Library of commonly used, famous, or interesting polytopes¶
This module gathers several constructors of polytopes that can be reached
through polytopes.<tab>
. For example, here is the hypercube in dimension 5:
sage: polytopes.hypercube(5)
A 5dimensional polyhedron in ZZ^5 defined as the convex hull of 32 vertices
The following constructions are available

class
sage.geometry.polyhedron.library.
Polytopes
¶ A class of constructors for commonly used, famous, or interesting polytopes.

Birkhoff_polytope
(n, backend=None)¶ Return the Birkhoff polytope with \(n!\) vertices.
The vertices of this polyhedron are the (flattened) \(n\) by \(n\) permutation matrices. So the ambient vector space has dimension \(n^2\) but the dimension of the polyhedron is \((n1)^2\).
INPUT:
n
– a positive integer giving the size of the permutation matrices.backend
– the backend to use to create the polytope.
See also
sage.matrix.matrix2.Matrix.as_sum_of_permutations()
– return the current matrix as a sum of permutation matricesEXAMPLES:
sage: b3 = polytopes.Birkhoff_polytope(3) sage: b3.f_vector() (1, 6, 15, 18, 9, 1) sage: b3.ambient_dim(), b3.dim() (9, 4) sage: b3.is_lattice_polytope() True sage: p3 = b3.ehrhart_polynomial() # optional  latte_int sage: p3 # optional  latte_int 1/8*t^4 + 3/4*t^3 + 15/8*t^2 + 9/4*t + 1 sage: [p3(i) for i in [1,2,3,4]] # optional  latte_int [6, 21, 55, 120] sage: [len((i*b3).integral_points()) for i in [1,2,3,4]] [6, 21, 55, 120] sage: b4 = polytopes.Birkhoff_polytope(4) sage: b4.n_vertices(), b4.ambient_dim(), b4.dim() (24, 16, 9)

Gosset_3_21
(backend=None)¶ Return the Gosset \(3_{21}\) polytope.
The Gosset \(3_{21}\) polytope is a uniform 7polytope. It has 56 vertices, and 702 facets: \(126\) \(3_{11}\) and \(576\) \(6\)simplex. For more information, see the Wikipedia article 3_21_polytope.
INPUT:
backend
– the backend to use to create the polytope.
EXAMPLES:
sage: g = polytopes.Gosset_3_21(); g A 7dimensional polyhedron in ZZ^8 defined as the convex hull of 56 vertices sage: g.f_vector() # not tested (~16s) (1, 56, 756, 4032, 10080, 12096, 6048, 702, 1)

Kirkman_icosahedron
(backend=None)¶ Return the Kirkman icosahedron.
The Kirkman icosahedron is a 3polytope with integer coordinates: \((\pm 9, \pm 6, \pm 6)\), \((\pm 12, \pm 4, 0)\), \((0, \pm 12, \pm 8)\), \((\pm 6, 0, \pm 12)\). See [Fe2012] for more information.
INPUT:
backend
– the backend to use to create the polytope.
EXAMPLES:
sage: ki = polytopes.Kirkman_icosahedron() sage: ki.f_vector() (1, 20, 38, 20, 1) sage: ki.volume() 6528 sage: vertices = ki.vertices() sage: edges = [[vector(edge[0]),vector(edge[1])] for edge in ki.bounded_edges()] sage: edge_lengths = [norm(edge[0]edge[1]) for edge in edges] sage: union(edge_lengths) [7, 8, 9, 11, 12, 14, 16]

static
associahedron
(cartan_type)¶ Construct an associahedron.
The generalized associahedron is a polytopal complex with vertices in onetoone correspondence with clusters in the cluster complex, and with edges between two vertices if and only if the associated two clusters intersect in codimension 1.
The associahedron of type \(A_n\) is one way to realize the classical associahedron as defined in the Wikipedia article Associahedron.
A polytopal realization of the associahedron can be found in [CFZ2002]. The implementation is based on [CFZ2002], Theorem 1.5, Remark 1.6, and Corollary 1.9.
EXAMPLES:
sage: Asso = polytopes.associahedron(['A',2]); Asso Generalized associahedron of type ['A', 2] with 5 vertices sage: sorted(Asso.Hrepresentation(), key=repr) [An inequality (1, 0) x + 1 >= 0, An inequality (0, 1) x + 1 >= 0, An inequality (0, 1) x + 1 >= 0, An inequality (1, 0) x + 1 >= 0, An inequality (1, 1) x + 1 >= 0] sage: Asso.Vrepresentation() (A vertex at (1, 1), A vertex at (1, 1), A vertex at (1, 1), A vertex at (1, 0), A vertex at (0, 1)) sage: polytopes.associahedron(['B',2]) Generalized associahedron of type ['B', 2] with 6 vertices
The two pictures of [CFZ2002] can be recovered with:
sage: Asso = polytopes.associahedron(['A',3]); Asso Generalized associahedron of type ['A', 3] with 14 vertices sage: Asso.plot() # long time Graphics3d Object sage: Asso = polytopes.associahedron(['B',3]); Asso Generalized associahedron of type ['B', 3] with 20 vertices sage: Asso.plot() # long time Graphics3d Object

buckyball
(exact=True, base_ring=None, backend=None)¶ Return the bucky ball.
The bucky ball, also known as the truncated icosahedron is an Archimedean solid. It has 32 faces and 60 vertices.
See also
INPUT:
exact
– (boolean, defaultTrue
) IfFalse
use an approximate ring for the coordinates.base_ring
– the ring in which the coordinates will belong to. If it is not provided andexact=True
it will be a the number field \(\QQ[\phi]\) where \(\phi\) is the golden ratio and ifexact=False
it will be the real double field.backend
– the backend to use to create the polytope.
EXAMPLES:
sage: bb = polytopes.buckyball() # long time  6secs sage: bb.f_vector() # long time (1, 60, 90, 32, 1) sage: bb.base_ring() # long time Number Field in sqrt5 with defining polynomial x^2  5
A much faster implementation using floating point approximations:
sage: bb = polytopes.buckyball(exact=False) sage: bb.f_vector() (1, 60, 90, 32, 1) sage: bb.base_ring() Real Double Field
Its faces are 5 regular pentagons and 6 regular hexagons:
sage: sum(1 for f in bb.faces(2) if len(f.vertices()) == 5) 12 sage: sum(1 for f in bb.faces(2) if len(f.vertices()) == 6) 20

