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¶
Bases:
object
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: sorted(set(edge_lengths)) [7, 8, 9, 11, 12, 14, 16]
 bitruncated_six_hundred_cell(exact=True, backend=None)¶
Return the bitruncated 600cell.
The bitruncated 600cell is a 4dimensional 4uniform polytope in the \(H_4\) family. It has 3600 vertices. For more information see Wikipedia article Bitruncated 600cell.
Warning
The coordinates are exact by default. The computation with inexact coordinates (using the backend
'cdd'
) returns a numerical inconsistency error, and thus cannot be computed.INPUT:
exact
 (boolean, defaultTrue
) ifTrue
use exact coordinates instead of floating point approximations.backend
– the backend to use to create the polytope.
EXAMPLES:
sage: polytopes.runcinated_six_hundred_cell(exact=True,backend='normaliz') # not tested  very long time A 4dimensional polyhedron in AA^4 defined as the convex hull of 3600 vertices
 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 # optional  sage.rings.number_field sage: bb.f_vector() # long time # optional  sage.rings.number_field (1, 60, 90, 32, 1) sage: bb.base_ring() # long time # optional  sage.rings.number_field Number Field in sqrt5 with defining polynomial x^2  5 with sqrt5 = 2.236067977499790?
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 facets are 5 regular pentagons and 6 regular hexagons:
sage: sum(1 for f in bb.facets() if len(f.vertices()) == 5) 12 sage: sum(1 for f in bb.facets() if len(f.vertices()) == 6) 20
 cantellated_one_hundred_twenty_cell(exact=True, backend=None)¶
Return the cantellated 120cell.
The cantellated 120cell is a 4dimensional 4uniform polytope in the \(H_4\) family. It has 3600 vertices. For more information see Wikipedia article Cantellated 120cell.
Warning
The coordinates are exact by default. The computation with inexact coordinates (using the backend
'cdd'
) returns a numerical inconsistency error, and thus cannot be computed.INPUT:
exact
 (boolean, defaultTrue
) ifTrue
use exact coordinates instead of floating point approximations.backend
– the backend to use to create the polytope.
EXAMPLES:
sage: polytopes.cantellated_one_hundred_twenty_cell(backend='normaliz') # not tested  long time A 4dimensional polyhedron in AA^4 defined as the convex hull of 3600 vertices
 cantellated_six_hundred_cell(exact=False, backend=None)¶
Return the cantellated 600cell.
The cantellated 600cell is a 4dimensional 4uniform polytope in the \(H_4\) family. It has 3600 vertices. For more information see Wikipedia article Cantellated 600cell.
Warning
The coordinates are inexact by default. The computation with inexact coordinates (using the backend
'cdd'
) issues a UserWarning on inconsistencies.INPUT:
exact
 (boolean, defaultFalse
) ifTrue
use exact coordinates instead of floating point approximations.backend
– the backend to use to create the polytope.
EXAMPLES:
sage: polytopes.cantellated_six_hundred_cell() # not tested  very long time 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. A 4dimensional polyhedron in RDF^4 defined as the convex hull of 3600 vertices
It is possible to use the backend
'normaliz'
to get an exact representation:sage: polytopes.cantellated_six_hundred_cell(exact=True,backend='normaliz') # not tested  long time A 4dimensional polyhedron in AA^4 defined as the convex hull of 3600 vertices
 cantitruncated_one_hundred_twenty_cell(exact=True, backend=None)¶
Return the cantitruncated 120cell.
The cantitruncated 120cell is a 4dimensional 4uniform polytope in the \(H_4\) family. It has 7200 vertices. For more information see Wikipedia article Cantitruncated 120cell.
Warning
The coordinates are exact by default. The computation with inexact coordinates (using the backend
'cdd'
) returns a numerical inconsistency error, and thus cannot be computed.INPUT:
exact
 (boolean, defaultTrue
) ifTrue
use exact coordinates instead of floating point approximations.backend
– the backend to use to create the polytope.
EXAMPLES:
sage: polytopes.cantitruncated_one_hundred_twenty_cell(exact=True,backend='normaliz') # not tested  very long time A 4dimensional polyhedron in AA^4 defined as the convex hull of 7200 vertices
 cantitruncated_six_hundred_cell(exact=True, backend=None)¶
Return the cantitruncated 600cell.
The cantitruncated 600cell is a 4dimensional 4uniform polytope in the \(H_4\) family. It has 7200 vertices. For more information see Wikipedia article Cantitruncated 600cell.
Warning
The coordinates are exact by default. The computation with inexact coordinates (using the backend
'cdd'
) returns a numerical inconsistency error, and thus cannot be computed.INPUT:
exact
 (boolean, defaultTrue
) ifTrue
use exact coordinates instead of floating point approximations.backend
– the backend to use to create the polytope.
EXAMPLES:
sage: polytopes.cantitruncated_six_hundred_cell(exact=True,backend='normaliz') # not tested  very long time A 4dimensional polyhedron in AA^4 defined as the convex hull of 7200 vertices
 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(intervals=None, backend=None)¶
Return the cube.
The cube is the Platonic solid that is obtained as the convex hull of the eight \(\pm 1\) vectors of length 3 (by default). Alternatively, the cube is the product of three intervals from
intervals
.See also
INPUT:
intervals
– list (default=None). It takes the following possible inputs:If the input is
None
(the default), returns the convex hull of the eight \(\pm 1\) vectors of length three.'zero_one'
– (string). Return the \(0/1\)cube.a list of 3 lists of length 2. The cube will be a product of these three intervals.
backend
– the backend to use to create the polytope.
OUTPUT:
A cube as a polyhedron object.
