# Constructions

The standard way to define a polyhedron is by either giving a $V$-representation or an $H$-representation. But polyhedra may also be constructed through other means: by name, via operations on other polyhedra, or from other objects in OSCAR.

## $H$- and $V$-representations

### Intersecting halfspaces: $H$-representation

`polyhedron`

— Method`polyhedron([::Union{Type{T}, Field},] A::AnyVecOrMat, b) where T<:scalar_types`

The (convex) polyhedron defined by

\[P(A,b) = \{ x | Ax ≤ b \}.\]

see Def. 3.35 and Section 4.1. of [JT13]

The first argument either specifies the `Type`

of its coefficients or their parent `Field`

.

**Examples**

The following lines define the square $[0,1]^2 \subset \mathbb{R}^2$:

```
julia> A = [1 0; 0 1; -1 0 ; 0 -1];
julia> b = [1, 1, 0, 0];
julia> polyhedron(A,b)
Polyhedron in ambient dimension 2
```

`polyhedron`

— Function`polyhedron(::Union{Type{T}, Field}, I::Union{Nothing, AbstractCollection[AffineHalfspace]}, E::Union{Nothing, AbstractCollection[AffineHyperplane]} = nothing) where T<:scalar_types`

The (convex) polyhedron obtained intersecting the halfspaces `I`

(inequalities) and the hyperplanes `E`

(equations). The first argument either specifies the `Type`

of its coefficients or their parent `Field`

.

**Examples**

The following lines define the square $[0,1]^2 \subset \mathbb{R}^2$:

```
julia> A = [1 0; 0 1; -1 0 ; 0 -1];
julia> b = [1, 1, 0, 0];
julia> polyhedron((A,b))
Polyhedron in ambient dimension 2
```

As an example for a polyhedron constructed from both inequalities and equations, we construct the polytope $[0,1]\times\{0\}\subset\mathbb{R}^2$

```
julia> P = polyhedron(([-1 0; 1 0], [0,1]), ([0 1], [0]))
Polyhedron in ambient dimension 2
julia> is_feasible(P)
true
julia> dim(P)
1
julia> vertices(P)
2-element SubObjectIterator{PointVector{QQFieldElem}}:
[1, 0]
[0, 0]
```

The complete $H$-representation can be retrieved using `facets`

and `affine_hull`

:

```
julia> P = polyhedron(([-1 0; 1 0], [0,1]), ([0 1], [0]))
Polyhedron in ambient dimension 2
julia> facets(P)
2-element SubObjectIterator{AffineHalfspace{QQFieldElem}} over the Halfspaces of R^2 described by:
-x_1 <= 0
x_1 <= 1
julia> affine_hull(P)
1-element SubObjectIterator{AffineHyperplane{QQFieldElem}} over the Hyperplanes of R^2 described by:
x_2 = 0
julia> Q0 = polyhedron(facets(P))
Polyhedron in ambient dimension 2
julia> P == Q0
false
julia> Q1 = polyhedron(facets(P), affine_hull(P))
Polyhedron in ambient dimension 2
julia> P == Q1
true
```

### Computing convex hulls: $V$-representation

`convex_hull`

— Method`convex_hull([::Union{Type{T}, Field} = QQFieldElem,] V [, R [, L]]; non_redundant::Bool = false)`

Construct the convex hull of the vertices `V`

, rays `R`

, and lineality `L`

. If `R`

or `L`

are omitted, then they are assumed to be zero.

**Arguments**

- The first argument either specifies the
`Type`

of its coefficients or their

parent `Field`

.

`V::AbstractCollection[PointVector]`

: Points whose convex hull is to be computed.`R::AbstractCollection[RayVector]`

: Rays completing the set of points.`L::AbstractCollection[RayVector]`

: Generators of the Lineality space.

If an argument is given as a matrix, its content has to be encoded row-wise.

`R`

can be given as an empty matrix or as `nothing`

if the user wants to compute the convex hull only from `V`

and `L`

.

If it is known that `V`

and `R`

only contain extremal points and that the description of the lineality space is complete, set `non_redundant = true`

to avoid unnecessary redundancy checks.

See Def. 2.11 and Def. 3.1 of [JT13].

**Examples**

The following lines define the square $[0,1]^2 \subset \mathbb{R}^2$:

```
julia> Square = convex_hull([0 0; 0 1; 1 0; 1 1])
Polyhedron in ambient dimension 2
```

To construct the positive orthant, rays have to be passed:

```
julia> V = [0 0];
julia> R = [1 0; 0 1];
julia> PO = convex_hull(V, R)
Polyhedron in ambient dimension 2
```

The closed-upper half plane can be constructed by passing rays and a lineality space:

```
julia> V = [0 0];
julia> R = [0 1];
julia> L = [1 0];
julia> UH = convex_hull(V, R, L)
Polyhedron in ambient dimension 2
```

To obtain the x-axis in $\mathbb{R}^2$:

```
julia> V = [0 0];
julia> R = nothing;
julia> L = [1 0];
julia> XA = convex_hull(V, R, L)
Polyhedron in ambient dimension 2
```

This is a standard triangle, defined via a (redundant) $V$-representation and its unique minimal $H$-representation:

```
julia> T = convex_hull([ 0 0 ; 1 0 ; 0 1; 0 1//2 ])
Polyhedron in ambient dimension 2
julia> halfspace_matrix_pair(facets(T))
(A = [-1 0; 0 -1; 1 1], b = QQFieldElem[0, 0, 1])
```

The complete $V$-representation can be retrieved using `minimal_faces`

, `rays_modulo_lineality`

and `lineality_space`

:

```
julia> P = convex_hull([0 0], [1 0], [0 1])
Polyhedron in ambient dimension 2
julia> Q0 = convex_hull(vertices(P))
Polyhedron in ambient dimension 2
julia> P == Q0
false
julia> mfP = minimal_faces(P)
(base_points = PointVector{QQFieldElem}[[0, 0]], lineality_basis = RayVector{QQFieldElem}[[0, 1]])
julia> rmlP = rays_modulo_lineality(P)
(rays_modulo_lineality = RayVector{QQFieldElem}[[1, 0]], lineality_basis = RayVector{QQFieldElem}[[0, 1]])
julia> Q1 = convex_hull(mfP.base_points, rmlP.rays_modulo_lineality)
Polyhedron in ambient dimension 2
julia> P == Q1
false
julia> Q0 == Q1
false
julia> Q2 = convex_hull(mfP.base_points, rmlP.rays_modulo_lineality, lineality_space(P))
Polyhedron in ambient dimension 2
julia> P == Q2
true
```

## Regular polytopes

A polytope is regular, in the strict sense, if it admits a flag-transtive group of (linear) automorphisms. There are three infinite families of regular polytopes which exist in each dimension: the (regular) simplices, cubes and cross polytopes. In addition there are two exceptional regular 3-polytopes (dodecahedron and icosahedron) plus three exceptional regular 4-polytopes (24-cell, 120-cell and 600-cell).

The regular 3-polytopes are also known as the Platonic solids. Here we also list the Archimedean, Catalan and Johnson solids, which form various generalizations of the Platonic solids. However, here we implement "disjoint families", i.e., the proper Archimedean solids exclude the Platonic solids; similarly, the proper Johnson solids exclude the Archmidean solids.