cross_polytope
(dim, backend=None)¶ Return a crosspolytope in dimension
dim
.A crosspolytope is a higher dimensional generalization of the octahedron. It is the convex hull of the \(2d\) points \((\pm 1, 0, \ldots, 0)\), \((0, \pm 1, \ldots, 0)\), ldots, \((0, 0, \ldots, \pm 1)\). See the Wikipedia article Crosspolytope for more information.
INPUT:
dim
– integer. The dimension of the crosspolytope.backend
– the backend to use to create the polytope.
EXAMPLES:
sage: four_cross = polytopes.cross_polytope(4) sage: four_cross.f_vector() (1, 8, 24, 32, 16, 1) sage: four_cross.is_simple() False

cube
(backend=None)¶ Return the cube.
The cube is the Platonic solid that is obtained as the convex hull of the points \((\pm 1, \pm 1, \pm 1)\). It generalizes into several dimension into hypercubes.
See also
INPUT:
backend
– the backend to use to create the polytope.
EXAMPLES:
sage: c = polytopes.cube() sage: c A 3dimensional polyhedron in ZZ^3 defined as the convex hull of 8 vertices sage: c.f_vector() (1, 8, 12, 6, 1) sage: c.volume() 8 sage: c.plot() Graphics3d Object

cuboctahedron
(backend=None)¶ Return the cuboctahedron.
The cuboctahedron is an Archimedean solid with 12 vertices and 14 faces dual to the rhombic dodecahedron. It can be defined as the convex hull of the twelve vertices \((0, \pm 1, \pm 1)\), \((\pm 1, 0, \pm 1)\) and \((\pm 1, \pm 1, 0)\). For more information, see the Wikipedia article Cuboctahedron.
INPUT:
backend
– the backend to use to create the polytope.
See also
EXAMPLES:
sage: co = polytopes.cuboctahedron() sage: co.f_vector() (1, 12, 24, 14, 1)
Its faces are 8 triangles and 6 squares:
sage: sum(1 for f in co.faces(2) if len(f.vertices()) == 3) 8 sage: sum(1 for f in co.faces(2) if len(f.vertices()) == 4) 6
Some more computation:
sage: co.volume() 20/3 sage: co.ehrhart_polynomial() # optional  latte_int 20/3*t^3 + 8*t^2 + 10/3*t + 1

cyclic_polytope
(dim, n, base_ring=Rational Field, backend=None)¶ Return a cyclic polytope.
A cyclic polytope of dimension
dim
withn
vertices is the convex hull of the points(t,t^2,...,t^dim)
with \(t \in \{0,1,...,n1\}\) . For more information, see the Wikipedia article Cyclic_polytope.INPUT:
dim
– positive integer. the dimension of the polytope.n
– positive integer. the number of vertices.base_ring
– eitherQQ
(default) orRDF
.backend
– the backend to use to create the polytope.
EXAMPLES:
sage: c = polytopes.cyclic_polytope(4,10) sage: c.f_vector() (1, 10, 45, 70, 35, 1)

dodecahedron
(exact=True, base_ring=None, backend=None)¶ Return a dodecahedron.
The dodecahedron is the Platonic solid dual to the
icosahedron()
.INPUT:
exact
– (boolean, defaultTrue
) IfFalse
use an approximate ring for the coordinates.base_ring
– (optional) the ring in which the coordinates will belong to. Note that this ring must contain \(\sqrt(5)\). If it is not provided andexact=True
it will be the number field \(\QQ[\sqrt(5)]\) and ifexact=False
it will be the real double field.backend
– the backend to use to create the polytope.
EXAMPLES:
sage: d12 = polytopes.dodecahedron() sage: d12.f_vector() (1, 20, 30, 12, 1) sage: d12.volume() 176*sqrt5 + 400 sage: numerical_approx(_) 6.45203596003699 sage: d12 = polytopes.dodecahedron(exact=False) sage: d12.base_ring() Real Double Field
Here is an error with a field that does not contain \(\sqrt(5)\):
sage: polytopes.dodecahedron(base_ring=QQ) Traceback (most recent call last): ... TypeError: unable to convert 1/4*sqrt(5) + 1/4 to a rational