EXAMPLES:
Return the \(\pm 1\)cube:
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() # optional  sage.plot Graphics3d Object
Return the \(0/1\)cube:
sage: cc = polytopes.cube(intervals ='zero_one') sage: cc.vertices_list() [[1, 0, 0], [1, 1, 0], [1, 1, 1], [1, 0, 1], [0, 0, 1], [0, 0, 0], [0, 1, 0], [0, 1, 1]]
 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 facets are 8 triangles and 6 squares:
sage: sum(1 for f in co.facets() if len(f.vertices()) == 3) 8 sage: sum(1 for f in co.facets() 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() # optional  sage.rings.number_field sage: d12.f_vector() # optional  sage.rings.number_field (1, 20, 30, 12, 1) sage: d12.volume() # optional  sage.rings.number_field 176*sqrt5 + 400 sage: numerical_approx(_) # optional  sage.rings.number_field 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) # optional  sage.symbolic Traceback (most recent call last): ... TypeError: unable to convert 1/4*sqrt(5) + 1/4 to a rational
 static edge_polytope(backend=None)¶
Return the edge polytope of
self
.The edge polytope (EP) of a Graph on \(n\) vertices is the polytope in \(\ZZ^{n}\) defined as the convex hull of \(e_i + e_j\) for each edge \((i, j)\). Here \(e_1, \dots, e_n\) denotes the standard basis.
INPUT:
backend
– string orNone
(default); the backend to use; seesage.geometry.polyhedron.constructor.Polyhedron()
EXAMPLES:
The EP of a \(4\)cycle is a square:
sage: G = graphs.CycleGraph(4) sage: P = G.edge_polytope(); P A 2dimensional polyhedron in ZZ^4 defined as the convex hull of 4 vertices
The EP of a complete graph on \(4\) vertices is cross polytope:
sage: G = graphs.CompleteGraph(4) sage: P = G.edge_polytope(); P A 3dimensional polyhedron in ZZ^4 defined as the convex hull of 6 vertices sage: P.is_combinatorially_isomorphic(polytopes.cross_polytope(3)) True
The EP of a graph is isomorphic to the subdirect sum of its connected components EPs:
sage: n = randint(3, 6) sage: G1 = graphs.RandomGNP(n, 0.2) sage: n = randint(3, 6) sage: G2 = graphs.RandomGNP(n, 0.2) sage: G = G1.disjoint_union(G2) sage: P = G.edge_polytope() sage: P1 = G1.edge_polytope() sage: P2 = G2.edge_polytope() sage: P.is_combinatorially_isomorphic(P1.subdirect_sum(P2)) True
All trees on \(n\) vertices have isomorphic EPs:
sage: n = randint(4, 10) sage: G1 = graphs.RandomTree(n) sage: G2 = graphs.RandomTree(n) sage: P1 = G1.edge_polytope() sage: P2 = G2.edge_polytope() sage: P1.is_combinatorially_isomorphic(P2) True
However, there are still many different EPs:
sage: len(list(graphs(5))) 34 sage: polys = [] sage: for G in graphs(5): ....: P = G.edge_polytope() ....: for P1 in polys: ....: if P.is_combinatorially_isomorphic(P1): ....: break ....: else: ....: polys.append(P) sage: len(polys) 19
 static flow_polytope(edges=None, ends=None, backend=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 [PS2002].
INPUT:
edges
– list (default:None
); a list of edges ofself
. If not specified, the list of all edges ofself
is used with the default ordering ofself.edges()
. 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 \(T\).backend
– string orNone
(default); the backend to use; seesage.geometry.polyhedron.constructor.Polyhedron()
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
A digraph with multiple edges (trac ticket #28837):
sage: G = DiGraph([(0, 1), (0,1)], multiedges=True) sage: G Multidigraph on 2 vertices sage: P = G.flow_polytope() sage: P A 1dimensional polyhedron in QQ^2 defined as the convex hull of 2 vertices sage: P.vertices() (A vertex at (1, 0), A vertex at (0, 1)) sage: P.lines() ()
 generalized_permutahedron(coxeter_type, point=None, exact=True, regular=False, backend=None)¶
Return the generalized permutahedron of type
coxeter_type
as the convex hull of the orbit ofpoint
in the fundamental cone.This generalized permutahedron lies in the vector space used in the geometric representation, that is, in the default case, the dimension of generalized permutahedron equals the dimension of the space.
INPUT:
coxeter_type
– a Coxeter type; given as a pair [type,rank], where type is a letter and rank is the number of generators.point
– a list (default:None
); a point given by its coordinates in the weight basis. IfNone
is given, the point \((1, 1, 1, \ldots)\) is used.exact
 (boolean, defaultTrue
) ifFalse
use floating point approximations instead of exact coordinatesregular
– boolean (default:False
); whether to apply a linear transformation making the vertex figures isometric.backend
– backend to use to create the polytope; (default:None
)
EXAMPLES:
sage: perm_a3 = polytopes.generalized_permutahedron(['A',3]); perm_a3 A 3dimensional polyhedron in QQ^3 defined as the convex hull of 24 vertices
You can put the starting point along the hyperplane of the first generator:
sage: perm_a3_011 = polytopes.generalized_permutahedron(['A',3],[0,1,1]); perm_a3_011 A 3dimensional polyhedron in QQ^3 defined as the convex hull of 12 vertices sage: perm_a3_110 = polytopes.generalized_permutahedron(['A',3],[1,1,0]); perm_a3_110 A 3dimensional polyhedron in QQ^3 defined as the convex hull of 12 vertices sage: perm_a3_110.is_combinatorially_isomorphic(perm_a3_011) True sage: perm_a3_101 = polytopes.generalized_permutahedron(['A',3],[1,0,1]); perm_a3_101 A 3dimensional polyhedron in QQ^3 defined as the convex hull of 12 vertices sage: perm_a3_110.is_combinatorially_isomorphic(perm_a3_101) False sage: perm_a3_011.f_vector() (1, 12, 18, 8, 1) sage: perm_a3_101.f_vector() (1, 12, 24, 14, 1)
