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 4.13, release 4.12, release 4.11, release 4.10, release 4.9, release 4.8, release 4.7, release 4.6, release 4.5, release 4.4, 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