static
flow_polytope
(edges=None, ends=None)¶ Return the flow polytope of a digraph.
The flow polytope of a directed graph is the polytope consisting of all nonnegative flows on the graph with a given set \(S\) of sources and a given set \(T\) of sinks.
A flow on a directed graph \(G\) with a given set \(S\) of sources and a given set \(T\) of sinks means an assignment of a nonnegative real to each edge of \(G\) such that the flow is conserved in each vertex outside of \(S\) and \(T\), and there is a unit of flow entering each vertex in \(S\) and a unit of flow leaving each vertex in \(T\). These flows clearly form a polytope in the space of all assignments of reals to the edges of \(G\).
The polytope is empty unless the sets \(S\) and \(T\) are equinumerous.
By default, \(S\) is taken to be the set of all sources (i.e., vertices of indegree \(0\)) of \(G\), and \(T\) is taken to be the set of all sinks (i.e., vertices of outdegree \(0\)) of \(G\). If a different choice of \(S\) and \(T\) is desired, it can be specified using the optional
ends
parameter.The polytope is returned as a polytope in \(\RR^m\), where \(m\) is the number of edges of the digraph
self
. The \(k\)th coordinate of a point in the polytope is the real assigned to the \(k\)th edge ofself
. The order of the edges is the one returned byself.edges()
. If a different order is desired, it can be specified using the optionaledges
parameter.The faces and volume of these polytopes are of interest. Examples of these polytopes are the ChanRobbinsYuen polytope and the PitmanStanley polytope [PitSta].
INPUT:
edges
– (optional, default:self.edges()
) a list or tuple of all edges ofself
(each only once). This determines which coordinate of a point in the polytope will correspond to which edge ofself
. It is also possible to specify a list which contains not all edges ofself
; this results in a polytope corresponding to the flows which are \(0\) on all remaining edges. Notice that the edges entered here must be in the precisely same format as outputted byself.edges()
; so, ifself.edges()
outputs an edge in the form(1, 3, None)
, then(1, 3)
will not do!ends
– (optional, default:(self.sources(), self.sinks())
) a pair \((S, T)\) of an iterable \(S\) and an iterable \(T\).
Note
Flow polytopes can also be built through the
polytopes.<tab>
object:sage: polytopes.flow_polytope(digraphs.Path(5)) A 0dimensional polyhedron in QQ^4 defined as the convex hull of 1 vertex
EXAMPLES:
A commutative square:
sage: G = DiGraph({1: [2, 3], 2: [4], 3: [4]}) sage: fl = G.flow_polytope(); fl A 1dimensional polyhedron in QQ^4 defined as the convex hull of 2 vertices sage: fl.vertices() (A vertex at (0, 1, 0, 1), A vertex at (1, 0, 1, 0))
Using a different order for the edges of the graph:
sage: fl = G.flow_polytope(edges=G.edges(key=lambda x: x[0]  x[1])); fl A 1dimensional polyhedron in QQ^4 defined as the convex hull of 2 vertices sage: fl.vertices() (A vertex at (0, 1, 1, 0), A vertex at (1, 0, 0, 1))
A tournament on 4 vertices:
sage: H = digraphs.TransitiveTournament(4) sage: fl = H.flow_polytope(); fl A 3dimensional polyhedron in QQ^6 defined as the convex hull of 4 vertices sage: fl.vertices() (A vertex at (0, 0, 1, 0, 0, 0), A vertex at (0, 1, 0, 0, 0, 1), A vertex at (1, 0, 0, 0, 1, 0), A vertex at (1, 0, 0, 1, 0, 1))
Restricting to a subset of the edges:
sage: fl = H.flow_polytope(edges=[(0, 1, None), (1, 2, None), ....: (2, 3, None), (0, 3, None)]) sage: fl A 1dimensional polyhedron in QQ^4 defined as the convex hull of 2 vertices sage: fl.vertices() (A vertex at (0, 0, 0, 1), A vertex at (1, 1, 1, 0))
Using a different choice of sources and sinks:
sage: fl = H.flow_polytope(ends=([1], [3])); fl A 1dimensional polyhedron in QQ^6 defined as the convex hull of 2 vertices sage: fl.vertices() (A vertex at (0, 0, 0, 1, 0, 1), A vertex at (0, 0, 0, 0, 1, 0)) sage: fl = H.flow_polytope(ends=([0, 1], [3])); fl The empty polyhedron in QQ^6 sage: fl = H.flow_polytope(ends=([3], [0])); fl The empty polyhedron in QQ^6 sage: fl = H.flow_polytope(ends=([0, 1], [2, 3])); fl A 3dimensional polyhedron in QQ^6 defined as the convex hull of 5 vertices sage: fl.vertices() (A vertex at (0, 0, 1, 1, 0, 0), A vertex at (0, 1, 0, 0, 1, 0), A vertex at (1, 0, 0, 2, 0, 1), A vertex at (1, 0, 0, 1, 1, 0), A vertex at (0, 1, 0, 1, 0, 1)) sage: fl = H.flow_polytope(edges=[(0, 1, None), (1, 2, None), ....: (2, 3, None), (0, 2, None), ....: (1, 3, None)], ....: ends=([0, 1], [2, 3])); fl A 2dimensional polyhedron in QQ^5 defined as the convex hull of 4 vertices sage: fl.vertices() (A vertex at (0, 0, 0, 1, 1), A vertex at (1, 2, 1, 0, 0), A vertex at (1, 1, 0, 0, 1), A vertex at (0, 1, 1, 1, 0))
A digraph with one source and two sinks:
sage: Y = DiGraph({1: [2], 2: [3, 4]}) sage: Y.flow_polytope() The empty polyhedron in QQ^3
A digraph with one vertex and no edge:
sage: Z = DiGraph({1: []}) sage: Z.flow_polytope() A 0dimensional polyhedron in QQ^0 defined as the convex hull of 1 vertex
REFERENCES:
[PitSta] Jim Pitman, Richard Stanley, “A polytope related to empirical distributions, plane trees, parking functions, and the associahedron”, Arxiv math/9908029

grand_antiprism
(exact=True, backend=None)¶ Return the grand antiprism.
The grand antiprism is a 4dimensional nonWythoffian uniform polytope. The coordinates were taken from http://eusebeia.dyndns.org/4d/gap. For more information, see the Wikipedia article Grand_antiprism.
Warning
The coordinates are exact by default. The computation with exact coordinates is not as fast as with floating point approximations. If you find this method to be too slow, consider using floating point approximations
INPUT:
exact
 (boolean, defaultTrue
) ifFalse
use floating point approximations instead of exact coordinatesbackend
– the backend to use to create the polytope.
EXAMPLES:
sage: gap = polytopes.grand_antiprism() # not tested  very long time sage: gap # not tested  very long time A 4dimensional polyhedron in (Number Field in sqrt5 with defining polynomial x^2  5)^4 defined as the convex hull of 100 vertices
Computation with approximated coordinates is much faster:
sage: gap = polytopes.grand_antiprism(exact=False) # random sage: gap A 4dimensional polyhedron in RDF^4 defined as the convex hull of 100 vertices sage: gap.f_vector() (1, 100, 500, 720, 320, 1) sage: len(list(gap.bounded_edges())) 500

great_rhombicuboctahedron
(exact=True, base_ring=None, backend=None)¶ Return the great rhombicuboctahedron.
The great rhombicuboctahedron (or truncated cuboctahedron) is an Archimedean solid with 48 vertices and 26 faces. For more information see the Wikipedia article Truncated_cuboctahedron.
INPUT:
exact
– (boolean, defaultTrue
) IfFalse
use an approximate ring for the coordinates.base_ring
– the ring in which the coordinates will belong to. If it is not provided andexact=True
it will be a the number field \(\QQ[\phi]\) where \(\phi\) is the golden ratio and ifexact=False
it will be the real double field.backend
– the backend to use to create the polytope.
EXAMPLES:
sage: gr = polytopes.great_rhombicuboctahedron() # long time ~ 3sec sage: gr.f_vector() # long time (1, 48, 72, 26, 1)
A faster implementation is obtained by setting
exact=False
:sage: gr = polytopes.great_rhombicuboctahedron(exact=False) sage: gr.f_vector() (1, 48, 72, 26, 1)
Its faces are 4 squares, 8 regular hexagons and 6 regular octagons:
sage: sum(1 for f in gr.faces(2) if len(f.vertices()) == 4) 12 sage: sum(1 for f in gr.faces(2) if len(f.vertices()) == 6) 8 sage: sum(1 for f in gr.faces(2) if len(f.vertices()) == 8) 6