The usual output does not necessarily give a polyhedron with isometric vertex figures:
sage: perm_a2 = polytopes.generalized_permutahedron(['A',2]) sage: perm_a2.vertices() (A vertex at (1, 1), A vertex at (1, 0), A vertex at (0, 1), A vertex at (0, 1), A vertex at (1, 0), A vertex at (1, 1))
Setting
regular=True
applies a linear transformation to get isometric vertex figures and the result is inscribed. Even though there are traces of small numbers, the internal computations are done using an exact embedded NumberField:sage: perm_a2_reg = polytopes.generalized_permutahedron(['A',2],regular=True) sage: V = sorted(perm_a2_reg.vertices()); V # random [A vertex at (1, 0), A vertex at (1/2, 0.866025403784439?), A vertex at (1/2, 0.866025403784439?), A vertex at (1/2, 0.866025403784439?), A vertex at (1/2, 0.866025403784439?), A vertex at (1.000000000000000?, 0.?e18)] sage: for v in V: ....: for x in v: ....: x.exactify() sage: V [A vertex at (1, 0), A vertex at (1/2, 0.866025403784439?), A vertex at (1/2, 0.866025403784439?), A vertex at (1/2, 0.866025403784439?), A vertex at (1/2, 0.866025403784439?), A vertex at (1, 0)] sage: perm_a2_reg.is_inscribed() True sage: perm_a3_reg = polytopes.generalized_permutahedron(['A',3],regular=True) # long time sage: perm_a3_reg.is_inscribed() # long time True
The same is possible with vertices in
RDF
:sage: perm_a2_inexact = polytopes.generalized_permutahedron(['A',2],exact=False) sage: sorted(perm_a2_inexact.vertices()) [A vertex at (1.0, 1.0), A vertex at (1.0, 0.0), A vertex at (0.0, 1.0), A vertex at (0.0, 1.0), A vertex at (1.0, 0.0), A vertex at (1.0, 1.0)] sage: perm_a2_inexact_reg = polytopes.generalized_permutahedron(['A',2],exact=False,regular=True) sage: sorted(perm_a2_inexact_reg.vertices()) [A vertex at (1.0, 0.0), A vertex at (0.5, 0.8660254038), A vertex at (0.5, 0.8660254038), A vertex at (0.5, 0.8660254038), A vertex at (0.5, 0.8660254038), A vertex at (1.0, 0.0)]
It works also with types with nonrational coordinates:
sage: perm_b3 = polytopes.generalized_permutahedron(['B',3]); perm_b3 # long time A 3dimensional polyhedron in (Number Field in a with defining polynomial x^2  2 with a = 1.414213562373095?)^3 defined as the convex hull of 48 vertices sage: perm_b3_reg = polytopes.generalized_permutahedron(['B',3],regular=True); perm_b3_reg # not tested  long time (12sec on 64 bits). A 3dimensional polyhedron in AA^3 defined as the convex hull of 48 vertices
It is faster with the backend
'normaliz'
:sage: perm_b3_reg_norm = polytopes.generalized_permutahedron(['B',3],regular=True,backend='normaliz') # optional  pynormaliz sage: perm_b3_reg_norm # optional  pynormaliz A 3dimensional polyhedron in AA^3 defined as the convex hull of 48 vertices
The backend
'normaliz'
allows further faster computation in the nonrational case:sage: perm_h3 = polytopes.generalized_permutahedron(['H',3],backend='normaliz') # optional  pynormaliz sage: perm_h3 # optional  pynormaliz A 3dimensional polyhedron in (Number Field in a with defining polynomial x^2  5 with a = 2.236067977499790?)^3 defined as the convex hull of 120 vertices sage: perm_f4 = polytopes.generalized_permutahedron(['F',4],backend='normaliz') # optional  pynormaliz, long time sage: perm_f4 # optional  pynormaliz, long time A 4dimensional polyhedron in (Number Field in a with defining polynomial x^2  2 with a = 1.414213562373095?)^4 defined as the convex hull of 1152 vertices
See also
permutahedron()
permutahedron()
 grand_antiprism(exact=True, backend=None, verbose=False)¶
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 with sqrt5 = 2.236067977499790?)^4 defined as the convex hull of 100 vertices
Computation with the backend
'normaliz'
is instantaneous:sage: gap_norm = polytopes.grand_antiprism(backend='normaliz') # optional  pynormaliz sage: gap_norm # optional  pynormaliz A 4dimensional polyhedron in (Number Field in sqrt5 with defining polynomial x^2  5 with sqrt5 = 2.236067977499790?)^4 defined as the convex hull of 100 vertices
Computation with approximated coordinates is also faster, but inexact:
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 # optional  sage.rings.number_field sage: gr.f_vector() # long time # optional  sage.rings.number_field (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 facets are 4 squares, 8 regular hexagons and 6 regular octagons:
sage: sum(1 for f in gr.facets() if len(f.vertices()) == 4) 12 sage: sum(1 for f in gr.facets() if len(f.vertices()) == 6) 8 sage: sum(1 for f in gr.facets() if len(f.vertices()) == 8) 6
 hypercube(dim, intervals=None, backend=None)¶
Return a hypercube of the given dimension.
The
dim
dimensional hypercube is by default the convex hull of the \(2^{\text{dim}}\) \(\pm 1\) vectors of lengthdim
. Alternatively, it is the product ofdim
line segments given in theintervals
. For more information see the wikipedia article Wikipedia article Hypercube.INPUT:
dim
– integer. The dimension of the hypercube.intervals
– (default = None). It takes the following possible inputs:If
None
(the default), it returns the \(\pm 1\)cube of dimensiondim
.'zero_one'
– (string). Return the \(0/1\)cube.a list of length
dim
. Its elements are pairs of numbers \((a,b)\) with \(a < b\). The cube will be the product of these intervals.
backend
– the backend to use to create the polytope.