`simplex`

— Function`simplex([::Union{Type{T}, Field} = QQFieldElem,] d::Int [,n])`

Construct the simplex which is the convex hull of the standard basis vectors along with the origin in $\mathbb{R}^d$, scaled by $n$. The first argument either specifies the `Type`

of its coefficients or their parent `Field`

.

**Examples**

Here we take a look at the facets of the 7-simplex and a scaled 7-simplex:

```
julia> s = simplex(7)
Polytope in ambient dimension 7
julia> facets(s)
8-element SubObjectIterator{AffineHalfspace{QQFieldElem}} over the Halfspaces of R^7 described by:
-x_1 <= 0
-x_2 <= 0
-x_3 <= 0
-x_4 <= 0
-x_5 <= 0
-x_6 <= 0
-x_7 <= 0
x_1 + x_2 + x_3 + x_4 + x_5 + x_6 + x_7 <= 1
julia> t = simplex(7, 5)
Polytope in ambient dimension 7
julia> facets(t)
8-element SubObjectIterator{AffineHalfspace{QQFieldElem}} over the Halfspaces of R^7 described by:
-x_1 <= 0
-x_2 <= 0
-x_3 <= 0
-x_4 <= 0
-x_5 <= 0
-x_6 <= 0
-x_7 <= 0
x_1 + x_2 + x_3 + x_4 + x_5 + x_6 + x_7 <= 5
```

`cross_polytope`

— Function`cross_polytope([::Union{Type{T}, Field} = QQFieldElem,] d::Int [,n])`

Construct a $d$-dimensional cross polytope around origin with vertices located at $\pm e_i$ for each unit vector $e_i$ of $R^d$, scaled by $n$. The first argument either specifies the `Type`

of its coefficients or their parent `Field`

.

**Examples**

Here we print the facets of a non-scaled and a scaled 3-dimensional cross polytope:

```
julia> C = cross_polytope(3)
Polytope in ambient dimension 3
julia> facets(C)
8-element SubObjectIterator{AffineHalfspace{QQFieldElem}} over the Halfspaces of R^3 described by:
x_1 + x_2 + x_3 <= 1
-x_1 + x_2 + x_3 <= 1
x_1 - x_2 + x_3 <= 1
-x_1 - x_2 + x_3 <= 1
x_1 + x_2 - x_3 <= 1
-x_1 + x_2 - x_3 <= 1
x_1 - x_2 - x_3 <= 1
-x_1 - x_2 - x_3 <= 1
julia> D = cross_polytope(3, 2)
Polytope in ambient dimension 3
julia> facets(D)
8-element SubObjectIterator{AffineHalfspace{QQFieldElem}} over the Halfspaces of R^3 described by:
x_1 + x_2 + x_3 <= 2
-x_1 + x_2 + x_3 <= 2
x_1 - x_2 + x_3 <= 2
-x_1 - x_2 + x_3 <= 2
x_1 + x_2 - x_3 <= 2
-x_1 + x_2 - x_3 <= 2
x_1 - x_2 - x_3 <= 2
-x_1 - x_2 - x_3 <= 2
```

`cube`

— Function`cube([::Union{Type{T}, Field} = QQFieldElem,] d::Int , [l::Rational = -1, u::Rational = 1])`

Construct the $[l,u]$-cube in dimension $d$. The first argument either specifies the `Type`

of its coefficients or their parent `Field`

.

**Examples**

In this example the 5-dimensional unit cube is constructed to ask for one of its properties:

```
julia> C = cube(5,0,1);
julia> normalized_volume(C)
120
```

`tetrahedron`

— Function`tetrahedron()`

Construct the regular tetrahedron, one of the Platonic solids.

`dodecahedron`

— Function`dodecahedron()`

Construct the regular dodecahedron, one out of two Platonic solids.

`icosahedron`

— Function`icosahedron()`

Construct the regular icosahedron, one out of two exceptional Platonic solids.

`platonic_solid`

— Function`platonic_solid(s)`

Construct a Platonic solid with the name given by String `s`

from the list below.

**Arguments**

`s::String`

: The name of the desired Archimedean solid. Possible values:- "tetrahedron" : Tetrahedron. Regular polytope with four triangular facets.
- "cube" : Cube. Regular polytope with six square facets.
- "octahedron" : Octahedron. Regular polytope with eight triangular facets.
- "dodecahedron" : Dodecahedron. Regular polytope with 12 pentagonal facets.
- "icosahedron" : Icosahedron. Regular polytope with 20 triangular facets.

**Examples**

```
julia> T = platonic_solid("icosahedron")
Polytope in ambient dimension 3 with EmbeddedAbsSimpleNumFieldElem type coefficients
julia> n_facets(T)
20
```

`archimedean_solid`

— Function`archimedean_solid(s)`

Construct an Archimedean solid with the name given by String `s`

from the list below. The polytopes are realized with floating point numbers and thus not exact; Vertex-facet-incidences are correct in all cases.

**Arguments**

`s::String`

: The name of the desired Archimedean solid. Possible values:- "truncated_tetrahedron" : Truncated tetrahedron. Regular polytope with four triangular and four hexagonal facets.
- "cuboctahedron" : Cuboctahedron. Regular polytope with eight triangular and six square facets.
- "truncated_cube" : Truncated cube. Regular polytope with eight triangular and six octagonal facets.
- "truncated_octahedron" : Truncated Octahedron. Regular polytope with six square and eight hexagonal facets.
- "rhombicuboctahedron" : Rhombicuboctahedron. Regular polytope with eight triangular and 18 square facets.
- "truncated_cuboctahedron" : Truncated Cuboctahedron. Regular polytope with 12 square, eight hexagonal and six octagonal facets.
- "snub_cube" : Snub Cube. Regular polytope with 32 triangular and six square facets. The vertices are realized as floating point numbers. This is a chiral polytope.
- "icosidodecahedron" : Icosidodecahedon. Regular polytope with 20 triangular and 12 pentagonal facets.
- "truncated_dodecahedron" : Truncated Dodecahedron. Regular polytope with 20 triangular and 12 decagonal facets.
- "truncated_icosahedron" : Truncated Icosahedron. Regular polytope with 12 pentagonal and 20 hexagonal facets.
- "rhombicosidodecahedron" : Rhombicosidodecahedron. Regular polytope with 20 triangular, 30 square and 12 pentagonal facets.
- "truncated_icosidodecahedron" : Truncated Icosidodecahedron. Regular polytope with 30 square, 20 hexagonal and 12 decagonal facets.
- "snub_dodecahedron" : Snub Dodecahedron. Regular polytope with 80 triangular and 12 pentagonal facets. The vertices are realized as floating point numbers. This is a chiral polytope.

**Examples**

```
julia> T = archimedean_solid("cuboctahedron")
Polytope in ambient dimension 3
julia> sum([n_vertices(F) for F in faces(T, 2)] .== 3)
8
julia> sum([n_vertices(F) for F in faces(T, 2)] .== 4)
6
julia> n_facets(T)
14
```

`johnson_solid`

— Function`johnson_solid(i::Int)`

Construct the `i`

-th proper Johnson solid.