hypercube
(dim, backend=None)¶ Return a hypercube in the given dimension.
The \(d\) dimensional hypercube is the convex hull of the points \((\pm 1, \pm 1, \ldots, \pm 1)\) in \(\RR^d\). For more information see the Wikipedia article Hypercube.
INPUT:
dim
– integer. The dimension of the cube.backend
– the backend to use to create the polytope.
EXAMPLES:
sage: four_cube = polytopes.hypercube(4) sage: four_cube.is_simple() True sage: four_cube.base_ring() Integer Ring sage: four_cube.volume() 16 sage: four_cube.ehrhart_polynomial() # optional  latte_int 16*t^4 + 32*t^3 + 24*t^2 + 8*t + 1

hypersimplex
(dim, k, project=False, backend=None)¶ Return the hypersimplex in dimension
dim
and parameterk
.The hypersimplex \(\Delta_{d,k}\) is the convex hull of the vertices made of \(k\) ones and \(dk\) zeros. It lies in the \(d1\) hyperplane of vectors of sum \(k\). If you want a projected version to \(\RR^{d1}\) (with floating point coordinates) then set
project=True
in the options.See also
INPUT:
dim
– the dimensionn
– the numbers(1,...,n)
are permutedproject
– (boolean, defaultFalse
) ifTrue
, the polytope is (isometrically) projected to a vector space of dimensiondim1
. This operation turns the coordinates into floating point approximations and corresponds to the projection given by the matrix fromzero_sum_projection()
.backend
– the backend to use to create the polytope.
EXAMPLES:
sage: h_4_2 = polytopes.hypersimplex(4, 2) sage: h_4_2 A 3dimensional polyhedron in ZZ^4 defined as the convex hull of 6 vertices sage: h_4_2.f_vector() (1, 6, 12, 8, 1) sage: h_4_2.ehrhart_polynomial() # optional  latte_int 2/3*t^3 + 2*t^2 + 7/3*t + 1 sage: h_7_3 = polytopes.hypersimplex(7, 3, project=True) sage: h_7_3 A 6dimensional polyhedron in RDF^6 defined as the convex hull of 35 vertices sage: h_7_3.f_vector() (1, 35, 210, 350, 245, 84, 14, 1)

icosahedron
(exact=True, base_ring=None, backend=None)¶ Return an icosahedron with edge length 1.
The icosahedron is one of the Platonic solids. It has 20 faces and is dual to the
dodecahedron()
.INPUT:
exact
– (boolean, defaultTrue
) IfFalse
use an approximate ring for the coordinates.base_ring
– (optional) the ring in which the coordinates will belong to. Note that this ring must contain \(\sqrt(5)\). If it is not provided andexact=True
it will be the number field \(\QQ[\sqrt(5)]\) and ifexact=False
it will be the real double field.backend
– the backend to use to create the polytope.
EXAMPLES:
sage: ico = polytopes.icosahedron() sage: ico.f_vector() (1, 12, 30, 20, 1) sage: ico.volume() 5/12*sqrt5 + 5/4
Its non exact version:
sage: ico = polytopes.icosahedron(exact=False) sage: ico.base_ring() Real Double Field sage: ico.volume() # known bug (trac 18214) 2.181694990...
A version using \(AA <sage.rings.qqbar.AlgebraicRealField>\):
sage: ico = polytopes.icosahedron(base_ring=AA) # long time sage: ico.base_ring() # long time Algebraic Real Field sage: ico.volume() # long time 2.181694990624913?
Note that if base ring is provided it must contain the square root of \(5\). Otherwise you will get an error:
sage: polytopes.icosahedron(base_ring=QQ) Traceback (most recent call last): ... TypeError: unable to convert 1/4*sqrt(5) + 1/4 to a rational

icosidodecahedron
(exact=True, backend=None)¶ Return the icosidodecahedron.
The Icosidodecahedron is a polyhedron with twenty triangular faces and twelve pentagonal faces. For more information see the Wikipedia article Icosidodecahedron.
INPUT:
exact
– (boolean, defaultTrue
) IfFalse
use an approximate ring for the coordinates.backend
– the backend to use to create the polytope.
EXAMPLES:
sage: id = polytopes.icosidodecahedron() sage: id.f_vector() (1, 30, 60, 32, 1)

icosidodecahedron_V2
(exact=True, base_ring=None, backend=None)¶ Return the icosidodecahedron.
The icosidodecahedron is an Archimedean solid. It has 32 faces and 30 vertices. For more information, see the Wikipedia article Icosidodecahedron.
INPUT:
exact
– (boolean, defaultTrue
) IfFalse
use an approximate ring for the coordinates.base_ring
– the ring in which the coordinates will belong to. If it is not provided andexact=True
it will be a the number field \(\QQ[\phi]\) where \(\phi\) is the golden ratio and ifexact=False
it will be the real double field.backend
– the backend to use to create the polytope.
EXAMPLES:
sage: id = polytopes.icosidodecahedron() # long time  6secs sage: id.f_vector() # long time (1, 30, 60, 32, 1) sage: id.base_ring() # long time Number Field in sqrt5 with defining polynomial x^2  5
A much faster implementation using floating point approximations:
sage: id = polytopes.icosidodecahedron(exact=False) sage: id.f_vector() (1, 30, 60, 32, 1) sage: id.base_ring() Real Double Field
Its faces are 20 triangles and 12 regular pentagons:
sage: sum(1 for f in id.faces(2) if len(f.vertices()) == 3) 20 sage: sum(1 for f in id.faces(2) if len(f.vertices()) == 5) 12

octahedron
(backend=None)¶ Return the octahedron.
The octahedron is a Platonic solid with 6 vertices and 8 faces dual to the cube. It can be defined as the convex hull of the six vertices \((0, 0, \pm 1)\), \((\pm 1, 0, 0)\) and \((0, \pm 1, 0)\). For more information, see the Wikipedia article Octahedron.
INPUT:
backend
– the backend to use to create the polytope.
EXAMPLES:
sage: co = polytopes.octahedron() sage: co.f_vector() (1, 6, 12, 8, 1)
Its faces are 8 triangles:
sage: sum(1 for f in co.faces(2) if len(f.vertices()) == 3) 8
Some more computation:
sage: co.volume() 4/3 sage: co.ehrhart_polynomial() # optional  latte_int 4/3*t^3 + 2*t^2 + 8/3*t + 1

