polytope: Standard Properties

These properties and pseudo-targets are known to polymake when the default set of rules poly.rules is used.

Basic sections
POINTS matrix	Points such that the polyhedron is their convex hull. Redundancies are allowed. Vector (x₀ x₁ .. x_d) represents a point in (d+1)-space (homogeneous coordinates.) Affine points are identified by x₀ > 0. Points with x₀ = 0 can be interpreted as rays. polymake automatically normalizes each coordinate vector, dividing them by the first non-zero element. The clients and rule subroutines can always assume that x₀ is either 0 or 1.
VERTICES matrix	Vertices of the polyhedron. No redundancies are allowed. The coordinates are normalized the same way as POINTS.
INEQUALITIES matrix	Inequalities that describe half-spaces such that the polyhedron is their intersection. Redundancies are allowed. Dual to POINTS. A vector (A₀ A₁ .. A_d) defines the (closed affine) half-space of points (1,x₁,..,x_d) such that A₀ + A₁ x₁ + .. + A_d x_d >= 0.
FACETS matrix	Facets of the polyhedron. Dual to VERTICES.
EQUATIONS matrix	Equations that hold for all points of the polyhedron. A vector (A₀ A₁ .. A_d) describes the hyperplane of all points (1,x₁,..,x_d) such that A₀ + A₁ x₁ + .. + A_d x_d = 0.
AFFINE_HULL matrix	Dual basis of the affine hull of the polyhedron.
VALID_POINT vector	Some point belonging to the polyhedron.
DIM cardinal	Dimension of the affine hull of the polyhedron = dimension of the polyhedron.
AMBIENT_DIM cardinal	Dimension of the space where the polyhedron lives in.
FEASIBLE boolean	true if the polyhedron is not empty.
POINTED boolean	true if the polyhedron does not contain an affine line.
FULL_DIM boolean	true if DIM == AMBIENT_DIM
FAR_FACE set	Indices of vertices that are rays.
BOUNDED boolean	true if the polyhedron is a bounded polytope.
CENTERED boolean	true if (1,0,0,..) is a relative interior point. Polar to BOUNDED.
POSITIVE boolean	true if all vertices of the polyhedron have non-negative coordinates, that is, it lies entirely in the positive orthant.
N_POINTS cardinal	Number of points.
VERTEX_BARYCENTER vector	The center of gravity of the vertices of a bounded polytope.
ZONOTOPE_INPUT_VECTORS matrix	Contains the vector configuration for which a zonotope can be built.
REVERSE_TRANSFORMATION matrix	Some invertible linear transformation that can be used to get back a previous coordinate repersentation of the polytope. It operates from the right on point row vectors (e.g. in sections like POINTS, VERTICES, REL_INT_POINT); its inverse operates from the left on hyperplane column vectors.
VERTEX_LABELS array of labels	Unique names assigned to the vertices. If specified, they are shown by visualization tools instead of vertex indices. Due to an obscure convention, labels beginning with an underscore are fully suppressed during the visualization. For a polytope build from scratch, you should create this section by yourself, either manually in a text editor, or with a client program. If you build a polytope with a construction client taking some other input polytope(s), you can create the labels automatically if you call the client with a `-relabel` option. The exact format of the labels is dependent on the construction, and is described by the corresponding client.
FACET_LABELS array of labels	Unique names assigned to the facets, analogous to VERTEX_LABELS.
SCHLEGEL_TRANSFORM matrix	Matrix of a projective transformation mapping the whole polytope into one of its facets (specified by SCHLEGEL_PARAMS). The points belonging to this facet stay fixed.
GALE_TRANSFORM matrix	Coordinates of the Gale transform.
STEINER_POINTS matrix	A weighted inner point depending on the outer angle called Steiner point for all faces of dimensions 2 to d.
Combinatorics
DIM cardinal	Dimension of a minimal embedding space deduced from the combinatorial structure.
