Perles Irrational Polytope (in polymake)

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Lai
Posts: 7
Joined: 02 Jul 2019, 02:26

Perles Irrational Polytope (in polymake)

Hello,

I have the following naive question.

The vertices of perles 8-dimensional irrational polytope are expressed in terms of the square root of 5, denoted r5.

For example one coordinate of one vertex is " -1/2 + 1/2r5".

I am not fully certain whether this expression is -1/2 + (1/2)r5 or -1/2 + 1/(2r5) , although I suspect the latter.

Thanks

gawrilow
Main Author
Posts: 386
Joined: 25 Dec 2010, 17:40

Re: Perles Irrational Polytope (in polymake)

I am not fully certain whether this expression is -1/2 + (1/2)r5 or -1/2 + 1/(2r5) , although I suspect the latter.
No, the former. The quadratic extension numbers are always normalized, having square roots in the numerator.

Lai
Posts: 7
Joined: 02 Jul 2019, 02:26

Re: Perles Irrational Polytope (in polymake)

I see, thanks.

Also, is it possible to compute the convex hull of irrational vectors (say with coordinates in a quadratic extension), and then compute the facets of the resulting polytope?

gawrilow
Main Author
Posts: 386
Joined: 25 Dec 2010, 17:40

Re: Perles Irrational Polytope (in polymake)

Yes, it's possible:

$P=new Polytope<QuadraticExtension>(POINTS=>$matrix);
print \$P->FACETS;

Just keep in mind that the support for quadratic extensions is currently rather rudimentary: only one distinct square root is allowed in all coordinates, and it has to be a root of a rational number, no iterated roots.

Lai
Posts: 7
Joined: 02 Jul 2019, 02:26

Great, thanks!