parallelotope
(generators, backend=None)¶ Return the zonotope, or parallelotope, spanned by the generators.
The parallelotope is the multidimensional generalization of a parallelogram (2 generators) and a parallelepiped (3 generators).
INPUT:
generators
– a list of vectors of same dimensionbackend
– the backend to use to create the polytope.
EXAMPLES:
sage: polytopes.parallelotope([ (1,0), (0,1) ]) A 2dimensional polyhedron in ZZ^2 defined as the convex hull of 4 vertices sage: polytopes.parallelotope([[1,2,3,4],[0,1,0,7],[3,1,0,2],[0,0,1,0]]) A 4dimensional polyhedron in ZZ^4 defined as the convex hull of 16 vertices sage: K = QuadraticField(2, 'sqrt2') sage: sqrt2 = K.gen() sage: polytopes.parallelotope([ (1,sqrt2), (1,1) ]) A 2dimensional polyhedron in (Number Field in sqrt2 with defining polynomial x^2  2)^2 defined as the convex hull of 4 vertices

pentakis_dodecahedron
(exact=True, base_ring=None, backend=None)¶ Return the pentakis dodecahedron.
The pentakis dodecahedron (orkisdodecahedron) is a faceregular, vertexuniform polytope dual to the truncated icosahedron. It has 60 faces and 32 vertices. See the Wikipedia article Pentakis_dodecahedron for more information.
INPUT:
exact
– (boolean, defaultTrue
) IfFalse
use an approximate ring for the coordinates.base_ring
– the ring in which the coordinates will belong to. If it is not provided andexact=True
it will be a the number field \(\QQ[\phi]\) where \(\phi\) is the golden ratio and ifexact=False
it will be the real double field.backend
– the backend to use to create the polytope.
EXAMPLES:
sage: pd = polytopes.pentakis_dodecahedron() # long time  ~10 sec sage: pd.n_vertices() # long time 32 sage: pd.n_inequalities() # long time 60
A much faster implementation is obtained when setting
exact=False
:sage: pd = polytopes.pentakis_dodecahedron(exact=False) sage: pd.n_vertices() 32 sage: pd.n_inequalities() 60
The 60 are triangles:
sage: all(len(f.vertices()) == 3 for f in pd.faces(2)) True

permutahedron
(n, project=False, backend=None)¶ Return the standard permutahedron of (1,…,n).
The permutahedron (or permutohedron) is the convex hull of the permutations of \(\{1,\ldots,n\}\) seen as vectors. The edges between the permutations correspond to multiplication on the right by an elementary transposition in the
SymmetricGroup
.If we take the graph in which the vertices correspond to vertices of the polyhedron, and edges to edges, we get the
BubbleSortGraph()
.INPUT:
n
– integerproject
– (boolean, defaultFalse
) ifTrue
, the polytope is (isometrically) projected to a vector space of dimensiondim1
. This operation turns the coordinates into floating point approximations and corresponds to the projection given by the matrix fromzero_sum_projection()
.backend
– the backend to use to create the polytope.
EXAMPLES:
sage: perm4 = polytopes.permutahedron(4) sage: perm4 A 3dimensional polyhedron in ZZ^4 defined as the convex hull of 24 vertices sage: perm4.is_lattice_polytope() True sage: perm4.ehrhart_polynomial() # optional  latte_int 16*t^3 + 15*t^2 + 6*t + 1 sage: perm4 = polytopes.permutahedron(4, project=True) sage: perm4 A 3dimensional polyhedron in RDF^3 defined as the convex hull of 24 vertices sage: perm4.plot() Graphics3d Object sage: perm4.graph().is_isomorphic(graphs.BubbleSortGraph(4)) True
See also

regular_polygon
(n, exact=True, base_ring=None, backend=None)¶ Return a regular polygon with \(n\) vertices.
INPUT:
n
– a positive integer, the number of vertices.exact
– (boolean, defaultTrue
) ifFalse
floating point numbers are used for coordinates.base_ring
– a ring in which the coordinates will lie. It isNone
by default. If it is not provided andexact
isTrue
then it will be the field of real algebraic number, ifexact
isFalse
it will be the real double field.backend
– the backend to use to create the polytope.
EXAMPLES:
sage: octagon = polytopes.regular_polygon(8) sage: octagon A 2dimensional polyhedron in AA^2 defined as the convex hull of 8 vertices sage: octagon.n_vertices() 8 sage: v = octagon.volume() sage: v 2.828427124746190? sage: v == 2*QQbar(2).sqrt() True
Its non exact version:
sage: polytopes.regular_polygon(3, exact=False).vertices() (A vertex at (0.0, 1.0), A vertex at (0.8660254038, 0.5), A vertex at (0.8660254038, 0.5)) sage: polytopes.regular_polygon(25, exact=False).n_vertices() 25

rhombic_dodecahedron
(backend=None)¶ Return the rhombic dodecahedron.
The rhombic dodecahedron is a a polytope dual to the cuboctahedron. It has 14 vertices and 12 faces. For more information see the Wikipedia article Rhombic_dodecahedron.
INPUT:
backend
– the backend to use to create the polytope.
See also
EXAMPLES:
sage: rd = polytopes.rhombic_dodecahedron() sage: rd.f_vector() (1, 14, 24, 12, 1)
Its faces are 12 quadrilaterals (not all identical):
sage: sum(1 for f in rd.faces(2) if len(f.vertices()) == 4) 12
Some more computations:
sage: p = rd.ehrhart_polynomial() # optional  latte_int sage: p # optional  latte_int 16*t^3 + 12*t^2 + 4*t + 1 sage: [p(i) for i in [1,2,3,4]] # optional  latte_int [33, 185, 553, 1233] sage: [len((i*rd).integral_points()) for i in [1,2,3,4]] [33, 185, 553, 1233]