EXAMPLES:
Create the \(\pm 1\)hypercube of dimension 4:
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
Return the \(0/1\)hypercube of dimension 4:
sage: z_cube = polytopes.hypercube(4,intervals = 'zero_one') sage: z_cube.vertices()[0] A vertex at (1, 0, 1, 1) sage: z_cube.is_simple() True sage: z_cube.base_ring() Integer Ring sage: z_cube.volume() 1 sage: z_cube.ehrhart_polynomial() # optional  latte_int t^4 + 4*t^3 + 6*t^2 + 4*t + 1
Return the 4dimensional combinatorial cube that is the product of [0,3]^4:
sage: t_cube = polytopes.hypercube(4, intervals = [[0,3]]*4)
Checking that t_cube is three times the previous \(0/1\)cube:
sage: t_cube == 3 * z_cube True
 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: TestSuite(h_4_2).run() 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) sage: TestSuite(h_7_3).run(skip=["_test_pyramid", "_test_lawrence"])
 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() # optional  sage.rings.number_field sage: ico.f_vector() # optional  sage.rings.number_field (1, 12, 30, 20, 1) sage: ico.volume() # optional  sage.rings.number_field 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 # optional  sage.rings.number_field sage: ico.base_ring() # long time # optional  sage.rings.number_field Algebraic Real Field sage: ico.volume() # long time # optional  sage.rings.number_field 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) # optional  sage.symbolic 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() # optional  sage.rings.number_field sage: id.f_vector() # optional  sage.rings.number_field (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_V2() # 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 with sqrt5 = 2.236067977499790?
A much faster implementation using floating point approximations:
sage: id = polytopes.icosidodecahedron_V2(exact=False) sage: id.f_vector() (1, 30, 60, 32, 1) sage: id.base_ring() Real Double Field
Its facets are 20 triangles and 12 regular pentagons:
sage: sum(1 for f in id.facets() if len(f.vertices()) == 3) 20 sage: sum(1 for f in id.facets() 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 facets are 8 triangles:
sage: sum(1 for f in co.facets() 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
 omnitruncated_one_hundred_twenty_cell(exact=True, backend=None)¶
Return the omnitruncated 120cell.
The omnitruncated 120cell is a 4dimensional 4uniform polytope in the \(H_4\) family. It has 14400 vertices. For more information see Wikipedia article Omnitruncated 120cell.
Warning
The coordinates are exact by default. The computation with inexact coordinates (using the backend
'cdd'
) returns a numerical inconsistency error, and thus cannot be computed.INPUT:
exact
 (boolean, defaultTrue
) ifTrue
use exact coordinates instead of floating point approximations.backend
– the backend to use to create the polytope.
EXAMPLES:
sage: polytopes.omnitruncated_one_hundred_twenty_cell(backend='normaliz') # not tested  very long time ~10min A 4dimensional polyhedron in AA^4 defined as the convex hull of 14400 vertices
 omnitruncated_six_hundred_cell(exact=True, backend=None)¶
Return the omnitruncated 120cell.
The omnitruncated 120cell is a 4dimensional 4uniform polytope in the \(H_4\) family. It has 14400 vertices. For more information see Wikipedia article Omnitruncated 120cell.
Warning
The coordinates are exact by default. The computation with inexact coordinates (using the backend
'cdd'
) returns a numerical inconsistency error, and thus cannot be computed.INPUT:
exact
 (boolean, defaultTrue
) ifTrue
use exact coordinates instead of floating point approximations.backend
– the backend to use to create the polytope.
EXAMPLES:
sage: polytopes.omnitruncated_one_hundred_twenty_cell(backend='normaliz') # not tested  very long time ~10min A 4dimensional polyhedron in AA^4 defined as the convex hull of 14400 vertices
 one_hundred_twenty_cell(exact=True, backend=None, construction='coxeter')¶
Return the 120cell.
The 120cell is a 4dimensional 4uniform polytope in the \(H_4\) family. It has 600 vertices and 120 facets. For more information see Wikipedia article 120cell.
Warning
The coordinates are exact by default. The computation with inexact coordinates (using the backend
'cdd'
) returns a numerical inconsistency error, and thus cannot be computed.INPUT:
exact
– (boolean, defaultTrue
) ifTrue
use exact coordinates instead of floating point approximations.backend
– the backend to use to create the polytope.construction
– the construction to use (string, default ‘coxeter’); the other possibility is ‘as_permutahedron’.
` EXAMPLES:
The classical construction given by Coxeter in [Cox1969] is given by:
sage: polytopes.one_hundred_twenty_cell() # not tested  long time ~15 sec. A 4dimensional polyhedron in (Number Field in sqrt5 with defining polynomial x^2  5 with sqrt5 = 2.236067977499790?)^4 defined as the convex hull of 600 vertices
The
'normaliz'
is faster:sage: P = polytopes.one_hundred_twenty_cell(backend='normaliz'); P # optional  pynormaliz A 4dimensional polyhedron in (Number Field in sqrt5 with defining polynomial x^2  5 with sqrt5 = 2.236067977499790?)^4 defined as the convex hull of 600 vertices
It is also possible to realize it using the generalized permutahedron of type \(H_4\):
sage: polytopes.one_hundred_twenty_cell(backend='normaliz',construction='as_permutahedron') # not tested  long time A 4dimensional polyhedron in AA^4 defined as the convex hull of 600 vertices
 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: P = polytopes.parallelotope([ (1,sqrt2), (1,1) ]); P A 2dimensional polyhedron in (Number Field in sqrt2 with defining polynomial x^2  2 with sqrt2 = 1.414213562373095?)^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 facets 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.facets()) 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() # optional  sage.plot Graphics3d Object sage: perm4.graph().is_isomorphic(graphs.BubbleSortGraph(4)) True
As both Hrepresentation an Vrepresentation are known, the permutahedron can be set up with both using the backend
field
. The following takes very very long time to recompute, e.g. with backendppl
:sage: polytopes.permutahedron(8, backend='field') # (~1s) A 7dimensional polyhedron in QQ^8 defined as the convex hull of 40320 vertices sage: polytopes.permutahedron(9, backend='field') # not tested (memory consumption) # (~5s) A 8dimensional polyhedron in QQ^9 defined as the convex hull of 362880 vertices
See also
 rectified_one_hundred_twenty_cell(exact=True, backend=None)¶
Return the rectified 120cell.
The rectified 120cell is a 4dimensional 4uniform polytope in the \(H_4\) family. It has 1200 vertices. For more information see Wikipedia article Rectified 120cell.
Warning
The coordinates are exact by default. The computation with inexact coordinates (using the backend
'cdd'
) returns a numerical inconsistency error, and thus cannot be computed.INPUT:
exact
 (boolean, defaultTrue
) ifTrue
use exact coordinates instead of floating point approximations.backend
– the backend to use to create the polytope.
