There are also some bug fixes and other improvements. = HyperplaneArrangements(QQ)Ī 2-dimensional polyhedron in QQ^2 defined as the convex hull of 1 vertex and 2 lines Hyperplane arrangements have a new method center: sage: H. Simplicial complex with vertex set (0, 1, 2, 3, 4, 5) and 8 facets There is a new method to obtain the boundary of complex simplicial polytopes: sage: oc = polytopes.octahedron() Note that this computation requires optional package latte_int or normaliz. Sage: %time P = polytopes.one_hundred_twenty_cell(backend='normaliz', construction='as_permutahedron')ĬPU times: user 18.6 s, sys: 137 ms, total: 18.8 sĮhrhart polynomials are computable for lattice polytopes defined with base ring QQ: sage: P = polytopes.cube()*1/1 Sage: %time P = polytopes.one_hundred_twenty_cell(backend='field')ĬPU times: user 15.9 s, sys: 87.2 ms, total: 16 s 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')ĬPU times: user 942 ms, sys: 81.8 ms, total: 1.02 s Sage has the classical construction of the 120-cell of Coxeter from 1969. It is implemented in the combinatorial polyhedron of P, which is newly exposed: sage: P.combinatorial_polyhedron()Ī 6-dimensional combinatorial polyhedron with 126 facets There is ongoing work to improve this implementation. 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)ĬPU times: user 679 ms, sys: 4.01 ms, total: 683 msĥ040 x 127 dense matrix over Integer Ring (use the '.str()' method to see the entries)ĬPU times: user 901 ms, sys: 16 ms, total: 917 ms Sage uses a new algorithm to obtain the f-vector for polyhedra. Note that Jmol is still available, via the option viewer='jmol' in the plot functions. Three.js has become the default viewer for 3D plots, in replacement of Jmol. Just in time for the new decade, this is the first version of Sage running on Python 3 by default. Availability of Sage 9.0 and installation help.
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