rhombicosidodecahedron
(exact=True, base_ring=None, backend=None)¶ Return the rhombicosidodecahedron.
The rhombicosidodecahedron is an Archimedean solid. It has 62 faces and 60 vertices. For more information, see the Wikipedia article Rhombicosidodecahedron.
INPUT:
exact
– (boolean, defaultTrue
) IfFalse
use an approximate ring for the coordinates.base_ring
– the ring in which the coordinates will belong to. If it is not provided andexact=True
it will be a the number field \(\QQ[\phi]\) where \(\phi\) is the golden ratio and ifexact=False
it will be the real double field.backend
– the backend to use to create the polytope.
EXAMPLES:
sage: rid = polytopes.rhombicosidodecahedron() # long time  6secs sage: rid.f_vector() # long time (1, 60, 120, 62, 1) sage: rid.base_ring() # long time Number Field in sqrt5 with defining polynomial x^2  5
A much faster implementation using floating point approximations:
sage: rid = polytopes.rhombicosidodecahedron(exact=False) sage: rid.f_vector() (1, 60, 120, 62, 1) sage: rid.base_ring() Real Double Field
Its faces are 20 triangles, 30 squares and 12 pentagons:
sage: sum(1 for f in rid.faces(2) if len(f.vertices()) == 3) 20 sage: sum(1 for f in rid.faces(2) if len(f.vertices()) == 4) 30 sage: sum(1 for f in rid.faces(2) if len(f.vertices()) == 5) 12

simplex
(dim=3, project=False, base_ring=None, backend=None)¶ Return the
dim
dimensional simplex.The \(d\)simplex is the convex hull in \(\RR^{d+1}\) of the standard basis \((1,0,\ldots,0)\), \((0,1,\ldots,0)\), ldots, \((0,0,\ldots,1)\). For more information, see the Wikipedia article Simplex.
INPUT:
dim
– The dimension of the simplex, a positive integer.project
– (boolean, defaultFalse
) ifTrue
, the polytope is (isometrically) projected to a vector space of dimensiondim1
. This corresponds to the projection given by the matrix fromzero_sum_projection()
. By default, this operation turns the coordinates into floating point approximations (seebase_ring
).base_ring
– the base ring to use to create the polytope. Ifproject
isFalse
, this defaults to \(\ZZ\). Otherwise, it defaults toRDF
.backend
– the backend to use to create the polytope.
See also
EXAMPLES:
sage: s5 = polytopes.simplex(5) sage: s5 A 5dimensional polyhedron in ZZ^6 defined as the convex hull of 6 vertices sage: s5.f_vector() (1, 6, 15, 20, 15, 6, 1) sage: s5 = polytopes.simplex(5, project=True) sage: s5 A 5dimensional polyhedron in RDF^5 defined as the convex hull of 6 vertices
Its volume is \(\sqrt{d+1} / d!\):
sage: s5 = polytopes.simplex(5, project=True) sage: s5.volume() # abs tol 1e10 0.0204124145231931 sage: sqrt(6.) / factorial(5) 0.0204124145231931 sage: s6 = polytopes.simplex(6, project=True) sage: s6.volume() # abs tol 1e10 0.00367465459870082 sage: sqrt(7.) / factorial(6) 0.00367465459870082
Computation in algebraic reals:
sage: s3 = polytopes.simplex(3, project=True, base_ring=AA) sage: s3.volume() == sqrt(3+1) / factorial(3) True

six_hundred_cell
(exact=False, backend=None)¶ Return the standard 600cell polytope.
The 600cell is a 4dimensional regular polytope. In many ways this is an analogue of the icosahedron.
Warning
The coordinates are not exact by default. The computation with exact coordinates takes a huge amount of time.
INPUT:
exact
 (boolean, defaultFalse
) ifTrue
use exact coordinates instead of floating point approximationsbackend
– the backend to use to create the polytope.
EXAMPLES:
sage: p600 = polytopes.six_hundred_cell() sage: p600 A 4dimensional polyhedron in RDF^4 defined as the convex hull of 120 vertices sage: p600.f_vector() # long time ~2sec (1, 120, 720, 1200, 600, 1)
Computation with exact coordinates is currently too long to be useful:
sage: p600 = polytopes.six_hundred_cell(exact=True) # not tested  very long time sage: len(list(p600.bounded_edges())) # not tested  very long time 120

small_rhombicuboctahedron
(exact=True, base_ring=None, backend=None)¶ Return the (small) rhombicuboctahedron.
The rhombicuboctahedron is an Archimedean solid with 24 vertices and 26 faces. See the Wikipedia article Rhombicuboctahedron for more information.
INPUT:
exact
– (boolean, defaultTrue
) IfFalse
use an approximate ring for the coordinates.base_ring
– the ring in which the coordinates will belong to. If it is not provided andexact=True
it will be a the number field \(\QQ[\phi]\) where \(\phi\) is the golden ratio and ifexact=False
it will be the real double field.backend
– the backend to use to create the polytope.
EXAMPLES:
sage: sr = polytopes.small_rhombicuboctahedron() sage: sr.f_vector() (1, 24, 48, 26, 1) sage: sr.volume() 80/3*sqrt2 + 32
The faces are \(8\) equilateral triangles and \(18\) squares:
sage: sum(1 for f in sr.faces(2) if len(f.vertices()) == 3) 8 sage: sum(1 for f in sr.faces(2) if len(f.vertices()) == 4) 18
Its non exact version:
sage: sr = polytopes.small_rhombicuboctahedron(False) sage: sr A 3dimensional polyhedron in RDF^3 defined as the convex hull of 24 vertices sage: sr.f_vector() (1, 24, 48, 26, 1)

snub_cube
(backend=None)¶ Return a snub cube.
The snub cube is an Archimedean solid. It has 24 vertices and 38 faces. For more information see the Wikipedia article Snub_cube.
It uses the real double field for the coordinates.
INPUT:
backend
– the backend to use to create the polytope.
EXAMPLES:
sage: sc = polytopes.snub_cube() sage: sc.f_vector() (1, 24, 60, 38, 1)

snub_dodecahedron
(base_ring=None, backend=None)¶ Return the snub dodecahedron.
The snub dodecahedron is an Archimedean solid. It has 92 faces and 60 vertices. For more information, see the Wikipedia article Snub_dodecahedron.
INPUT:
base_ring
– the ring in which the coordinates will belong to. If it is not provided it will be the real double field.backend
– the backend to use to create the polytope.
EXAMPLES:
Unfortunately, no polyhedra backend supports the construction of the snub dodecahedron at the moment:
sage: sd = polytopes.snub_dodecahedron() Traceback (most recent call last): ... ValueError: *Error: Numerical inconsistency is found. Use the GMP exact arithmetic. sage: sd.f_vector() # not tested (1, 60, 150, 92, 1) sage: sd.base_ring() # not tested Real Double Field
Its faces are 80 triangles and 12 pentagons:
sage: sum(1 for f in sd.faces(2) if len(f.vertices()) == 3) # not tested 80 sage: sum(1 for f in sd.faces(2) if len(f.vertices()) == 5) # not tested 12