EXAMPLES:
sage: polytopes.rectified_one_hundred_twenty_cell(backend='normaliz') # not tested  long time A 4dimensional polyhedron in AA^4 defined as the convex hull of 1200 vertices
 rectified_six_hundred_cell(exact=True, backend=None)¶
Return the rectified 600cell.
The rectified 600cell is a 4dimensional 4uniform polytope in the \(H_4\) family. It has 720 vertices. For more information see Wikipedia article Rectified 600cell.
Warning
The coordinates are exact by default. The computation with inexact coordinates (using the backend
'cdd'
) returns a numerical inconsistency error, and thus cannot be computed.INPUT:
exact
 (boolean, defaultTrue
) ifTrue
use exact coordinates instead of floating point approximations.backend
– the backend to use to create the polytope.
EXAMPLES:
sage: polytopes.rectified_six_hundred_cell(backend='normaliz') # not tested  long time ~14sec A 4dimensional polyhedron in AA^4 defined as the convex hull of 720 vertices
 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) # optional  sage.rings.number_field sage: octagon # optional  sage.rings.number_field A 2dimensional polyhedron in AA^2 defined as the convex hull of 8 vertices sage: octagon.n_vertices() # optional  sage.rings.number_field 8 sage: v = octagon.volume() # optional  sage.rings.number_field sage: v # optional  sage.rings.number_field 2.828427124746190? sage: v == 2*QQbar(2).sqrt() # optional  sage.rings.number_field 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 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 facets are 12 quadrilaterals (not all identical):
sage: sum(1 for f in rd.facets() 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 with sqrt5 = 2.236067977499790?
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 facets are 20 triangles, 30 squares and 12 pentagons:
sage: sum(1 for f in rid.facets() if len(f.vertices()) == 3) 20 sage: sum(1 for f in rid.facets() if len(f.vertices()) == 4) 30 sage: sum(1 for f in rid.facets() if len(f.vertices()) == 5) 12
 runcinated_one_hundred_twenty_cell(exact=False, backend=None)¶
Return the runcinated 120cell.
The runcinated 120cell is a 4dimensional 4uniform polytope in the \(H_4\) family. It has 2400 vertices. For more information see Wikipedia article Runcinated 120cell.
Warning
The coordinates are inexact by default. The computation with inexact coordinates (using the backend
'cdd'
) issues a UserWarning on inconsistencies.INPUT:
exact
 (boolean, defaultFalse
) ifTrue
use exact coordinates instead of floating point approximations.backend
– the backend to use to create the polytope.
EXAMPLES:
sage: polytopes.runcinated_one_hundred_twenty_cell(exact=False) # not tested  very long time 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. A 4dimensional polyhedron in RDF^4 defined as the convex hull of 2400 vertices
It is possible to use the backend
'normaliz'
to get an exact representation:sage: polytopes.runcinated_one_hundred_twenty_cell(exact=True,backend='normaliz') # not tested  very long time A 4dimensional polyhedron in AA^4 defined as the convex hull of 2400 vertices
 runcitruncated_one_hundred_twenty_cell(exact=False, backend=None)¶
Return the runcitruncated 120cell.
The runcitruncated 120cell is a 4dimensional 4uniform polytope in the \(H_4\) family. It has 7200 vertices. For more information see Wikipedia article Runcitruncated 120cell.
Warning
The coordinates are inexact by default. The computation with inexact coordinates (using the backend
'cdd'
) issues a UserWarning on inconsistencies.INPUT:
exact
 (boolean, defaultFalse
) ifTrue
use exact coordinates instead of floating point approximations.backend
– the backend to use to create the polytope.
EXAMPLES:
sage: polytopes.runcitruncated_one_hundred_twenty_cell(exact=False) # not tested  very long time 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.
It is possible to use the backend
'normaliz'
to get an exact representation:sage: polytopes.runcitruncated_one_hundred_twenty_cell(exact=True,backend='normaliz') # not tested  very long time A 4dimensional polyhedron in AA^4 defined as the convex hull of 7200 vertices
 runcitruncated_six_hundred_cell(exact=True, backend=None)¶
Return the runcitruncated 600cell.
The runcitruncated 600cell is a 4dimensional 4uniform polytope in the \(H_4\) family. It has 7200 vertices. For more information see Wikipedia article Runcitruncated 600cell.
Warning
The coordinates are exact by default. The computation with inexact coordinates (using the backend
'cdd'
) returns a numerical inconsistency error, and thus cannot be computed.INPUT:
exact
 (boolean, defaultTrue
) ifTrue
use exact coordinates instead of floating point approximations.backend
– the backend to use to create the polytope.
EXAMPLES:
sage: polytopes.runcitruncated_six_hundred_cell(backend='normaliz') # not tested  very long time A 4dimensional polyhedron in AA^4 defined as the convex hull of 7200 vertices
 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) # optional  sage.rings.number_field sage: s3.volume() == sqrt(3+1) / factorial(3) # optional  sage.rings.number_field 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 720
 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() # optional  sage.rings.number_field sage: sr.f_vector() # optional  sage.rings.number_field (1, 24, 48, 26, 1) sage: sr.volume() # optional  sage.rings.number_field 80/3*sqrt2 + 32
The faces are \(8\) equilateral triangles and \(18\) squares:
sage: sum(1 for f in sr.facets() if len(f.vertices()) == 3) # optional  sage.rings.number_field 8 sage: sum(1 for f in sr.facets() if len(f.vertices()) == 4) # optional  sage.rings.number_field 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(exact=False, base_ring=None, backend=None, verbose=False)¶
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.
The constant \(z\) used in constructing this polytope is the reciprocal of the tribonacci constant, that is, the solution of the equation \(x^3 + x^2 + x  1 = 0\). See Wikipedia article Generalizations_of_Fibonacci_numbers#Tribonacci_numbers.
