Sage 9.0 Release Tour

released Jan 1, 2020

Python 3 transition

Just in time for the new decade, this is the first version of Sage running on Python 3 by default.

See Python3-Switch for more details

Three.js is the new default 3D viewer

Three.js has become the default viewer for 3D plots, in replacement of Jmol. Note that Jmol is still available, via the option viewer='jmol' in the plot functions.

New plotting capabilities

Polyhedral Geometry

Sage uses a new algorithm to obtain the f-vector for polyhedra. This is the only memory efficient implementation for the f-vector at the time of writing and it is as fast or faster as other implementations:

sage: P = polytopes.permutahedron(7)
sage: %time P.incidence_matrix()
CPU times: user 679 ms, sys: 4.01 ms, total: 683 ms
Wall time: 682 ms
5040 x 127 dense matrix over Integer Ring (use the '.str()' method to see the entries)
sage: %time P.f_vector()
CPU times: user 901 ms, sys: 16 ms, total: 917 ms
Wall time: 916 ms
(1, 5040, 15120, 16800, 8400, 1806, 126, 1)

There is ongoing work to improve this implementation. It is implemented in the combinatorial polyhedron of P, which is newly exposed:

sage: P.combinatorial_polyhedron()
A 6-dimensional combinatorial polyhedron with 126 facets

Sage has the classical construction of the 120-cell of Coxeter from 1969. This construction is much faster than to realize it as generalized permutahedron so that even without the optional package normaliz you won't waste much time:

sage: %time P = polytopes.one_hundred_twenty_cell(backend='normaliz')
CPU times: user 942 ms, sys: 81.8 ms, total: 1.02 s
Wall time: 457 ms
sage: %time P = polytopes.one_hundred_twenty_cell(backend='field')
CPU times: user 15.9 s, sys: 87.2 ms, total: 16 s
Wall time: 16 s
sage: %time P = polytopes.one_hundred_twenty_cell(backend='normaliz', construction='as_permutahedron')
CPU times: user 18.6 s, sys: 137 ms, total: 18.8 s
Wall time: 18 s

Ehrhart polynomials are computable for lattice polytopes defined with base ring QQ:

sage: P = polytopes.cube()*1/1
sage: P.base_ring()
Rational Field
sage: P.ehrhart_polynomial()
8*t^3 + 12*t^2 + 6*t + 1

Note that this computation requires optional package latte_int or normaliz.

There is a new method to obtain the boundary of complex simplicial polytopes:

sage: oc = polytopes.octahedron()
sage: oc.boundary_complex()
Simplicial complex with vertex set (0, 1, 2, 3, 4, 5) and 8 facets

Hyperplane arrangements have a new method center:

sage: H.<x,y> = HyperplaneArrangements(QQ)
sage: A = H()
A 2-dimensional polyhedron in QQ^2 defined as the convex hull of 1 vertex and 2 lines

There are also some bug fixes and other improvements. For more details see the release notes for optimization and polyhedral geometry softwares interactions in Sage.

Group Theory

Cubic Braid Groups

Factor groups of the Artin braid groups such that their generators have order three can easily be declared, now. They show a lot of their known properties:

sage: C4 = CubicBraidGroup(4)
sage: C4Cl = C4.as_classical_group(); C4Cl
Subgroup with 3 generators (
[  E(3)^2        0        0]  [       1 -E(12)^7        0]
[-E(12)^7        1        0]  [       0   E(3)^2        0]
[       0        0        1], [       0 -E(12)^7        1],

[       1        0        0]
[       0        1 -E(12)^7]
[       0        0   E(3)^2]
) of General Unitary Group of degree 3 over Universal Cyclotomic Field with respect to positive definite hermitian form
[-E(12)^7 + E(12)^11                  -1                   0]
[                 -1 -E(12)^7 + E(12)^11                  -1]
[                  0                  -1 -E(12)^7 + E(12)^11]

See also the reference manual.


The main novelty is the introduction of vector bundles (ticket #28159). For instance:

sage: M = Manifold(2, 'M')
sage: X.<x,y> = M.chart()
sage: TM = M.tangent_bundle(); TM
Tangent bundle TM over the 2-dimensional differentiable manifold M
sage: v = TM.section([-y, x], name='v'); v
Vector field v on the 2-dimensional differentiable manifold M
sage: v.display()
v = -y d/dx + x d/dy
sage: p = M((2,3), name='p'); p
Point p on the 2-dimensional differentiable manifold M
sage: TM.fiber(p)
Tangent space at Point p on the 2-dimensional differentiable manifold M
sage: in TM.fiber(p)
v = -3 d/dx + 2 d/dy

Other new features are

See the full changelog for details, as well as the list of improvements and bug fixes in this release.

EdgesView for graphs

An EdgesView is a read-only iterable container of edges enabling operations like for e in E and e in E. An EdgesView can be iterated multiple times, and checking membership is done in constant time. It avoids the construction of edge lists and so consumes little memory. It is updated as the graph is updated. Hence, the graph should not be updated while iterating through an EdgesView.

sage: G = Graph([(0, 1, 'C'), (0, 2, 'A'), (1, 2, 'B')])
sage: E = G.edges()
sage: E
[(0, 1, 'C'), (0, 2, 'A'), (1, 2, 'B')]
sage: type(E)
<class 'sage.graphs.views.EdgesView'>
sage: (0, 2) in E
sage: (0, 2, 'A') in E
sage: (2, 0, 'A') in E
sage: for e in E:
....:     for ee in E:
....:         print(e, ee)
(0, 1, 'C') (0, 1, 'C')
(0, 1, 'C') (0, 2, 'A')
(0, 1, 'C') (1, 2, 'B')
(0, 2, 'A') (0, 1, 'C')
(0, 2, 'A') (0, 2, 'A')
(0, 2, 'A') (1, 2, 'B')
(1, 2, 'B') (0, 1, 'C')
(1, 2, 'B') (0, 2, 'A')
(1, 2, 'B') (1, 2, 'B')

See for more details.

Availability of Sage 9.0 and installation help

Installation FAQ

More details

ReleaseTours/sage-9.0 (last edited 2021-09-29 20:33:45 by soehms)