====== Random Constructions ======

===== Random points on the unit sphere =====

The easiest way to randomly construct a polytope is by sampling points on the unit sphere. The following chooses 100 points on the units sphere in 3-space.

<code perl>
> $p1=rand_sphere(3,100);
> print $p1->SIMPLICIAL;
true
</code>
With probability one such polytopes are simplicial.

===== Random polytopes with are neither simplicial nor simple =====

<code perl>
> ($d,$m,$n) = (4,50,30);
> $p1=rand_sphere($d,$m);
> $p2=polarize($p1);
> $p3=new Polytope(POINTS=>rand_vert($p2->VERTICES,$n));
> print $p3->SIMPLICIAL, " ", $p3->SIMPLE, "\n", $p3->F_VECTOR;
false false
30 163 256 123
</code>