A Johnson solid is a 3-polytope whose facets are regular polygons, of various gonalities. It is proper if it is not an Archimedean solid. Up to scaling there are exactly 92 proper Johnson solids.

`catalan_solid`

— Function`catalan_solid(s::String)`

Construct a Catalan solid with the name `s`

from the list below. The polytopes are realized with floating point coordinates and thus are not exact. However, vertex-facet-incidences are correct in all cases.

**Arguments**

`s::String`

: The name of the desired Archimedean solid. Possible values:- "triakis_tetrahedron" : Triakis Tetrahedron. Dual polytope to the Truncated Tetrahedron, made of 12 isosceles triangular facets.
- "triakis_octahedron" : Triakis Octahedron. Dual polytope to the Truncated Cube, made of 24 isosceles triangular facets.
- "rhombic_dodecahedron" : Rhombic dodecahedron. Dual polytope to the cuboctahedron, made of 12 rhombic facets.
- "tetrakis_hexahedron" : Tetrakis hexahedron. Dual polytope to the truncated octahedron, made of 24 isosceles triangluar facets.
- "disdyakis_dodecahedron" : Disdyakis dodecahedron. Dual polytope to the truncated cuboctahedron, made of 48 scalene triangular facets.
- "pentagonal_icositetrahedron" : Pentagonal Icositetrahedron. Dual polytope to the snub cube, made of 24 irregular pentagonal facets. The vertices are realized as floating point numbers.
- "pentagonal_hexecontahedron" : Pentagonal Hexecontahedron. Dual polytope to the snub dodecahedron, made of 60 irregular pentagonal facets. The vertices are realized as floating point numbers.
- "rhombic_triacontahedron" : Rhombic triacontahedron. Dual polytope to the icosidodecahedron, made of 30 rhombic facets.
- "triakis_icosahedron" : Triakis icosahedron. Dual polytope to the icosidodecahedron, made of 30 rhombic facets.
- "deltoidal_icositetrahedron" : Deltoidal Icositetrahedron. Dual polytope to the rhombicubaoctahedron, made of 24 kite facets.
- "pentakis_dodecahedron" : Pentakis dodecahedron. Dual polytope to the truncated icosahedron, made of 60 isosceles triangular facets.
- "deltoidal_hexecontahedron" : Deltoidal hexecontahedron. Dual polytope to the rhombicosidodecahedron, made of 60 kite facets.
- "disdyakis_triacontahedron" : Disdyakis triacontahedron. Dual polytope to the truncated icosidodecahedron, made of 120 scalene triangular facets.

**Examples**

```
julia> T = catalan_solid("triakis_tetrahedron");
julia> count(F -> n_vertices(F) == 3, faces(T, 2))
12
julia> n_facets(T)
12
```

`regular_24_cell`

— Function`regular_24_cell()`

Construct the regular 24-cell, one out of three exceptional regular 4-polytopes.

`regular_120_cell`

— Function`regular_120_cell()`

Construct the regular 120-cell, one out of three exceptional regular 4-polytopes.

`regular_600_cell`

— Function`regular_600_cell()`

Construct the regular 600-cell, one out of three exceptional regular 4-polytopes.

## Other polytope constructions

`SIM_body_polytope`

— Function`SIM_body_polytope(alpha::AbstractVector)`

Produce an $n$-dimensional SIM-body as generalized permutahedron in $(n+1)$-space. SIM-bodies are defined in [GK14], but the input needs to be descending instead of ascending, as used in [JKS22], i.e. `alpha`

has parameters $(a_1,\dots,a_n)$ such that $a_1 \geq \dots \geq a_n \geq 0$.

**Example**

To produce a $2$-dimensional SIM-body, use for example the following code. Note that the polytope lives in $3$-space, so we project it down to $2$-space by eliminating the last coordinate.

```
julia> s = SIM_body_polytope([3,1])
Polyhedron in ambient dimension 3
julia> p = convex_hull(map(x->x[1:dim(s)],vertices(s)))
Polyhedron in ambient dimension 2
julia> vertices(p)
5-element SubObjectIterator{PointVector{QQFieldElem}}:
[0, 0]
[3, 0]
[3, 1]
[0, 3]
[1, 3]
```

`associahedron`

— Function`associahedron(d::Int)`

Produce a $d$-dimensional associahedron (or Stasheff polytope). We use the facet description given in section 9.2. of [Zie95].

Note that in polymake, this function has an optional Boolean parameter `group`

, to also construct the symmetry group of the polytope. For details, see [CSZ15].

**Example**

Produce the $2$-dimensional associahedron is a polygon in $\mathbb{R}⁴$ having $5$ vertices and $5$ facets.

```
julia> A = associahedron(2)
Polyhedron in ambient dimension 4
julia> vertices(A)
5-element SubObjectIterator{PointVector{QQFieldElem}}:
[9, 4, 1, 10]
[10, 1, 4, 9]
[1, 10, 1, 9]
[1, 4, 9, 6]
[4, 1, 10, 6]
julia> facets(A)
5-element SubObjectIterator{AffineHalfspace{QQFieldElem}} over the Halfspaces of R^4 described by:
-x_1 <= -1
-2*x_1 - 2*x_2 <= -10
-x_2 <= -1
-2*x_2 - 2*x_3 <= -10
-x_3 <= -1
```

`billera_lee_polytope`

— Function`billera_lee_polytope(h::AbstractVector)`

Construct a simplicial polytope whose h-vector is $h$. The corresponding g-vector must be an M-sequence. The ambient dimension equals the length of $h$, and the polytope lives in codimension one.

- [BL81]

**Examples**

```
julia> BL = billera_lee_polytope([1,3,3,1])
Polyhedron in ambient dimension 4
julia> f_vector(BL)
3-element Vector{ZZRingElem}:
6
12
8
```

`binary_markov_graph_polytope`

— Function`binary_markov_graph_polytope(observation::AbstractVector)`

Defines a very simple graph for a polytope propagation related to a Hidden Markov Model. The length of `observation`

is the number of possible oberservations. Its elements are of types `Bool`

or `Int`

. The propagated polytope is always a polygon. For a detailed description see [Jos05].

**Examples**

```
julia> P = binary_markov_graph_polytope([1,1,1,1])
Polyhedron in ambient dimension 2
julia> vertices(P)
4-element SubObjectIterator{PointVector{QQFieldElem}}:
[3, 0]
[1, 1]
[0, 2]
[0, 7]
```

`birkhoff_polytope`

— Function`birkhoff_polytope(n::Integer, even::Bool = false)`

Construct the Birkhoff polytope of dimension $n^2$.

This is the polytope of $n \times n$ stochastic matrices (encoded as row vectors of length $n^2$), i.e., the matrices with non-negative real entries whose row and column entries sum up to one. Its vertices are the permutation matrices.

Use `even = true`

to get the vertices only for the even permutation matrices.

