`Polyhedron`

and `polymake`

's `Polytope`

Many polyhedral computations are done through `polymake`

. `polymake`

(Ewgenij Gawrilow, Michael Joswig (2000), polymake.org) is open source software for research in polyhedral geometry and is attached to Julia via `Polymake.jl`

(Marek Kaluba, Benjamin Lorenz, Sascha Timme (2020), Polymake.jl). This is visible in the structure `Polyhedron`

via a pointer `pm_polytope`

to the corresponding `polymake`

object. Using `Polymake.jl`

one can apply all functionality of `polymake`

to the `polymake`

object hidden behind this pointer.

Sometimes it can be necessary to directly invoke some `polymake`

functions on an OSCAR `Polyhedron`

object (e.g. because some functionality has not yet been made available via OSCAR's interface). In that case, the following two functions allow extracting the underlying `Polymake.jl`

object from a `Polyhedron`

, respectively wrapping a `Polymake.jl`

object representing a polyhedron into a high-level `Polyhedron`

object.

`Polyhedron`

— Method`Polyhedron{T}(P::Polymake.BigObject, F::Field) where T<:scalar_types`

Construct a `Polyhedron`

corresponding to a `Polymake.BigObject`

of type `Polytope`

with scalars from `Field`

`F`

.

`polyhedron`

— Method`polyhedron(P::Polymake.BigObject)`

Construct a `Polyhedron`

corresponding to a `Polymake.BigObject`

of type `Polytope`

. Scalar type and parent field will be detected automatically. To improve type stability and performance, please use `Polyhedron{T}(p::Polymake.BigObject, f::Field) where T<:scalar_types`

instead, where possible.

The following shows all the data currently known for a `Polyhedron`

.

```
julia> C = cube(3)
Polyhedron in ambient dimension 3
julia> C.pm_polytope
type: Polytope<Rational>
description: cube of dimension 3
AFFINE_HULL
BOUNDED
true
CONE_AMBIENT_DIM
4
CONE_DIM
4
FACETS
1 1 0 0
1 -1 0 0
1 0 1 0
1 0 -1 0
1 0 0 1
1 0 0 -1
VERTICES_IN_FACETS
{0 2 4 6}
{1 3 5 7}
{0 1 4 5}
{2 3 6 7}
{0 1 2 3}
{4 5 6 7}
```

`polymake`

allows for an interactive visualization of 3-dimensional polytopes in the browser: `Polymake.visual(C.pm_polytope)`

.

There are several design differences between `polymake`

and `OSCAR`

. Polyhedra in `polymake`

and `Polymake.jl`

use homogeneous coordinates. The polyhedra in `OSCAR`

use affine coordinates. Indices in `polymake`

are zero-based, whereas in `OSCAR`

they are one-based.

The next example shows a purely combinatorial construction of a polytope (here: a square). In spite of being given no coordinates, `polymake`

can check for us that this is a simple polytope; i.e., each vertex is contained in dimension many facets.

```
julia> Q = Polymake.polytope.Polytope(VERTICES_IN_FACETS=[[0,2],[1,3],[0,1],[2,3]]);
julia> Q.SIMPLE
true
```

However, without coordinates, some operations such as computing the volume cannot work:

```
julia> Q.VOLUME
polymake: WARNING: available properties insufficient to compute 'VOLUME'
```