tetrahedron
(backend=None)¶ Return the tetrahedron.
The tetrahedron is a Platonic solid with 4 vertices and 4 faces dual to itself. It can be defined as the convex hull of the 4 vertices \((0, 0, 0)\), \((1, 1, 0)\), \((1, 0, 1)\) and \((0, 1, 1)\). For more information, see the Wikipedia article Tetrahedron.
INPUT:
backend
– the backend to use to create the polytope.
See also
EXAMPLES:
sage: co = polytopes.tetrahedron() sage: co.f_vector() (1, 4, 6, 4, 1)
Its faces are 4 triangles:
sage: sum(1 for f in co.faces(2) if len(f.vertices()) == 3) 4
Some more computation:
sage: co.volume() 1/3 sage: co.ehrhart_polynomial() # optional  latte_int 1/3*t^3 + t^2 + 5/3*t + 1

truncated_cube
(exact=True, base_ring=None, backend=None)¶ Return the truncated cube.
The truncated cube is an Archimedean solid with 24 vertices and 14 faces. It can be defined as the convex hull of the 24 vertices \((\pm x, \pm 1, \pm 1), (\pm 1, \pm x, \pm 1), (\pm 1, \pm 1, \pm x)\) where \(x = \sqrt(2)  1\). For more information, see the Wikipedia article Truncated_cube.
INPUT:
exact
– (boolean, defaultTrue
) IfFalse
use an approximate ring for the coordinates.base_ring
– the ring in which the coordinates will belong to. If it is not provided andexact=True
it will be a the number field \(\QQ[\sqrt{2}]\) and ifexact=False
it will be the real double field.backend
– the backend to use to create the polytope.
EXAMPLES:
sage: co = polytopes.truncated_cube() sage: co.f_vector() (1, 24, 36, 14, 1)
Its faces are 8 triangles and 6 octogons:
sage: sum(1 for f in co.faces(2) if len(f.vertices()) == 3) 8 sage: sum(1 for f in co.faces(2) if len(f.vertices()) == 8) 6
Some more computation:
sage: co.volume() 56/3*sqrt2  56/3

truncated_dodecahedron
(exact=True, base_ring=None, backend=None)¶ Return the truncated dodecahedron.
The truncated dodecahedron is an Archimedean solid. It has 32 faces and 60 vertices. For more information, see the Wikipedia article Truncated dodecahedron.
INPUT:
exact
– (boolean, defaultTrue
) IfFalse
use an approximate ring for the coordinates.base_ring
– the ring in which the coordinates will belong to. If it is not provided andexact=True
it will be a the number field \(\QQ[\phi]\) where \(\phi\) is the golden ratio and ifexact=False
it will be the real double field.backend
– the backend to use to create the polytope.
EXAMPLES:
sage: td = polytopes.truncated_dodecahedron() sage: td.f_vector() (1, 60, 90, 32, 1) sage: td.base_ring() Number Field in sqrt5 with defining polynomial x^2  5
Its faces are 20 triangles and 12 regular decagons:
sage: sum(1 for f in td.faces(2) if len(f.vertices()) == 3) 20 sage: sum(1 for f in td.faces(2) if len(f.vertices()) == 10) 12
The faster implementation using floating point approximations does not fully work unfortunately, see https://github.com/cddlib/cddlib/pull/7 for a detailed discussion of this case:
sage: td = polytopes.truncated_dodecahedron(exact=False) # random doctest:warning ... UserWarning: This polyhedron data is numerically complicated; cdd could not convert between the inexact V and H representation without loss of data. The resulting object might show inconsistencies. sage: td.f_vector() Traceback (most recent call last): ... KeyError: ... sage: td.base_ring() Real Double Field

truncated_icosidodecahedron
(exact=True, base_ring=None, backend=None)¶ Return the truncated icosidodecahedron.
The truncated icosidodecahedron is an Archimedean solid. It has 62 faces and 120 vertices. For more information, see the Wikipedia article Truncated_icosidodecahedron.
INPUT:
exact
– (boolean, defaultTrue
) IfFalse
use an approximate ring for the coordinates.base_ring
– the ring in which the coordinates will belong to. If it is not provided andexact=True
it will be a the number field \(\QQ[\phi]\) where \(\phi\) is the golden ratio and ifexact=False
it will be the real double field.backend
– the backend to use to create the polytope.
EXAMPLES:
sage: ti = polytopes.truncated_icosidodecahedron() # long time sage: ti.f_vector() # long time (1, 120, 180, 62, 1) sage: ti.base_ring() # long time Number Field in sqrt5 with defining polynomial x^2  5
The implementation using floating point approximations is much faster:
sage: ti = polytopes.truncated_icosidodecahedron(exact=False) # random sage: ti.f_vector() (1, 120, 180, 62, 1) sage: ti.base_ring() Real Double Field
Its faces are 30 squares, 20 hexagons and 12 decagons:
sage: sum(1 for f in ti.faces(2) if len(f.vertices()) == 4) 30 sage: sum(1 for f in ti.faces(2) if len(f.vertices()) == 6) 20 sage: sum(1 for f in ti.faces(2) if len(f.vertices()) == 10) 12

truncated_octahedron
(backend=None)¶ Return the truncated octahedron.
The truncated octahedron is an Archimedean solid with 24 vertices and 14 faces. It can be defined as the convex hull off all the permutations of \((0, \pm 1, \pm 2)\). For more information, see the Wikipedia article Truncated_octahedron.
This is also known as the permutohedron of dimension 3.
INPUT:
backend
– the backend to use to create the polytope.
EXAMPLES:
sage: co = polytopes.truncated_octahedron() sage: co.f_vector() (1, 24, 36, 14, 1)
Its faces are 6 squares and 8 hexagons:
sage: sum(1 for f in co.faces(2) if len(f.vertices()) == 4) 6 sage: sum(1 for f in co.faces(2) if len(f.vertices()) == 6) 8
Some more computation:
sage: co.volume() 32 sage: co.ehrhart_polynomial() # optional  latte_int 32*t^3 + 18*t^2 + 6*t + 1