**Examples**

```
julia> b = birkhoff_polytope(3)
Polytope in ambient dimension 9
julia> vertices(b)
6-element SubObjectIterator{PointVector{QQFieldElem}}:
[1, 0, 0, 0, 1, 0, 0, 0, 1]
[0, 1, 0, 1, 0, 0, 0, 0, 1]
[0, 0, 1, 1, 0, 0, 0, 1, 0]
[1, 0, 0, 0, 0, 1, 0, 1, 0]
[0, 1, 0, 0, 0, 1, 1, 0, 0]
[0, 0, 1, 0, 1, 0, 1, 0, 0]
```

`cyclic_caratheodory_polytope`

— Function`cyclic_caratheodory_polytope(d::Int, n::Int)`

Produce a $d$-dimensional cyclic polytope with $n$ points. Clearly $n\geq d$ is required. It is a prototypical example of a neighborly polytope whose combinatorics completely known due to Gale's evenness criterion. The coordinates are chosen on the trigonometric moment curve.

**Example**

```
julia> C= cyclic_caratheodory_polytope(4,5)
Polytope in ambient dimension 4
julia> vertices(C)
5-element SubObjectIterator{PointVector{QQFieldElem}}:
[1, 0, 1, 0]
[347922205179541//1125899906842624, 8566355544790271//9007199254740992, -7286977268806823//9007199254740992, 5294298886396511//9007199254740992]
[-7286977268806823//9007199254740992, 5294298886396511//9007199254740992, 1391688820718163//4503599627370496, -33462326346837//35184372088832]
[-7286977268806825//9007199254740992, -5294298886396509//9007199254740992, 5566755282872661//18014398509481984, 8566355544790271//9007199254740992]
[1391688820718163//4503599627370496, -33462326346837//35184372088832, -3643488634403413//4503599627370496, -5294298886396507//9007199254740992]
```

`cyclic_polytope`

— Function`cyclic_polytope(d::Int, n::Int)`

Construct the cyclic polytope that is the convex hull of $n$ points on the moment curve in dimension $d$.

**Examples**

```
julia> cp = cyclic_polytope(3, 20)
Polytope in ambient dimension 3
julia> n_vertices(cp)
20
```

`del_pezzo_polytope`

— Function`del_pezzo_polytope(d::Int)`

Produce the d-dimensional del Pezzo polytope, which is the convex hull of the cross polytope together with the all-ones and minus all-ones vector.

**Examples**

```
julia> DP = del_pezzo_polytope(4)
Polytope in ambient dimension 4
julia> f_vector(DP)
4-element Vector{ZZRingElem}:
10
40
60
30
```

`dwarfed_cube`

— Function`dwarfed_cube(d::Int)`

Produce the $d$-dimensional dwarfed cube as defined in [ABS97].

**Example**

The $3$-dimensional dwarfed cube is illustrated in [Jos03].

```
julia> c = dwarfed_cube(3)
Polytope in ambient dimension 3
julia> vertices(c)
10-element SubObjectIterator{PointVector{QQFieldElem}}:
[1//2, 0, 1]
[1//2, 1, 0]
[1, 0, 1//2]
[1, 1//2, 0]
[1, 0, 0]
[0, 0, 0]
[0, 1, 0]
[0, 1//2, 1]
[0, 0, 1]
[0, 1, 1//2]
```

`dwarfed_product_polygons`

— Function`dwarfed_product_polygons(d::Int, s::Int)`

Produce a $d$-dimensional dwarfed product of polygons of size $s$ as defined in [ABS97]. It must be $d\geq4$ and even as well as $s\geq 3$.

**Example**

```
julia> p = dwarfed_product_polygons(4,3)
Polytope in ambient dimension 4
julia> vertices(p)
11-element SubObjectIterator{PointVector{QQFieldElem}}:
[5, 3, 0, 0]
[5, 0, 0, 0]
[2, 0, 3, 9]
[0, 0, 5, 3]
[0, 0, 3, 9]
[2, 6, 3, 9]
[0, 0, 5, 0]
[0, 0, 0, 0]
[3, 9, 2, 6]
[3, 9, 2, 0]
[3, 9, 0, 0]
```

`explicit_zonotope`

— Function`explicit_zonotope(zones::Matrix; rows_are_points::Bool=true)`

Produce the points of a zonotope as the iterated Minkowski sum of all intervals $[-x,x]$, where $x$ ranges over the rows of the input matrix `zones`

. If `rows_are_points`

is `true`

(default), the rows of the input matrix represent affine points, otherwise they represent linear vectors.

**Examples**

```
julia> Z = explicit_zonotope([1 1; 1 -1], rows_are_points=false)
Polyhedron in ambient dimension 2
julia> vertices(Z)
4-element SubObjectIterator{PointVector{QQFieldElem}}:
[2, 0]
[0, -2]
[0, 2]
[-2, 0]
```

`fano_simplex`

— Function`fano_simplex(d::Int)`

Construct a lattice simplex such that the origin is the unique interior lattice point. The normal toric variety associated with its face fan is smooth.

**Keywords**

`d::Int`

: the dimension.

**Examples**

```
julia> S = fano_simplex(3)
Polytope in ambient dimension 3
julia> X = normal_toric_variety(face_fan(S))
Normal toric variety
julia> is_smooth(X)
true
```

`fractional_cut_polytope`

— Function`fractional_cut_polytope(G::Graph{Undirected})`

Construct the fractional cut polytope of the graph $G$.

**Examples**

```
julia> G = complete_graph(4);
julia> fractional_cut_polytope(G)
Polytope in ambient dimension 6
```

`fractional_knapsack_polytope`

— Function`fractional_knapsack_polytope(b::AbstractVector{<:Base.Number})`

Produce a knapsack polytope defined by one linear inequality (and non-negativity constraints).

**Example**

```
julia> f = fractional_knapsack_polytope([10,-2,-3,-5])
Polytope in ambient dimension 3
julia> print_constraints(f)
2*x_1 + 3*x_2 + 5*x_3 <= 10
-x_1 <= 0
-x_2 <= 0
-x_3 <= 0
```

`fractional_matching_polytope`

— Function`fractional_matching_polytope(G::Graph{Undirected})`

Construct the fractional matching polytope of the graph $G$.

**Examples**

```
julia> G = complete_graph(4);
julia> fractional_matching_polytope(G)
Polytope in ambient dimension 6
```

`gelfand_tsetlin_polytope`

— Function`gelfand_tsetlin_polytope(lambda::AbstractVector)`

Construct the Gelfand-Tsetlin polytope indexed by a weakly decreasing vector `lambda`

.

**Examples**

```
julia> P = gelfand_tsetlin_polytope([5,3,2])
Polyhedron in ambient dimension 6
julia> is_fulldimensional(P)
false
julia> p = project_full(P)
Polyhedron in ambient dimension 3
julia> is_fulldimensional(p)
true
julia> volume(p)
3
```

`gelfand_tsetlin_polytope(lambda::AbstractVector, sigma::PermGroupElem)`

Construct the generalized Gelfand-Tsetlin polytope indexed by a weakly decreasing vector `lambda`

and a permutation `sigma`

.

- [PS09]

```
julia> P = gelfand_tsetlin_polytope([5,3,2], @perm (1,3,2))
Polyhedron in ambient dimension 6
```

`goldfarb_cube`

— Function`goldfarb_cube(d::Int, e::Number, g::Number)`

Produce a `d`

-dimensional Goldfarb cube. The first parameter of deformation `e`

must be $<\frac{1}{2}$, the second parameter of deformation `d`

must be $\geq \frac{\texttt{e}}{4}$.

