices \n", $p->VERTICES, "\n";
print "facets \n", $p->FACETS, "\n";

$m = new Matrix<Int>([
[0,1,0,0,0,0,0],
[0,0,1,0,0,0,0],
[0,0,0,1,0,0,0],
[0,0,0,0,1,0,0],
[0,0,0,0,0,1,0],
[0,0,0,0,0,0,1],
[1,-1,0,0,0,0,0],
[1,0,-1,0,0,0,0],
[1,0,0,-1,0,0,0],
[1,0,0,0,-1,0,0],
[1,0,0,0,0,-1,0],
[1,0,0,0,0,0,-1],
[1,-1,-1,-1,0,0,0],
[1,0,0,-1,-1,-1,0],
[1,0,0,0,-1,-1,-1]
]);
$p = new Polytope(INEQUALITIES=>$m);
print "lattice polytope? ", $p->LATTICE, "\n";
print "volume: ", $p->VOLUME, "\n";
print "h*-vecotr: ", $p->H_STAR_VECTOR, "\n";
print("if lattice, normal? ", $p->NORMAL, "\n") if ($p->LATTICE);
print "vertices \n", $p->VERTICES, "\n";
print "facets \n", $p->FACETS, "\n";