This tutorial is probably also available as a Jupyter notebook in the demo folder in the polymake source and on github.
Different versions of this tutorial: latest release, release 4.3, release 4.2, release 4.1, release 4.0, release 3.6, nightly master
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 unit sphere in 3-space.
> $p1=rand_sphere(3,100); > print $p1->SIMPLICIAL; true
Random points sampled from a normal distribution
A second way to randomly construct a polytope is by sampling points by a standard normal distribution. The following chooses 100 points sampled from the standard normal distribution in 3-space.
> $p2 = rand_normal(3, 100); > print $p2 -> SIMPLICIAL, "\n", $p2 -> F_VECTOR; true 24 66 44
With probability one such polytopes under either distribution are simplicial.
Random polytopes with are neither simplicial nor simple
> ($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 30 162 251 119