The Goldfarb cube is a combinatorial cube and yields a bad example for the Simplex Algorithm using the Shadow Vertex Pivoting Strategy. Here we use the description as a deformed product due to [AZ99]. For $g=0$ we obtain a Klee-Minty cube, in particular for $e=g=0$ we obtain the standard cube.

**Example**

The following produces a $3$-dimensional Klee-Minty cube for $e=\frac{1}{3}$.

```
julia> c = goldfarb_cube(3,1//3,0)
Polytope in ambient dimension 3
julia> vertices(c)
8-element SubObjectIterator{PointVector{QQFieldElem}}:
[1, 1//3, 8//9]
[1, 2//3, 7//9]
[1, 2//3, 2//9]
[1, 1//3, 1//9]
[0, 0, 0]
[0, 1, 1//3]
[0, 1, 2//3]
[0, 0, 1]
```

`goldfarb_sit_cube`

— Function`goldfarb_sit_cube(d::Int, eps::Number, delta::Number)`

Produces a `d`

-dimensional variation of the Klee-Minty cube, which is scaled in direction $x_{d-i}$ by `eps*delta^i`

. The first parameter of deformation `eps`

must be $<\frac{1}{2}$, the second parameter of deformation `delta`

must be $\geq \frac{1}{2}$. This cube is a combinatorial cube and yields a bad example for the Simplex Algorithm using the Steepest Edge Pivoting Strategy. Here we use a scaled description of the construction of Goldfarb and Sit, see [GS79].

**Examples**

```
julia> c = goldfarb_sit_cube(3,1//3,1//2)
Polytope in ambient dimension 3
julia> vertices(c)
8-element SubObjectIterator{PointVector{QQFieldElem}}:
[1//36, 1//18, 8//9]
[1//36, 1//9, 7//9]
[1//36, 1//9, 2//9]
[1//36, 1//18, 1//9]
[0, 0, 0]
[0, 1//6, 1//3]
[0, 1//6, 2//3]
[0, 0, 1]
```

`hypersimplex`

— Function`hypersimplex(k::Int, d::Int; no_vertices::Bool=false, no_facets::Bool=false, no_vif::Bool=false)`

Produce the hypersimplex $\Delta(k,d)$, that is the the convex hull of all $0/1$-vector in $\mathbb{R}^d$ with exactly $k$ ones. Note that the output is never full-dimensional.

**Optional Arguments**

`no_vertices::Bool`

: If set equal to`true`

, vertices of the underlying`polymake`

object are not computed.`no_facets::Bool`

: If set equal to`true`

, facets of the underlying`polymake`

object are not computed.`no_vif::Bool`

: If set equal to`true`

, vertices in facets of the underlying`polymake`

object are not computed.

**Example**

```
julia> H = hypersimplex(3,4)
Polytope in ambient dimension 4
julia> G = hypersimplex(3,4,no_facets=true)
Polytope in ambient dimension 4
julia> facets(G)
4-element SubObjectIterator{AffineHalfspace{QQFieldElem}} over the Halfspaces of R^4 described by:
x_4 <= 1
x_3 <= 1
-x_1 - x_3 - x_4 <= -2
x_1 <= 1
julia> facets(H)
4-element SubObjectIterator{AffineHalfspace{QQFieldElem}} over the Halfspaces of R^4 described by:
x_4 <= 1
x_3 <= 1
-x_1 - x_3 - x_4 <= -2
x_1 <= 1
```

`hypertruncated_cube`

— Function`hypertruncated_cube(d::Int, k::Number, lambda::Number)`

Produce a $d$-dimensional hypertruncated cube with symmetric linear objective function $(1,1,…,1)$.

**Arguments**

`k`

: cutoff parameter`lambda`

: scaling of extra vertex

**Example**

```
julia> H = hypertruncated_cube(3,2,3)
Polytope in ambient dimension 3
julia> print_constraints(H)
-x_1 <= 0
-x_2 <= 0
-x_3 <= 0
x_1 <= 1
x_2 <= 1
x_3 <= 1
5*x_1 - 2*x_2 - 2*x_3 <= 3
-2*x_1 + 5*x_2 - 2*x_3 <= 3
-2*x_1 - 2*x_2 + 5*x_3 <= 3
```

`k_cyclic_polytope`

— Function`k_cyclic_polytope(n::Int, s::Vector)`

Produce a (rounded) $2*k$-dimensional $k$-cyclic polytope with `n`

points, where $k$ is the length of the input vector `s`

. Special cases are the bicyclic ($k=2$) and tricyclic ($k=3$) polytopes. Only possible in even dimensions.

The parameters $\texttt{s}_i$ can be integers, floating-points or rational numbers. The $i$-th vertex then is: $(\cos(\texttt{s}_1 * 2\pi i/\texttt{n}), \sin(\texttt{s}_1 * 2\pi i/\texttt{n}), ... , \cos(\texttt{s}_k * 2\pi i/\texttt{n}), \sin(\texttt{s}_k * 2\pi i/\texttt{n}))$.

Warning: Some of the $k-$cyclic polytopes are not simplicial. Since the components are rounded, this function might output a polytope which is not a $k-$cyclic polytope! More information see [Sch95].

**Example**

To produce a (not exactly) regular pentagon, type this:

```
julia> p = k_cyclic_polytope(5,[1])
Polytope in ambient dimension 2
julia> dim(p)
2
julia> n_vertices(p)
5
```

`klee_minty_cube`

— Function`klee_minty_cube(d::Int, e::Number)`

Produces a $d-$dimensional Klee-Minty-cube if $\texttt{e} < 1/2$. Uses the `goldfarb_cube`

method with the argument $\texttt{g} = 0$.

#Example

```
julia> k = klee_minty_cube(3,1//8)
Polytope in ambient dimension 3
julia> print_constraints(k)
-x_1 <= 0
x_1 <= 1
1//8*x_1 - x_2 <= 0
1//8*x_1 + x_2 <= 1
1//8*x_2 - x_3 <= 0
1//8*x_2 + x_3 <= 1
```

`lecture_hall_simplex`

— Function`lecture_hall_simplex(d::Int)`

Produce the $d$-dimensional lecture hall simplex for the sequence $(s_i)=i$ for $1\geq i \geq d$ as defined in [SS12].

Note that in polymake, this function has an optional Boolean parameter `group`

, to also construct the symmetry group of the simplex.

**Example**

The $3$-dimensional lecture hall simplex:

```
julia> S = lecture_hall_simplex(3)
Polytope in ambient dimension 3
julia> vertices(S)
4-element SubObjectIterator{PointVector{QQFieldElem}}:
[0, 0, 0]
[0, 0, 3]
[0, 2, 3]
[1, 2, 3]
```

`max_GC_rank_polytope`

— Function`max_GC_rank_polytope(d::Int)`

Produce a `d`

-dimensional polytope of maximal Gomory-Chvatal rank $\Omega(d/\log(d))$, integrally infeasible. With symmetric linear objective function $(1,1..,1)$. Construction due to Pokutta and Schulz, see [PS11].