N_VERTICES cardinal	Number of vertices.
N_FACETS cardinal	Number of facets.
N_VERTEX_FACET_INC cardinal	Number of pairs of incident vertices and facets.
VERTICES_IN_FACETS incidence matrix	Vertex-facet incidence matrix, with rows corresponding to facets and columns to vertices. Vertices and facets are numbered from 0 to N_VERTICES-1 rsp. N_FACETS-1, according to their order in VERTICES rsp. FACETS.
FACETS_THRU_VERTICES	transposed VERTICES_IN_FACETS
HASSE_DIAGRAM face lattice	The face lattice of the polytope organized as a directed graph. Each node corresponds to some proper face of the polytope. The nodes corresponding to the vertices and facets appear in the same order as the elements of VERTICES and FACETS sections. Two special nodes represent the whole polytope and the empty face.
FACE_LATTICE pseudo	A pseudo-target causing the face lattice to be printed on the standard output. Unlike HASSE_DIAGRAM, the computed face lattice is just dumped out, but not stored in the data file. Thus you can apply it to polytopes with rather many faces without the risk of exhausting your disk space. The output is ordered by dimension. Each row starts with [ -k : f_d-k ], where k is the co-dimension, and f_d-k the number of faces (as in F_VECTOR.) The faces are represented as lists of vertex indices. They appear lexicographically sorted.
DUAL_FACE_LATTICE pseudo	Dual to FACE_LATTICE. The dual faces are represented as lists of facet indices.
VERTEX_SIZES array of cardinals	Number of incident facets for each vertex.
FACET_SIZES array of cardinals	Number of incident vertices for each facet.
GRAPH graph	Vertex-edge graph.
DUAL_GRAPH graph	Facet-ridge graph. Dual to GRAPH.
N_EDGES cardinal	Number of edges.
N_RIDGES cardinal	Number of ridges.
VERTEX_DEGREES array of cardinals	Degrees of vertices in the GRAPH.
FACET_DEGREES array of cardinals	Degrees of facets in the DUAL_GRAPH.
DIAMETER cardinal	Graph theoretical diameter of GRAPH.
DUAL_DIAMETER cardinal	Graph theoretical diameter of DUAL_GRAPH.
TRIANGLE_FREE boolean	true if GRAPH does not contain a triangle.
DUAL_TRIANGLE_FREE boolean	true if DUAL_GRAPH does not contain a triangle.
ALTSHULER_DET cardinal	Let M be the vertex-facet incidence matrix, then the Altshulter determinant is defined as max{det(M∗M^T), det(M^T∗M)}.
F_VECTOR vector of cardinals	f_k is the number of k-faces.
F2_VECTOR matrix of cardinals	f_ik is the number of incident pairs of i-faces and k-faces; the main diagonal contains the F_VECTOR.
H_VECTOR vector of cardinals	Simplicial h-vector. Defined for simplicial polytopes and also for their duals.
ESSENTIALLY_GENERIC boolean	All intermediate polytopes (with respect to the given insertion order) in the beneath-and-beyond algorithm are simplicial. We have the implications: VERTICES in general position => ESSENTIALLY_GENERIC => SIMPLICIAL
SIMPLICIAL boolean	true if the polytope is simplicial.
SIMPLE boolean	true if the polytope is simple. Dual to SIMPLICIAL.
EVEN boolean	true if the GRAPH of the polytope is bipartite.
DUAL_EVEN boolean	true if the DUAL_GRAPH of the polytope is bipartite. Dual to EVEN.
CONNECTIVITY cardinal	Node connectivity of the GRAPH of the polytope, that is, the minimal number of nodes to be removed from the graph such that the result is disconnected.
DUAL_CONNECTIVITY cardinal	Node connectivity of the DUAL_GRAPH of the polytope. Dual to CONNECTIVITY
SIMPLICIALITY cardinal	Maximal dimension in which all faces are simplices.