INPUT:
exact
– (boolean, defaultFalse
) ifTrue
use exact coordinates instead of floating point approximationsbase_ring
– the field to use. IfNone
(the default), construct the exact number field needed (ifexact
isTrue
) or default toRDF
(ifexact
isTrue
).backend
– the backend to use to create the polytope. IfNone
(the default), the backend will be selected automatically.
EXAMPLES:
sage: sc_inexact = polytopes.snub_cube(exact=False); sc_inexact # optional  sage.groups A 3dimensional polyhedron in RDF^3 defined as the convex hull of 24 vertices sage: sc_inexact.f_vector() # optional  sage.groups (1, 24, 60, 38, 1) sage: sc_exact = polytopes.snub_cube(exact=True) # long time # optional  sage.groups sage.rings.number_field sage: sc_exact.f_vector() # long time # optional  sage.groups sage.rings.number_field (1, 24, 60, 38, 1) sage: sorted(sc_exact.vertices()) # long time # optional  sage.groups sage.rings.number_field [A vertex at (1, z, z^2), A vertex at (1, z^2, z), A vertex at (1, z^2, z), A vertex at (1, z, z^2), A vertex at (z, 1, z^2), A vertex at (z, z^2, 1), A vertex at (z, z^2, 1), A vertex at (z, 1, z^2), A vertex at (z^2, 1, z), A vertex at (z^2, z, 1), A vertex at (z^2, z, 1), A vertex at (z^2, 1, z), A vertex at (z^2, 1, z), A vertex at (z^2, z, 1), A vertex at (z^2, z, 1), A vertex at (z^2, 1, z), A vertex at (z, 1, z^2), A vertex at (z, z^2, 1), A vertex at (z, z^2, 1), A vertex at (z, 1, z^2), A vertex at (1, z, z^2), A vertex at (1, z^2, z), A vertex at (1, z^2, z), A vertex at (1, z, z^2)] sage: sc_exact.is_combinatorially_isomorphic(sc_inexact) # long time # optional  sage.groups sage.rings.number_field True
 snub_dodecahedron(base_ring=None, backend=None, verbose=False)¶
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:
Only the backend using the optional normaliz package can construct the snub dodecahedron in reasonable time:
sage: sd = polytopes.snub_dodecahedron(base_ring=AA, backend='normaliz') # optional  pynormaliz, long time sage: sd.f_vector() # optional  pynormaliz, long time (1, 60, 150, 92, 1) sage: sd.base_ring() # optional  pynormaliz, long time Algebraic Real Field
Its facets are 80 triangles and 12 pentagons:
sage: sum(1 for f in sd.facets() if len(f.vertices()) == 3) # optional  pynormaliz, long time 80 sage: sum(1 for f in sd.facets() if len(f.vertices()) == 5) # optional  pynormaliz, long time 12
 static symmetric_edge_polytope(backend=None)¶
Return the symmetric edge polytope of
self
.The symmetric edge polytope (SEP) of a Graph on \(n\) vertices is the polytope in \(\ZZ^{n}\) defined as the convex hull of \(e_i  e_j\) and \(e_j  e_i\) for each edge \((i, j)\). Here \(e_1, \dots, e_n\) denotes the standard basis.
INPUT:
backend
– string orNone
(default); the backend to use; seesage.geometry.polyhedron.constructor.Polyhedron()
EXAMPLES:
The SEP of a \(4\)cycle is a cube:
sage: G = graphs.CycleGraph(4) sage: P = G.symmetric_edge_polytope(); P A 3dimensional polyhedron in ZZ^4 defined as the convex hull of 8 vertices sage: P.is_combinatorially_isomorphic(polytopes.cube()) True
The SEP of a complete graph on \(4\) vertices is a cuboctahedron:
sage: G = graphs.CompleteGraph(4) sage: P = G.symmetric_edge_polytope(); P A 3dimensional polyhedron in ZZ^4 defined as the convex hull of 12 vertices sage: P.is_combinatorially_isomorphic(polytopes.cuboctahedron()) True
The SEP of a graph with edges on \(n\) vertices has dimension \(n\) minus the number of connected components:
sage: n = randint(5, 12) sage: G = Graph() sage: while not G.num_edges(): ....: G = graphs.RandomGNP(n, 0.2) sage: P = G.symmetric_edge_polytope() sage: P.ambient_dim() == n True sage: P.dim() == n  G.connected_components_number() True
The SEP of a graph is isomorphic to the subdirect sum of its connected components SEP’s:
sage: n = randint(3, 6) sage: G1 = graphs.RandomGNP(n, 0.2) sage: n = randint(3, 6) sage: G2 = graphs.RandomGNP(n, 0.2) sage: G = G1.disjoint_union(G2) sage: P = G.symmetric_edge_polytope() sage: P1 = G1.symmetric_edge_polytope() sage: P2 = G2.symmetric_edge_polytope() sage: P.is_combinatorially_isomorphic(P1.subdirect_sum(P2)) True
All trees on \(n\) vertices have isomorphic SEPs:
sage: n = randint(4, 10) sage: G1 = graphs.RandomTree(n) sage: G2 = graphs.RandomTree(n) sage: P1 = G1.symmetric_edge_polytope() sage: P2 = G2.symmetric_edge_polytope() sage: P1.is_combinatorially_isomorphic(P2) True
However, there are still many different SEPs:
sage: len(list(graphs(5))) 34 sage: polys = [] sage: for G in graphs(5): ....: P = G.symmetric_edge_polytope() ....: for P1 in polys: ....: if P.is_combinatorially_isomorphic(P1): ....: break ....: else: ....: polys.append(P) sage: len(polys) 25
A nontrivial example of two graphs with isomorphic SEPs:
sage: G1 = graphs.CycleGraph(4) sage: G1.add_edges([[0, 5], [5, 2], [1, 6], [6, 2]]) sage: G2 = copy(G1) sage: G1.add_edges([[2, 7], [7, 3]]) sage: G2.add_edges([[0, 7], [7, 3]]) sage: G1.is_isomorphic(G2) False sage: P1 = G1.symmetric_edge_polytope() sage: P2 = G2.symmetric_edge_polytope() sage: P1.is_combinatorially_isomorphic(P2) True
Apparently, glueing two graphs together on a vertex gives isomorphic SEPs:
sage: n = randint(3, 7) sage: g1 = graphs.RandomGNP(n, 0.2) sage: g2 = graphs.RandomGNP(n, 0.2) sage: G = g1.disjoint_union(g2) sage: H = copy(G) sage: G.merge_vertices(((0, randrange(n)), (1, randrange(n)))) sage: H.merge_vertices(((0, randrange(n)), (1, randrange(n)))) sage: PG = G.symmetric_edge_polytope() sage: PH = H.symmetric_edge_polytope() sage: PG.is_combinatorially_isomorphic(PH) True
 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 facets are 4 triangles:
sage: sum(1 for f in co.facets() 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() # optional  sage.rings.number_field sage: co.f_vector() # optional  sage.rings.number_field (1, 24, 36, 14, 1)
Its facets are 8 triangles and 6 octogons:
sage: sum(1 for f in co.facets() if len(f.vertices()) == 3) # optional  sage.rings.number_field 8 sage: sum(1 for f in co.facets() if len(f.vertices()) == 8) # optional  sage.rings.number_field 6
Some more computation:
sage: co.volume() # optional  sage.rings.number_field 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 with sqrt5 = 2.236067977499790?