**Example**

```
julia> c = max_GC_rank_polytope(3)
Polytope in ambient dimension 3
julia> vertices(c)
6-element SubObjectIterator{PointVector{QQFieldElem}}:
[0, 1//2, 1//2]
[1//2, 0, 1//2]
[1//2, 1//2, 0]
[1//2, 1, 1//2]
[1//2, 1//2, 1]
[1, 1//2, 1//2]
```

`n_gon`

— Function`n_gon(n::Int; r::RationalUnion=1, alpha_0::RationalUnion=0)`

Produce a regular `n`

-gon. All vertices lie on a circle of radius `r`

(defaults to $1$) and initial angle divided by pi `alpha_0`

(defaults to $0$).

**Examples**

To store the regular pentagon in the variable p, do this:

```
julia> p = n_gon(3)
Polytope in ambient dimension 2
julia> n_gon(3,r=3)
Polytope in ambient dimension 2
```

`newton_polytope`

— Function`newton_polytope(poly::Polynomial)`

Compute the Newton polytope of the multivariate polynomial `poly`

.

**Examples**

```
julia> S, (x, y) = polynomial_ring(ZZ, ["x", "y"])
(Multivariate polynomial ring in 2 variables over ZZ, ZZMPolyRingElem[x, y])
julia> f = x^3*y + 3x*y^2 + 1
x^3*y + 3*x*y^2 + 1
julia> NP = newton_polytope(f)
Polyhedron in ambient dimension 2
julia> vertices(NP)
3-element SubObjectIterator{PointVector{QQFieldElem}}:
[3, 1]
[1, 2]
[0, 0]
```

`orbit_polytope`

— Function`orbit_polytope(V::AbstractCollection[PointVector], G::PermGroup)`

Construct the convex hull of the orbit of one or several points (given row-wise in `V`

) under the action of `G`

.

**Examples**

This will construct the $3$-dimensional permutahedron:

```
julia> V = [1 2 3];
julia> G = symmetric_group(3);
julia> P = orbit_polytope(V, G)
Polyhedron in ambient dimension 3
julia> vertices(P)
6-element SubObjectIterator{PointVector{QQFieldElem}}:
[1, 2, 3]
[1, 3, 2]
[2, 1, 3]
[2, 3, 1]
[3, 1, 2]
[3, 2, 1]
```

`permutahedron`

— Function`permutahedron(d::Int)`

Produce a `d`

-dimensional permutahedron. The vertices correspond to the elements of the symmetric group of degree `d`

$+1$.

#Example

```
julia> p = permutahedron(2)
Polytope in ambient dimension 3
julia> vertices(p)
6-element SubObjectIterator{PointVector{QQFieldElem}}:
[1, 2, 3]
[1, 3, 2]
[2, 1, 3]
[2, 3, 1]
[3, 1, 2]
[3, 2, 1]
```

`pile_polytope`

— Function`pile_polytope(sizes::Vector{Int})`

Produce a $($`d`

$+1)$-dimensional polytope from a pile of cubes. Start with a `d`

-dimensional pile of cubes. Take a generic convex function to lift this polytopal complex to the boundary of a $($`d`

$+1)$–polytope. The argument `sizes`

is a vector $(s_1,…,s_d)$ where $s_i$ specifies the number of boxes in the $i$-th dimension.

`pitman_stanley_polytope`

— Function`pitman_stanley_polytope(y::AbstractVector)`

Produce a Pitman-Stanley polytope of dimension $n-1$, where `y`

is a Vector of $n$ positive parameters. Does not check if the parameters are actually positive; negative values are legal but that do not yield a Pitman-Stanley polytope. Zeros just reduce the dimension; negative numbers may produce unbounded polyhedra.

**Example:**

Pitman-Stanley polytopes are combinatorial cubes:

```
julia> p = pitman_stanley_polytope([1,2,3])
Polyhedron in ambient dimension 3
julia> f_vector(p)
2-element Vector{ZZRingElem}:
4
4
```

`perles_nonrational_8_polytope`

— Function`perles_nonrational_8_polytope()`

Create an $8$-dimensional polytope without rational realizations due to Perles. See [Gru03].

**Example**

```
julia> perles_nonrational_8_polytope()
Polytope in ambient dimension 8 with EmbeddedAbsSimpleNumFieldElem type coefficients
```

`pseudo_del_pezzo_polytope`

— Function`pseudo_del_pezzo_polytope(d::Int)`

Produce a `d`

-dimensional del-Pezzo polytope, which is the convex hull of the cross polytope together with the all-ones vector. All coordinates are plus or minus one.

**Example**

```
julia> DP = pseudo_del_pezzo_polytope(4)
Polytope in ambient dimension 4
julia> f_vector(DP)
4-element Vector{ZZRingElem}:
9
32
46
23
```

`rand01_polytope`

— Function`rand01_polytope(d::Int, n::Int; seed=nothing)`

Produce a `d`

-dimensional $0/1$-polytope with `n`

random vertices. Uniform distribution.

**Optional Argument**

-`seed::Int`

: Seed for random number generation

**Example**

```
julia> s = rand01_polytope(2, 4; seed=3)
Polytope in ambient dimension 2
julia> vertices(s)
4-element SubObjectIterator{PointVector{QQFieldElem}}:
[1, 1]
[1, 0]
[0, 0]
[0, 1]
```

`rand_box_polytope`

— Function`rand_box_polytope(d::Int, n::Int, b::Int; seed::Int=nothing)`

Computes the convex hull of `n`

points sampled uniformly at random from the integer points in the cube $\[0,\texttt{b}\]^{\textttt{d}}$.

**Optional Argument**

-`seed`

: Seed for random number generation.

**Example**

```
julia> r = rand_box_polytope(3, 10, 3, seed=1)
Polyhedron in ambient dimension 3
julia> vertices(r)
8-element SubObjectIterator{PointVector{QQFieldElem}}:
[3, 2, 3]
[0, 3, 0]
[3, 3, 0]
[3, 0, 1]
[1, 1, 0]
[2, 0, 3]
[0, 3, 3]
[0, 1, 2]
```

`rand_cyclic_polytope`

— Function`rand_cyclic_polytope(d::Int, n::Int; seed::Int=nothing)`

Computes a random instance of a cyclic polytope of dimension `d`

on `n`

vertices by randomly generating a Gale diagram whose cocircuits have alternating signs.

**Optional Argument**

-`seed`

: Seed for random number generation

**Examples**

```
julia> r = rand_cyclic_polytope(3, 5)
Polytope in ambient dimension 3
julia> f_vector(r)
3-element Vector{ZZRingElem}:
5
9
6
```

`rand_metric`

— Function`rand_metric(n::Int; seed=nothing)`

Produce a rational n-point metric with random distances. The values are uniformily distributed in $\[1,2\]$.

**Examples**

```
julia> rand_metric(3, seed=132)
[ 0 260222460282405//140737488355328 371474612593257//281474976710656]
[260222460282405//140737488355328 0 388326899436839//281474976710656]
[371474612593257//281474976710656 388326899436839//281474976710656 0]
```

`rand_metric_int`

— Function`rand_metric_int(n::Int, digits::Int; seed=nothing)`

Produce a `n`

-point metric with random integral distances. The values are uniformily distributed in $\[1,2\]$. The distances are integers and lie in $[10^digits, 10^(digits+1)[$.