SIMPLICITY	Maximal dimension in which all dual faces are simplices.
FACE_SIMPLICITY cardinal	Maximal dimension in which all faces are simple polytopes.
CUBICAL boolean	true if all facets are cubes.
CUBICALITY cardinal	Maximal dimension in which all facets are cubes.
COCUBICAL boolean	dual to CUBICAL.
COCUBICALITY cardinal	dual to CUBICALITY.
NEIGHBORLY boolean	true if the polytope is neighborly.
NEIGHBORLINESS cardinal	Maximal dimension in which all facets are neighborly.
BALANCED boolean	dual to NEIGHBORLY.
BALANCE cardinal	Maximal dimension in which all facets are balanced.
Convex hull computation Note that an explicit request for any pseudo-target from this group will overwrite the VERTICES or FACETS sections created formerly, as well as rearrange or delete other sections depending on these (VERTICES_IN_FACETS, GRAPH, etc.)
POINTS_IN_FACETS incidence matrix	Similar to VERTICES_IN_FACETS, but with columns corresponding to POINTS instead of VERTICES. This section is a byproduct of convex hull computation algorithms. It is discarded as soon as VERTICES_IN_FACETS is computed.
VERTICES_IN_INEQUALITIES incidence matrix	Similar to VERTICES_IN_FACETS, but with rows corresponding to INEQUALITIES instead of FACETS. This section is a byproduct of convex hull computation algorithms. It is discarded as soon as VERTICES_IN_FACETS is computed.
cdd pseudo	Use the double description method as implemented in cddlib. It is the default algorithm for computation of facets from points or dually. It operates with arbitrary precision arithmetic (GMP) or floating-point numbers, depending upon which rule set is being used.
lrs pseudo	Use the reverse search method, as implemented in lrslib
beneath_beyond pseudo	Use the sequential (beneath-beyond) convex hull algorithm. It performs well at lower dimensions and produces a triangulation of the polytope as a byproduct. There is no dual (vertex enumeration) implementation of this algorithm.
porta pseudo	Run the porta program, implementing the Fourier-Motzkin elimination method. The essential drawback of this tool is that it employs a limited-precision arithmetic, and therefore can fail on numerically difficult problems.
Triangulation and volume Everything in this group is defined for bounded polytopes only.
VOLUME scalar	Volume of the polytope.
TRIANGULATION array of sets	Some triangulation of the polytope using only its vertices. Each line contains indices from VERTICES comprising a simplex.
TRIANGULATION_INT array of sets	Similar to TRIANGULATION, but using POINTS.
TRIANGULATION_BOUNDARY array of powersets	Intersection of TRIANGULATION with polytope boundary. Each line describes the triangulation of the corresponding facet as a list of simplices.
TRIANGULATION_SIGNS vector	For each simplex in TRIANGULATION, contains the sign of determinant of its coordinate matrix, thus telling about its orientation.
TRIANGULATION_INT_SIGNS vector	Similar to TRIANGULATION_SIGNS, but describes TRIANGULATION_INT.
Visualization
VIF_CYCLIC_NORMAL array of lists of cardinals	Reordered VERTICES_IN_FACETS for 2-d and 3-d polytopes. Vertices are listed in the order of their appearance when traversing the facet border counterclockwise seen from outside of the polytope. For a 2-d polytope (which is a closed polygon), lists all vertices in the border traversing order.
NEIGHBOR_FACETS_CYCLIC_NORMAL array of lists of cardinals	Reordered DUAL_GRAPH for 3-d polytopes. The neighbor facets are listed in the order corresponding to VIF_CYCLIC_NORMAL, so that the first two vertices in VIF_CYCLIC_NORMAL make up the ridge to the first neighbor facet and so on.
FTV_CYCLIC_NORMAL array of lists of cardinals	Reordered transposed VERTICES_IN_FACETS. Dual to VIF_CYCLIC_NORMAL.