Its facets are 20 triangles and 12 regular decagons:
sage: sum(1 for f in td.facets() if len(f.vertices()) == 3) 20 sage: sum(1 for f in td.facets() 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): ... ValueError: not all vertices are intersections of facets 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 with sqrt5 = 2.236067977499790?
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 facets are 30 squares, 20 hexagons and 12 decagons:
sage: sum(1 for f in ti.facets() if len(f.vertices()) == 4) 30 sage: sum(1 for f in ti.facets() if len(f.vertices()) == 6) 20 sage: sum(1 for f in ti.facets() 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 facets are 6 squares and 8 hexagons:
sage: sum(1 for f in co.facets() if len(f.vertices()) == 4) 6 sage: sum(1 for f in co.facets() 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_one_hundred_twenty_cell(exact=True, backend=None)¶
Return the truncated 120cell.
The truncated 120cell is a 4dimensional 4uniform polytope in the \(H_4\) family. It has 2400 vertices. For more information see Wikipedia article Truncated 120cell.
Warning
The coordinates are exact by default. The computation with inexact coordinates (using the backend
'cdd'
) returns a numerical inconsistency error, and thus cannot be computed.INPUT:
exact
 (boolean, defaultTrue
) ifTrue
use exact coordinates instead of floating point approximations.backend
– the backend to use to create the polytope.
EXAMPLES:
sage: polytopes.truncated_one_hundred_twenty_cell(backend='normaliz') # not tested  long time A 4dimensional polyhedron in AA^4 defined as the convex hull of 2400 vertices
 truncated_six_hundred_cell(exact=False, backend=None)¶
Return the truncated 600cell.
The truncated 600cell is a 4dimensional 4uniform polytope in the \(H_4\) family. It has 1440 vertices. For more information see Wikipedia article Truncated 600cell.
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: polytopes.truncated_six_hundred_cell() # not tested  long time A 4dimensional polyhedron in RDF^4 defined as the convex hull of 1440 vertices
It is possible to use the backend
'normaliz'
to get an exact representation:sage: polytopes.truncated_six_hundred_cell(exact=True,backend='normaliz') # not tested  long time ~16sec A 4dimensional polyhedron in AA^4 defined as the convex hull of 1440 vertices
 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 facets are 4 triangles and 4 hexagons:
sage: sum(1 for f in co.facets() if len(f.vertices()) == 3) 4 sage: sum(1 for f in co.facets() 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: P = polytopes.parallelotope([ (1,sqrt2), (1,1) ]); P A 2dimensional polyhedron in (Number Field in sqrt2 with defining polynomial x^2  2 with sqrt2 = 1.414213562373095?)^2 defined as the convex hull of 4 vertices
 sage.geometry.polyhedron.library.gale_transform_to_polytope(vectors, base_ring=None, backend=None)¶
Return the polytope associated to the list of vectors forming a Gale transform.
This function is the inverse of
gale_transform()
up to projective transformation.INPUT:
vectors
– the vectors of the Gale transformbase_ring
– string (default: \(None\)); the base ring to be used for the constructionbackend
– string (default: \(None\)); the backend to use to create the polytope
Note
The order of the input vectors will not be preserved.
If the center of the (input) vectors is the origin, the function is much faster and might give a nicer representation of the polytope.
If this is not the case, the vectors will be scaled (each by a positive scalar) accordingly to obtain the polytope.
See also
:func`~sage.geometry.polyhedron.library.gale_transform_to_primal`.
EXAMPLES:
sage: from sage.geometry.polyhedron.library import gale_transform_to_polytope sage: points = polytopes.octahedron().gale_transform() sage: points ((0, 1), (1, 0), (1, 1), (1, 1), (1, 0), (0, 1)) sage: P = gale_transform_to_polytope(points); P A 3dimensional polyhedron in ZZ^3 defined as the convex hull of 6 vertices sage: P.vertices() (A vertex at (1, 0, 0), A vertex at (0, 1, 0), A vertex at (0, 0, 1), A vertex at (0, 0, 1), A vertex at (0, 1, 0), A vertex at (1, 0, 0))
One can specify the base ring:
sage: gale_transform_to_polytope( ....: [(1,1), (1,1), (1,0), ....: (1,0), (1,1), (2,1)]).vertices() (A vertex at (25, 0, 0), A vertex at (15, 50, 60), A vertex at (0, 25, 0), A vertex at (0, 0, 25), A vertex at (16, 35, 54), A vertex at (24, 10, 31)) sage: gale_transform_to_polytope( ....: [(1,1), (1,1), (1,0), ....: (1,0), (1,1), (2,1)], ....: base_ring=RDF).vertices() (A vertex at (0.64, 1.4, 2.16), A vertex at (0.96, 0.4, 1.24), A vertex at (0.6, 2.0, 2.4), A vertex at (1.0, 0.0, 0.0), A vertex at (0.0, 1.0, 0.0), A vertex at (0.0, 0.0, 1.0))
One can also specify the backend:
sage: gale_transform_to_polytope( ....: [(1,1), (1,1), (1,0), ....: (1,0), (1,1), (2,1)], ....: backend='field').backend() 'field' sage: gale_transform_to_polytope( ....: [(1,1), (1,1), (1,0), ....: (1,0), (1,1), (2,1)], ....: backend='cdd', base_ring=RDF).backend() 'cdd'
A gale transform corresponds to a polytope if and only if every oriented (linear) hyperplane has at least two vectors on each side. See Theorem 6.19 of [Zie2007]. If this is not the case, one of two errors is raised.