`rand_normal_polytope`

— Function`rand_normal_polytope(d::Int, n::Int; seed=nothing, precision=nothing)`

Produce a rational d-dimensional polytope from `n`

random points approximately normally distributed in the unit ball.

**Optional Arguments**

-`seed`

: controls the outcome of the random number generator; fixing a seed number guarantees the same outcome -`precision`

: number of bits for MPFR sphere approximation

**Example**

```
julia> rnp = rand_normal_polytope(2,4; seed=42, precision=4)
Polytope in ambient dimension 2
julia> is_simplicial(rnp)
true
julia> sort(map(x->dot(x,x), vertices(rnp)))
4-element Vector{QQFieldElem}:
1417//4096
481//1024
225//256
101//32
```

`rand_spherical_polytope`

— Function```
rand_spherical_polytope([rng::AbstractRNG,] d::Int, n::Int;
distribution=:uniform, precision=nothing, seed=nothing)
```

Construct the convex hull of $n$ points on the unit sphere in $\mathbb{R}^d$. Almost surely this is a simplicial polytope.

**Keywords**

`distribution::Symbol`

: One of the following two options:`:uniform`

(default): Use intermediate floating point numbers for an almost uniform distribution on the sphere. The points will not be exactly on the sphere.`:exact`

: Create exact rational points on the unit sphere, this works at the expense of both uniformity and log-height of the points.

`precision::Int64`

: Precision in bits during floating point approximation for uniform distribution.`seed::Int64`

: Seed for random number generation. Cannot be used together with the`AbstractRNG`

argument.

**Examples**

```
julia> rsph = rand_spherical_polytope(3, 20)
Polytope in ambient dimension 3
julia> is_simplicial(rsph)
true
julia> rsph = rand_spherical_polytope(3, 4; precision=5, seed=132)
Polytope in ambient dimension 3
julia> map(x->dot(x,x), vertices(rsph))
4-element Vector{QQFieldElem}:
4306545//4194304
15849//16384
4165//4096
8281//8192
julia> rsph = rand_spherical_polytope(3, 4; distribution=:exact)
Polytope in ambient dimension 3
julia> map(x->dot(x,x), vertices(rsph))
4-element Vector{QQFieldElem}:
1
1
1
1
```

`rand_subpolytope`

— Function`rand_subpolytope(P::Polyhedron, n::Int; seed=nothing)`

Construct a subpolytope of $P$ as the convex hull of $n$ vertices, chosen uniformly at random. The polyhedron $P$ must be bounded, and the number $n$ must not exceed the number of vertices.

**Keywords**

`seed::Int64`

: Seed for random number generation.

**Examples**

```
julia> n_vertices(rand_subpolytope(cube(3), 5))
5
```

`rss_associahedron`

— Functionrss_associahedron(n::Int)

Produce a polytope of constrained expansions in ambient dimension `n`

according to [RSS03].

**Examples:**

To produce a $3$-dimensional associahedron in $5$-space, do:

```
julia> a= rss_associahedron(5)
Polyhedron in ambient dimension 5
julia> vertices(a)
14-element SubObjectIterator{PointVector{QQFieldElem}}:
[0, 1, 12, 13, 16]
[0, 7, 8, 11, 16]
[0, 1, 4, 9, 16]
[0, 3, 4, 9, 16]
[0, 5, 6, 9, 16]
[0, 5, 8, 9, 16]
[0, 1, 8, 9, 16]
[0, 7, 10, 11, 16]
[0, 7, 12, 13, 16]
[0, 7, 12, 15, 16]
[0, 1, 12, 15, 16]
[0, 7, 8, 15, 16]
[0, 1, 4, 15, 16]
[0, 3, 4, 15, 16]
julia> facets(a)
9-element SubObjectIterator{AffineHalfspace{QQFieldElem}} over the Halfspaces of R^5 described by:
x_1 - x_2 <= -1
x_1 - x_3 <= -4
x_1 - x_4 <= -9
x_2 - x_3 <= -1
x_2 - x_4 <= -4
x_2 - x_5 <= -9
x_3 - x_4 <= -1
x_3 - x_5 <= -4
x_4 - x_5 <= -1
```

`signed_permutahedron`

— Function`signed_permutahedron(d::Int)`

Produce the `d`

-dimensional signed permutahedron. I.e. for all possible permutations of the vector $(1,\dots,d)$, all possible sign patterns define vertices of this polytope. Contrary to the classical permutahedron, the signed permutahedron is full-dimensional.

**Examples:**

To produce the $2$-dimensional signed permutahedron, do:

```
julia> P = signed_permutahedron(2)
Polytope in ambient dimension 2
julia> vertices(P)
8-element SubObjectIterator{PointVector{QQFieldElem}}:
[1, 2]
[-1, 2]
[1, -2]
[-1, -2]
[2, 1]
[-2, 1]
[2, -1]
[-2, -1]
```

`stable_set_polytope`

— Function`stable_set_polytope(G::Graph{Undirected})`

Produces the stable set polytope from an undirected graph `G`

$=(V,E)$. The stable set Polytope has the following inequalities: $x_i + x_j \leq 1 \forall \{i,j\} \in E$, $x_i \geq 0 \forall i \in V$ and $x_i \leq 1 \forall i \in V \text{ with } \mathrm{deg}(i)=0$

**Example:**

The following produces first the standard cube in $3$ dimensions, and then a bipyramid over the convex hull of the unit vectors.

```
julia> G = Graph{Undirected}(3)
Undirected graph with 3 nodes and no edges
julia> S = stable_set_polytope(G)
Polytope in ambient dimension 3
julia> vertices(S)
8-element SubObjectIterator{PointVector{QQFieldElem}}:
[1, 0, 0]
[1, 1, 0]
[1, 1, 1]
[1, 0, 1]
[0, 0, 1]
[0, 0, 0]
[0, 1, 0]
[0, 1, 1]
julia> add_edge!(G, 1, 2);
julia> add_edge!(G, 1, 3);
julia> add_edge!(G, 2, 3);
julia> S = stable_set_polytope(G)
Polytope in ambient dimension 3
julia> vertices(S)
5-element SubObjectIterator{PointVector{QQFieldElem}}:
[0, 0, 1]
[0, 1, 0]
[1, 0, 0]
[1//2, 1//2, 1//2]
[0, 0, 0]
```

`transportation_polytope`

— Function`transportation_polytope(r::AbstractVector, c::AbstractVector)`

Produce the transportation polytope from two vectors `r`

of length $m$ and `c`

of length $n$, i.e. all positive $m\times n$ Matrizes with row sums equal to `r`

and column sums equal to `c`

.

**Example:**

We can see that the set of $3\times 3$ magic squares with magic constant $15$ is a $4$-dimensional polytope.