NEIGHBOR_VERTICES_CYCLIC_NORMAL array of lists of cardinals	Reordered GRAPH. Dual to NEIGHBOR_FACETS_CYCLIC_NORMAL
SCHLEGEL_PARAMS tuple(cardinal , scalar , vector , vector)	Describes the parameter settings for the Schlegel diagram. The first line contains the index of the projection facet and the zoom factor. The following two coordinate vectors specify a point on the projection facet and an interior point of the polytope (in this order!) The line joining these two points is the view direction in the diagram.
SCHLEGEL_VERTICES matrix	Coordinates in affine 3-space of the vertices which correspond to a 3-dimensional (Schlegel-) projection of a 4-polytope.
GALE_VERTICES matrix	Coordinates of points for an affine Gale diagram.
GALE pseudo	Generates the Gale diagram of a d-polyhedron with at most d+4 vertices as a PostScript drawing and shows it via `gv`. You can set up your favorite PostScript renderer and tune some layout details such as paper format or font size in the customization file `postscript.def`.
SCHLEGEL_STEINER_POINTS matrix	Coordinates in affine 2- resp. 3-space of the vertices which correspond to a 2- resp. 3-dimensional (Schlegel-) projection of a 3 resp. 4-polytope.
steiner pseudo	Use the connecting line between the Steiner points as a view direction for Schlegel projection
GraphVisualizer word	Name of a package implementing the graph visualization interface.
DAGVisualizer word	Name of a package implementing the directed acyclic graph visualization.
SELECTED_FACES array of sets	Vertices to be visibly set off from the rest. Depending on the graph visualization method, the nodes corresponding to the selected vertices can be drawn bold, pulled aside in the 3-d embedding, or something else. Currently there are no rules producing this section; it should be filled in manually when desired. Please remember the valid syntax: each line must be enclosed in curly braces.
VISUAL_GRAPH pseudo	Visualization of the GRAPH of a polyhedron.
VISUAL_DUAL_GRAPH pseudo	Visualization of the DUAL_GRAPH of a polyhedron.
VISUAL_FACE_LATTICE pseudo	Visualization of the HASSE_DIAGRAM of a polyhedron as a multi-layer graph.
VISUAL_DUAL_FACE_LATTICE pseudo	Visualization of the DUAL_FACE_LATTICE of a polyhedron as a multi-layer graph.
VISUAL pseudo	Visualization of a 3-dimensional polytope as a solid body.
VISUAL_DUAL pseudo	Visualization of the dual polytope as a solid body. The polytope must be bounded and centered.
VISUAL_TRIANGULATION pseudo	Visualization of the TRIANGULATION of a 3-dimensional polytope. Hint: Use the method Method -> Effect -> Explode Group of Geometries of javaview for better insight in the internal structure.
VISUAL_TRIANGULATION_BOUNDARY pseudo	Visualization of the TRIANGULATION_BOUNDARY of a 3-dimensional polytope.
VISUAL_TRIANGULATION_BOUNDARY_WIRE pseudo	Visualization of the TRIANGULATION_BOUNDARY of a 3-dimensional polytope.
SCHLEGEL pseudo	Visualization of a Schlegel diagram of the 4-dimensional polytope as a wire skeleton.
SCHLEGEL_TRIANGULATION pseudo	Visualization of a Schlegel diagram of TRIANGULATION_BOUNDARY of a 4-dimensional polytope.
VISUAL_STEINER pseudo	Visualization of a 3-dimensional polytope as a wire skeleton with Steiner points.
SCHLEGEL_STEINER pseudo	Visualization of a Schlegel diagram of the 3- or 4-dimensional polytope as a wire skeleton with Steiner points.
geomview pseudo	Runs geomview - an alternative tool for the visualization of 3- or 4-dimensional polytopes (in the meantime obsolete).
javaview pseudo	Runs javaview program - default tool for the visualization of 3- or 4-dimensional polytopes with known coordinates. Recently you can also use it for graph visualization. The 3-d spring embedder starts with random node placement, so you have a chance to get various embeddings when trying it several times.
interactive pseudo	Runs javaview program in the interactive mode, allowing to modify the parameters of the Schlegel projection.
