extension:
application: common
Property Types
application: group
Objects
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Properties of GroupOfCone
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N_REPRESENTATIVE_SIMPLICES: common::Array<Int>
An array of size COMBINATORIAL_DIM whose k-th entry stores the number of symmetry classes of k-dimensional simplices in the cone.
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REPRESENTATIVE_SIMPLICES: common::Array<Array<boost_dynamic_bitset>>
An array of size COMBINATORIAL_DIM whose k-th entry stores one representative for each symmetry class of k-dimensional simplices in the cone.
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SYMMETRIZED_COCIRCUIT_EQUATIONS: common::SparseMatrix<Rational, NonSymmetric>
a SparseMatrix whose rows are the sum of all cocircuit equations corresponding to a fixed symmetry class of interior ridge
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UNDOCUMENTED
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REPRESENTATIVE_INTERIOR_RIDGE_SIMPLICES: common::Array<boost_dynamic_bitset>
The (d-1)-dimensional simplices in the interior.
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REPRESENTATIVE_MAX_BOUNDARY_SIMPLICES: common::Array<boost_dynamic_bitset>
The boundary (d-1)-dimensional simplices of a cone of combinatorial dimension d
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REPRESENTATIVE_MAX_INTERIOR_SIMPLICES: common::Array<boost_dynamic_bitset>
The interior d-dimensional simplices of a cone of combinatorial dimension d
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User Functions
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group_from_cyclic_notation0 (group) → Group
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group_from_cyclic_notation1 (group) → Group
application: polytope
Objects
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Specializations of Cone
Properties of Cone
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COCIRCUIT_EQUATIONS: common::SparseMatrix<Rational, NonSymmetric>
A matrix whose rows contain the cocircuit equations of P. The columns correspond to the MAX_INTERIOR_SIMPLICES.
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FOLDABLE_COCIRCUIT_EQUATIONS: common::SparseMatrix<Rational, NonSymmetric>
A matrix whose rows contain the foldable cocircuit equations of P. The columns correspond to 2 * MAX_INTERIOR_SIMPLICES. col 0 = 0, col 1 = first simplex (black copy), col 2 = first simplex (white copy), col 3 = second simplex (black copy), ...
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MAX_BOUNDARY_SIMPLICES: common::Array<Set<Int>>
The boundary (d-1)-dimensional simplices of a cone of combinatorial dimension d
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MAX_INTERIOR_SIMPLICES: common::Array<Set<Int>>
The interior d-dimensional simplices of a cone of combinatorial dimension d
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Properties of PointConfiguration
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COCIRCUIT_EQUATIONS: common::SparseMatrix<Rational, NonSymmetric>
Tells the cocircuit equations that hold for the configuration, one for each interior ridge
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MAX_BOUNDARY_SIMPLICES: common::Array<Set<Int>>
Tells the full-dimensional simplices on the boundary that contain no points except for the vertices.
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MAX_INTERIOR_SIMPLICES: common::Array<Set<Int>>
Tells the full-dimensional simplices that contain no points except for the vertices.
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SIMPLEXITY_LOWER_BOUND: common::Int
A lower bound for the minimal number of simplices in a triangulation
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Specializations of Polytope
Properties of Polytope
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FOLDABLE_MAX_SIGNATURE_UPPER_BOUND: common::Int
An upper bound for the maximal signature of a foldable triangulation of a polytope The signature is the absolute difference of the normalized volumes of black minus white maximal simplices, where only odd normalized volumes are taken into account.
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SIMPLEXITY_LOWER_BOUND: common::Int
A lower bound for the minimal number of simplices in a triangulation
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A topological quotient space obtained from a polytope by identifying faces. This object will sit inside a Polytope.
Properties of QuotientSpace
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UNDOCUMENTED
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IDENTIFICATION_GROUP: group::Group
The group encoding the quotient space. The faces of the space are the orbits of the faces of the polytope under the group.
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UNDOCUMENTED
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COCIRCUIT_EQUATIONS: common::SparseMatrix<Rational, NonSymmetric>
a SparseMatrix whose rows are the sum of all cocircuit equations corresponding to a fixed symmetry class of interior ridge
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FACES: common::Array<Array<Set<Int>>>
The faces of the quotient space, ordered by dimension. One representative of each orbit class is kept.
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FACE_ORBITS: common::Array<Set<Array<Set<Int>>>>
The orbits of faces of the quotient space, ordered by dimension.
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N_SIMPLICES: common::Array<Int>
The simplices made from points of the quotient space (also internal simplices, not just faces)
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REPRESENTATIVE_INTERIOR_RIDGE_SIMPLICES: common::Array<boost_dynamic_bitset>
The (d-1)-dimensional simplices in the interior.
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REPRESENTATIVE_MAX_BOUNDARY_SIMPLICES: common::Array<boost_dynamic_bitset>
The boundary (d-1)-dimensional simplices of a cone of combinatorial dimension d
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REPRESENTATIVE_MAX_INTERIOR_SIMPLICES: common::Array<boost_dynamic_bitset>
The interior d-dimensional simplices of a cone of combinatorial dimension d
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SIMPLEXITY_LOWER_BOUND: common::Int
A lower bound for the number of simplices needed to triangulate the quotient space
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SIMPLICIAL_COMPLEX: topaz::SimplicialComplex
A simplicial complex obtained by two stellar subdivisions of the defining polytope.
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SYMMETRY_GROUP: group::Group
The symmetry group induced by the symmetry group of the polytope on the @see FACES of the quotient space
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User Functions
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alternating_group (degree, domain) → GroupOfPolytope
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cs_quotient (Polytope)
For a centrally symmetric polytope, return the quotient space obtained by dividing out the central symmetry, i.e, identifying diametrically opposite faces
Parameters
P Polytope , centrally symmetric -
cyclic_group (degree, domain) → GroupOfPolytope
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cylinder_2 ()
Return a 2-dimensional cylinder obtained by identifying two opposite sides of a square
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group_from_cyclic_notation0 (group, domain) → GroupOfPolytope
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group_from_cyclic_notation1 (group, domain) → GroupOfPolytope
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max_interior_simplices (P) → Array<Set>
Find the maximal interior simplices of a polytope P. Symmetries of P are NOT taken into account.
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max_interior_simplices (P)
find the maximal interior simplices of a point configuration that DO NOT contain any point in their closure, except for the vertices. Symmetries of the configuration are NOT taken into account.
Parameters
PointConfiguration P the input point configuration -
quarter_turn_manifold ()
Return the 3-dimensional Euclidean manifold obtained by identifying opposite faces of a 3-dimensional cube by a quarter turn. After identification, two classes of vertices remain.
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symmetric_group (degree, domain) → GroupOfPolytope
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write_foldable_max_signature_ilp (P, outfile_name)
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write_quotient_space_simplexity_ilp ()
outputs a linear program whose optimal value is a lower bound for the number of simplices necessary to triangulate the polytope in such a way that its symmetries respect the triangulation of the boundary.
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write_simplexity_ilp (P, outfile_name)