```
julia> r = c = [15,15,15]
3-element Vector{Int64}:
15
15
15
julia> t = transportation_polytope(r,c)
Polytope in ambient dimension 9
julia> dim(t)
4
julia> is_bounded(t)
true
```

`zonotope`

— Function`zonotope(M::Matrix{<:Number}; centered::Bool=true)`

Create a zonotope from a matrix whose rows are input points.

**Optional Arguments**

`centered::Bool`

: This is`true`

if the output should be centered; the default is`true`

.

**Examples**

The following produces a parallelogram with the origin as its vertex barycenter:

```
julia> Z = zonotope([1 0; 1 1])
Polyhedron in ambient dimension 2
julia> vertices(Z)
4-element SubObjectIterator{PointVector{QQFieldElem}}:
[-1, -1//2]
[0, -1//2]
[0, 1//2]
[1, 1//2]
```

The following produces a parallelogram with the origin being a vertex (not centered case):

```
julia> Z = zonotope([1 0; 1 1], centered = false)
Polyhedron in ambient dimension 2
julia> vertices(Z)
4-element SubObjectIterator{PointVector{QQFieldElem}}:
[0, 0]
[1, 0]
[1, 1]
[2, 1]
```

`zonotope_vertices_fukuda_matrix`

— Function`zonotope_vertices_fukuda(M::Matrix)`

Create the vertices of a zonotope from a matrix whose rows are input points or vectors.

**Examples**

The following creates the vertices of a parallelogram with the origin as its vertex barycenter.

```
julia> zonotope_vertices_fukuda_matrix([1 1 0; 1 1 1])
pm::Matrix<pm::Rational>
-1 -1 -1/2
0 0 -1/2
0 0 1/2
1 1 1/2
```

## Operations on polyhedra

Polyhedra can be produced through operations on other polyhedra. For example, they can be added using Minkowski addition or scaled; each of which results in a new polyhedron.

`+`

— Method`+(P::Polyhedron, Q::Polyhedron)`

Return the Minkowski sum $P + Q = \{ x+y\ |\ x∈P, y∈Q\}$ of `P`

and `Q`

(see also `minkowski_sum`

).

**Examples**

The Minkowski sum of a square and the 2-dimensional cross-polytope is an octagon:

```
julia> P = cube(2);
julia> Q = cross_polytope(2);
julia> M = minkowski_sum(P, Q)
Polyhedron in ambient dimension 2
julia> n_vertices(M)
8
```

`*`

— Method`*(k::Union{Number, FieldElem}, Q::Polyhedron)`

Return the scaled polyhedron $kQ = \{ kx\ |\ x∈Q\}$.

Note that `k*Q = Q*k`

.

**Examples**

Scaling an $n$-dimensional bounded polyhedron by the factor $k$ results in the volume being scaled by $k^n$. This example confirms the statement for the 6-dimensional cube and $k = 2$.

```
julia> C = cube(6);
julia> SC = 2*C
Polyhedron in ambient dimension 6
julia> volume(SC)//volume(C)
64
```

`*`

— Method`*(P::Polyhedron, Q::Polyhedron)`

Return the Cartesian product of `P`

and `Q`

(see also `product`

).

**Examples**

The Cartesian product of a triangle and a line segment is a triangular prism.

```
julia> T=simplex(2)
Polytope in ambient dimension 2
julia> S=cube(1)
Polytope in ambient dimension 1
julia> length(vertices(T*S))
6
```

`bipyramid`

— Function`bipyramid(P::Polyhedron, z::Union{Number, FieldElem} = 1, z_prime::Union{Number, FieldElem} = -z)`

Make a bipyramid over a pointed polyhedron `P`

.

The bipyramid is the convex hull of the input polyhedron `P`

and two apexes (`v`

, `z`

), (`v`

, `z_prime`

) on both sides of the affine span of `P`

. For bounded polyhedra, the projections of the apexes `v`

to the affine span of `P`

is the vertex barycenter of `P`

.

**Examples**

```
julia> c = cube(2)
Polytope in ambient dimension 2
julia> vertices(bipyramid(c,2))
6-element SubObjectIterator{PointVector{QQFieldElem}}:
[-1, -1, 0]
[1, -1, 0]
[-1, 1, 0]
[1, 1, 0]
[0, 0, 2]
[0, 0, -2]
```

`intersect`

— Method`intersect(P::Polyhedron...)`

Return the intersection $\bigcap\limits_{p \in P} p$.

**Examples**

The positive orthant of the plane is the intersection of the two halfspaces with $x≥0$ and $y≥0$ respectively.

```
julia> UH1 = convex_hull([0 0],[1 0],[0 1]);
julia> UH2 = convex_hull([0 0],[0 1],[1 0]);
julia> PO = intersect(UH1, UH2)
Polyhedron in ambient dimension 2
julia> rays(PO)
2-element SubObjectIterator{RayVector{QQFieldElem}}:
[1, 0]
[0, 1]
```

`pyramid`

— Function`pyramid(P::Polyhedron, z::Union{Number, FieldElem} = 1)`

Make a pyramid over the given polyhedron `P`

.

The pyramid is the convex hull of the input polyhedron `P`

and a point `v`

outside the affine span of `P`

. For bounded polyhedra, the projection of `v`

to the affine span of `P`

coincides with the vertex barycenter of `P`

. The scalar `z`

is the distance between the vertex barycenter and `v`

.

**Examples**

```
julia> c = cube(2)
Polytope in ambient dimension 2
julia> vertices(pyramid(c,5))
5-element SubObjectIterator{PointVector{QQFieldElem}}:
[-1, -1, 0]
[1, -1, 0]
[-1, 1, 0]
[1, 1, 0]
[0, 0, 5]
```

`vertex_figure`

— Function`vertex_figure(P::Polyhedron, n::Int; cutoff=1//2)`

Construct the vertex figure of the vertex `n`

of a bounded polytope. The vertex figure is dual to a facet of the dual polytope.

**Optional Arguments**

`cutoff::Number`

: controls the exact location of the cutting hyperplane. It should lie in the open Interval $(0,1)$. Value $0$ would let the hyperplane go through the chosen vertex, thus degenerating the vertex figure to a single point. Value $1$ would let the hyperplane touch the nearest neighbor vertex of a polyhedron. Default value is $\frac{1}{2}$.

**Example**

To produce a triangular vertex figure of a $3$-dimensional cube in the positive orthant, do:

```
julia> T = vertex_figure(cube(3), 8)
Polyhedron in ambient dimension 3
julia> vertices(T)
3-element SubObjectIterator{PointVector{QQFieldElem}}:
[1, 1, 0]
[1, 0, 1]
[0, 1, 1]
julia> T = vertex_figure(cube(3), 8, cutoff = 1/4)
Polyhedron in ambient dimension 3
julia> vertices(T)
3-element SubObjectIterator{PointVector{QQFieldElem}}:
[1, 1, 1//2]
[1, 1//2, 1]
[1//2, 1, 1]
```

The convex hull of two polytopes can be computed via `convex_hull`

.

`convex_hull`

— Method`convex_hull(P::Polyhedron, Q::Polyhedron)`

Return the convex_hull of `P`

and `Q`

.

**Examples**

The convex hull of the following two line segments in $R^3$ is a tetrahedron.

```
julia> L₁ = convex_hull([-1 0 0; 1 0 0])
Polyhedron in ambient dimension 3
julia> L₂ = convex_hull([0 -1 0; 0 1 0])
Polyhedron in ambient dimension 3
julia> T=convex_hull(L₁,L₂);
julia> f_vector(T)
2-element Vector{ZZRingElem}:
4
4
```