If there is such a hyperplane with no vector on one side, the vectors are not totally cyclic:
sage: gale_transform_to_polytope([(0,1), (1,1), (1,0), (1,0)]) Traceback (most recent call last): ... ValueError: input vectors not totally cyclic
If every hyperplane has at least one vector on each side, then the gale transform corresponds to a point configuration. It corresponds to a polytope if and only if this point configuration is convex and if and only if every hyperplane contains at least two vectors of the gale transform on each side.
If this is not the case, an error is raised:
sage: gale_transform_to_polytope([(0,1), (1,1), (1,0), (1,1)]) Traceback (most recent call last): ... ValueError: the gale transform does not correspond to a polytope
 sage.geometry.polyhedron.library.gale_transform_to_primal(vectors, base_ring=None, backend=None)¶
Return a point configuration dual to a totally cyclic vector configuration.
This is the dehomogenized vector configuration dual to the input. The dual vector configuration is acyclic and can therefore be dehomogenized as the input is totally cyclic.
INPUT:
vectors
– the ordered vectors of the Gale transformbase_ring
– string (default: \(None\)); the base ring to be used for the constructionbackend
– string (default: \(None\)); the backend to be use to construct a polyhedral, used internally in case the center is not the origin, seePolyhedron()
OUTPUT: An ordered point configuration as list of vectors.
Note
If the center of the (input) vectors is the origin, the function is much faster and might give a nicer representation of the point configuration.
If this is not the case, the vectors will be scaled (each by a positive scalar) accordingly.
ALGORITHM:
Step 1: If the center of the (input) vectors is not the origin, we do an appropriate transformation to make it so.
Step 2: We add a row of ones on top of
Matrix(vectors)
. The right kernel of this larger matrix is the dual configuration space, and a basis of this space provides the dual point configuration.More concretely, the dual vector configuration (inhomogeneous) is obtained by taking a basis of the right kernel of
Matrix(vectors)
. If the center of the (input) vectors is the origin, there exists a basis of the right kernel of the form[[1], [V]]
, where[1]
represents a row of ones. Then,V
is a dehomogenization and thus the dual point configuration.To extend
[1]
to a basis ofMatrix(vectors)
, we add a row of ones toMatrix(vectors)
and calculate a basis of the right kernel of the obtained matrix.REFERENCES:
For more information, see Section 6.4 of [Zie2007] or Definition 2.5.1 and Definition 4.1.35 of [DLRS2010].
See also
:func`~sage.geometry.polyhedron.library.gale_transform_to_polytope`.
EXAMPLES:
sage: from sage.geometry.polyhedron.library import gale_transform_to_primal sage: points = ((0, 1), (1, 0), (1, 1), (1, 1), (1, 0), (0, 1)) sage: gale_transform_to_primal(points) [(0, 0, 1), (0, 1, 0), (1, 0, 0), (1, 0, 0), (0, 1, 0), (0, 0, 1)]
One can specify the base ring:
sage: gale_transform_to_primal( ....: [(1,1), (1,1), (1,0), ....: (1,0), (1,1), (2,1)]) [(16, 35, 54), (24, 10, 31), (15, 50, 60), (25, 0, 0), (0, 25, 0), (0, 0, 25)] sage: gale_transform_to_primal( ....: [(1,1),(1,1),(1,0),(1,0),(1,1),(2,1)], base_ring=RDF) [(0.6400000000000001, 1.4, 2.1600000000000006), (0.9600000000000002, 0.39999999999999997, 1.2400000000000002), (0.6000000000000001, 2.0, 2.4000000000000004), (1.0, 0.0, 0.0), (0.0, 1.0, 0.0), (0.0, 0.0, 1.0)]
One can also specify the backend to be used internally:
sage: gale_transform_to_primal( ....: [(1,1), (1,1), (1,0), ....: (1,0), (1,1), (2,1)], backend='field') [(48, 71, 88), (84, 28, 99), (77, 154, 132), (55, 0, 0), (0, 55, 0), (0, 0, 55)] sage: gale_transform_to_primal( # optional  pynormaliz ....: [(1,1), (1,1), (1,0), ....: (1,0), (1,1), (2,1)], backend='normaliz') [(16, 35, 54), (24, 10, 31), (15, 50, 60), (25, 0, 0), (0, 25, 0), (0, 0, 25)]
The input vectors should be totally cyclic:
sage: gale_transform_to_primal([(0,1), (1,0), (1,1), (1,0)]) Traceback (most recent call last): ... ValueError: input vectors not totally cyclic sage: gale_transform_to_primal( ....: [(1,1,0), (1,1,0), (1,0,0), ....: (1,0,0), (1,1,0), (2,1,0)], backend='field') Traceback (most recent call last): ... ValueError: input vectors not totally cyclic
 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))) # optional  sage.combinat sage: P = project_points(*V) # optional  sage.combinat sage: for i in range(21): # optional  sage.combinat ....: 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) ] # optional  sage.combinat sage: P = project_points(*V, base_ring=AA) # optional  sage.combinat sage.rings.number_field sage: for i in range(len(V)): # optional  sage.combinat sage.rings.number_field ....: 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=None)¶
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: zero_sum_projection(3, base_ring=AA) # optional  sage.rings.number_field [ 0.7071067811865475? 0.7071067811865475? 0] [ 0.4082482904638630? 0.4082482904638630? 0.8164965809277260?]