SCHLEGEL_JVX	JVX file stencil prepared for interactive Schlegel diagram visualization.
graphlet pseudo	Runs graphlet - the default tool for the visualization of graphs of polyhedra. Hint: use "Layout:Spring Embedder with Constraints" for GRAPH or DUAL_GRAPH; try also "Layout:DAG" for DIRECTED_GRAPH.
povray pseudo	Generates a scene file for povray describing a 3- or 4-dimensional polytope as a wire model.
Optimization
LINEAR_OBJECTIVE vector	Linear objective function.
ABSTRACT_OBJECTIVE vector	Abstract objective function. The i-th element is the value of the objective function at vertex number i. Only defined for bounded polytopes.
MAXIMAL_FACE set	Indices of vertices at which the maximum of the objective function is attained.
MINIMAL_FACE set	Similar to MAXIMAL_FACE
MAXIMAL_VERTEX vector	Coordinates of a (possible not unique) vertex at which the maximum of the objective function is attained.
MINIMAL_VERTEX vector	Similar to MAXIMAL_VERTEX
MAXIMAL_VALUE scalar	Maximum value of the objective function. Negated if linear problem is unbounded.
MINIMAL_VALUE scalar	Similar to MAXIMAL_VALUE
DIRECTED_GRAPH graph	Subgraph of GRAPH. Consists only of directed arcs along which the value of the objective function increases.
VERTEX_OUT_DEGREES array of cardinals	Number of outgoing arcs for each vertex in DIRECTED_GRAPH.
VERTEX_IN_DEGREES array of cardinals	Number of incoming arcs for each vertex in DIRECTED_GRAPH.
VERTEX_COLORS array of RGB	Approximate color encoding of objective function values. For each vertex, the color is chosen by linear interpolation in HSV system between fixed "maximal" and "minimal" colors, with the same ratio as the objective function value has in relation to the global maximum and mininum on the entire polytope.
RANDOM_EDGE_EPL scalar	Expected average path length for a simplex algorithm employing "random edge" pivoting strategy.
VARIABLE_NAMES array of labels	You can give a list of variable names here (this amounts to naming the columns of the coordinate vectors for VERTICES). This is of not much use for polymake. But this information will be used by programs which convert polymake format to other formats (such as LP).
VISUAL_COLORED_GRAPH pseudo	Visualization of the graph of the polyhedron, with nodes colored according to the objective function values (VERTEX_COLORS.) The spring embedder used in javaview mode tries to arrange the nodes so that their z coordinates approximately correspond to the objective function values.
VISUAL_DIRECTED_GRAPH pseudo	The same as VISUAL_GRAPH, showing the objective function growth direction via arrowed edges.
VISUAL_COLORED_DIRECTED_GRAPH pseudo	The same as VISUAL_COLORED_GRAPH, showing the objective function growth direction via arrowed edges.
VISUAL_DIRECTED pseudo	Visualization of a 3-dimensional polytope with superposed DIRECTED_GRAPH.
SCHLEGEL_DIRECTED pseudo	Visualization of a Schlegel diagram of a 4-dimensional polytope with superposed DIRECTED_GRAPH.
Polarization
REL_INT_POINT vector	Relatively interior point of the polyhedron.
FAR_HYPERPLANE vector	Valid strict inequality for all affine points of the polyhedron.
Oriented matroids
CHIROTOPE chirotope	Chirotope corresponding to VERTICES.
CHIROTOPE_INT chirotope	Chirotope corresponding to POINTS.
topcom pseudo	Runs the triangulation program from topcom package.
Integral lattice points
N_NON_NEG_INT cardinal	Number of non-negative interior lattice points.