This is a first IJulia notebook to demonstrate features of Polymake.jl.

Author: Michael Joswig

In :
using Cxx

In :
using Polymake

polymake version 3.1.1
Ewgenij Gawrilow, Michael Joswig (TU Berlin)
http://www.polymake.org

This is free software licensed under GPL; see the source for copying conditions.
There is NO warranty; not even for MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.


Produce a 6-dimensional polytope with 20 vertices, drawn uniformly at random on the unit sphere, and print its f-vector.

In :
P = rand_sphere(6,20)
give( P, "F_VECTOR" )

polymake: used package cdd
cddlib
Implementation of the double description method of Motzkin et al.
http://www-oldurls.inf.ethz.ch/personal/fukudak/cdd_home/


Out:
pm::Vector<pm::Integer>
20 167 636 1173 1026 342

Now for the g-vector of the same polytope, but this time it is converted to a Julia Array.

In :
g = convert(Array{Int32}, give( P, "G_VECTOR" ))

Out:
4-element Array{Int32,1}:
1
13
68
66

Let's do a census of 100 random samples and plot the result. Since g_0=1 and g_1=#vertices-dimension-1 are constant, it suffices to plot g_2 versus g_3.

In :
n_vertices=20
n_samples=100
g_vectors = Array{Int32}(n_samples,2)

for i=1:n_samples
RS = rand_sphere(6,n_vertices)
g = convert(Array{Int32}, give( RS, "G_VECTOR" ))
g_vectors[i,1] = g # notice index shift as Julia counts from 1
g_vectors[i,2] = g
end

In :
using Gadfly
plot(x=g_vectors[1:n_samples,1], y=g_vectors[1:n_samples,2],
Guide.xlabel("g_2"), Guide.ylabel("g_3"))

Out:

Comparing with the maximal g-vector from McMullen's upper bound theorem shows that the random polytopes have rather few faces.

In :
give( upper_bound_theorem(6,n_vertices), "G_VECTOR" )

Out:
pm::Vector<pm::Integer>
1 13 91 455

The lesson to learn here is the following.

By just looking at this class of random polytopes, one cannot get an adequate intuition about the set of all 6-polytopes with 20 vertices, not even the simplicial ones.