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I do not know the details, but it is pretty easy to apply the matrix to all the vertices of the polyhedron.

I do not know the details, but it is pretty easy to apply the matrix to all the vertices of the polyhedron.

sage: D.ambient_space()
Vector space of dimension 3 over Number Field in sqrt5 with defining polynomial x^2 - 5 with sqrt5 = 2.236067977499790?

I do Regarding your first question, Sage relies on various libraries to represent and deal with polyhedras, so there are various corresponding backends, like ppl, cdd, normaliz, polymake. In the case of your docahedron, the backend is named field, which means that it relies on Sage own code, not know external library:

sage: D.backend()
'field'

Now, to see the details, but it corresponding source code, you can do:

sage: from sage.geometry.polyhedron.backend_field import Polyhedron_field
sage: Polyhedron_field??

As you can see, the internal representation of your polyhedron is pretty easy to apply both the matrix to all V-reprentation and the vertices of the polyhedron.H-representation.

Regarding your second question, D is already a subset of some 3d space :

sage: D.ambient_space()
Vector space of dimension 3 over Number Field in sqrt5 with defining polynomial x^2 - 5 with sqrt5 = 2.236067977499790?

Regarding your first question, Sage relies on various libraries to represent and deal with polyhedras, so there are various corresponding backends, like ppl, cdd, normaliz, polymake. In the case of your docahedron, the backend is named field, which means that it relies on Sage own code, not external library:

sage: D.backend()
'field'

Now, to see the corresponding source code, you can do:

sage: from sage.geometry.polyhedron.backend_field import Polyhedron_field
sage: Polyhedron_field??

As you can see, the internal representation of your polyhedron is both the V-reprentation and the H-representation.H-representation. In both cases, it is pretty easy so see how a matrix acts on them.

Regarding your second question, D is already a subset of some 3d space :

sage: D.ambient_space()
Vector space of dimension 3 over Number Field in sqrt5 with defining polynomial x^2 - 5 with sqrt5 = 2.236067977499790?

Regarding your first question, Sage relies on various libraries to represent and deal with polyhedras, so there are various corresponding backends, like ppl, cdd, normaliz, polymake. In the case of your docahedron, the backend is named field, which means that it relies on Sage own code, not external library:

sage: D.backend()
'field'

Now, to see the corresponding source code, you can do:

sage: from sage.geometry.polyhedron.backend_field import Polyhedron_field
sage: Polyhedron_field??

As you can see, the internal representation of your polyhedron is both the V-reprentation and the H-representation. H-representation, simultaneously. In both cases, it is pretty easy so see how a matrix acts on them.

Regarding your second question, D is already a subset of some 3d space :

sage: D.ambient_space()
Vector space of dimension 3 over Number Field in sqrt5 with defining polynomial x^2 - 5 with sqrt5 = 2.236067977499790?