truncated_tetrahedron
(backend=None)¶ Return the truncated tetrahedron.
The truncated tetrahedron is an Archimedean solid with 12 vertices and 8 faces. It can be defined as the convex hull off all the permutations of \((\pm 1, \pm 1, \pm 3)\) with an even number of minus signs. For more information, see the Wikipedia article Truncated_tetrahedron.
INPUT:
backend
– the backend to use to create the polytope.
EXAMPLES:
sage: co = polytopes.truncated_tetrahedron() sage: co.f_vector() (1, 12, 18, 8, 1)
Its faces are 4 triangles and 4 hexagons:
sage: sum(1 for f in co.faces(2) if len(f.vertices()) == 3) 4 sage: sum(1 for f in co.faces(2) if len(f.vertices()) == 6) 4
Some more computation:
sage: co.volume() 184/3 sage: co.ehrhart_polynomial() # optional  latte_int 184/3*t^3 + 28*t^2 + 26/3*t + 1

twenty_four_cell
(backend=None)¶ Return the standard 24cell polytope.
The 24cell polyhedron (also called icositetrachoron or octaplex) is a regular polyhedron in 4dimension. For more information see the Wikipedia article 24cell.
INPUT:
backend
– the backend to use to create the polytope.
EXAMPLES:
sage: p24 = polytopes.twenty_four_cell() sage: p24.f_vector() (1, 24, 96, 96, 24, 1) sage: v = next(p24.vertex_generator()) sage: for adj in v.neighbors(): print(adj) A vertex at (1/2, 1/2, 1/2, 1/2) A vertex at (1/2, 1/2, 1/2, 1/2) A vertex at (1, 0, 0, 0) A vertex at (1/2, 1/2, 1/2, 1/2) A vertex at (0, 1, 0, 0) A vertex at (0, 0, 1, 0) A vertex at (0, 0, 0, 1) A vertex at (1/2, 1/2, 1/2, 1/2) sage: p24.volume() 2

zonotope
(generators, backend=None)¶ Return the zonotope, or parallelotope, spanned by the generators.
The parallelotope is the multidimensional generalization of a parallelogram (2 generators) and a parallelepiped (3 generators).
INPUT:
generators
– a list of vectors of same dimensionbackend
– the backend to use to create the polytope.
EXAMPLES:
sage: polytopes.parallelotope([ (1,0), (0,1) ]) A 2dimensional polyhedron in ZZ^2 defined as the convex hull of 4 vertices sage: polytopes.parallelotope([[1,2,3,4],[0,1,0,7],[3,1,0,2],[0,0,1,0]]) A 4dimensional polyhedron in ZZ^4 defined as the convex hull of 16 vertices sage: K = QuadraticField(2, 'sqrt2') sage: sqrt2 = K.gen() sage: polytopes.parallelotope([ (1,sqrt2), (1,1) ]) A 2dimensional polyhedron in (Number Field in sqrt2 with defining polynomial x^2  2)^2 defined as the convex hull of 4 vertices


sage.geometry.polyhedron.library.
project_points
(*points, **kwds)¶ Projects a set of points into a vector space of dimension one less.
INPUT:
points
… – the points to project.base_ring
– (defaults toRDF
if keyword isNone
or not provided inkwds
) the base ring to use.
The projection is isometric to the orthogonal projection on the hyperplane made of zero sum vector. Hence, if the set of points have all equal sums, then their projection is isometric (as a set of points).
The projection used is the matrix given by
zero_sum_projection()
.EXAMPLES:
sage: from sage.geometry.polyhedron.library import project_points sage: project_points([2,1,3,2]) # abs tol 1e15 [(2.1213203435596424, 2.041241452319315, 0.577350269189626)] sage: project_points([1,2,3],[3,3,5]) # abs tol 1e15 [(0.7071067811865475, 1.2247448713915892), (0.0, 1.6329931618554523)]
These projections are compatible with the restriction. More precisely, given a vector \(v\), the projection of \(v\) restricted to the first \(i\) coordinates will be equal to the projection of the first \(i+1\) coordinates of \(v\):
sage: project_points([1,2]) # abs tol 1e15 [(0.7071067811865475)] sage: project_points([1,2,3]) # abs tol 1e15 [(0.7071067811865475, 1.2247448713915892)] sage: project_points([1,2,3,4]) # abs tol 1e15 [(0.7071067811865475, 1.2247448713915892, 1.7320508075688776)] sage: project_points([1,2,3,4,0]) # abs tol 1e15 [(0.7071067811865475, 1.2247448713915892, 1.7320508075688776, 2.23606797749979)]
Check that it is (almost) an isometry:
sage: V = list(map(vector, IntegerVectors(n=5,length=3))) sage: P = project_points(*V) sage: for i in range(21): ....: for j in range(21): ....: assert abs((V[i]V[j]).norm()  (P[i]P[j]).norm()) < 0.00001
Example with exact computation:
sage: V = [ vector(v) for v in IntegerVectors(n=4,length=2) ] sage: P = project_points(*V, base_ring=AA) sage: for i in range(len(V)): ....: for j in range(len(V)): ....: assert (V[i]V[j]).norm() == (P[i]P[j]).norm()

sage.geometry.polyhedron.library.
zero_sum_projection
(d, base_ring=Real Double Field)¶ Return a matrix corresponding to the projection on the orthogonal of \((1,1,\ldots,1)\) in dimension \(d\).
The projection maps the orthonormal basis \((1,1,0,\ldots,0) / \sqrt(2)\), \((1,1,1,0,\ldots,0) / \sqrt(3)\), ldots, \((1,1,\ldots,1,1) / \sqrt(d)\) to the canonical basis in \(\RR^{d1}\).
OUTPUT:
A matrix of dimensions \((d1)\times d\) defined over
base_ring
(default:RDF
).EXAMPLES:
sage: from sage.geometry.polyhedron.library import zero_sum_projection sage: zero_sum_projection(2) [ 0.7071067811865475 0.7071067811865475] sage: zero_sum_projection(3) [ 0.7071067811865475 0.7071067811865475 0.0] [ 0.4082482904638631 0.4082482904638631 0.8164965809277261]
Exact computation in \(AA <sage.rings.qqbar.AlgebraicRealField>`\):
sage: zero_sum_projection(3, base_ring=AA) [ 0.7071067811865475? 0.7071067811865475? 0] [ 0.4082482904638630? 0.4082482904638630? 0.8164965809277260?]