The categories in this variety correspond to some classes of classical geometries.
Finite planes
Incidence geometries
Polyhedra
Although finite planes correspond to particular families of designs, separate categories are provided for both projective and affine planes in order to exploit the rich structure possessed by these objects.
Creation of classical and non-classical finite projective and affine planes
Subplanes, dual of a projective plane
Numerical invariants: order, p-rank
Properties: Desarguesian, self-dual
Parallel classes of an affine plane
k-arcs: testing, complete, tangents, secants, passants
Conics: through given points, knot, exterior, interior
Unitals: testing, tangents, feet
Affine to projective planes and vice versa
Related structures: design, incidence matrix, incidence graph, linear code
Collineation group, isomorphism testing (optimized algorithm for projective planes)
Central collineations: testing, groups
Group actions on a plane: orbits and stabilizers of points and lines
Symmetry properties: point transitive, line transitive
Apart from elementary invariants, a reasonably fast method is available for testing whether a plane is desarguesian. Among special configurations of interest, a search procedure for k-arcs is provided. A specialized algorithm developed by Jeff Leon is used to compute the collineation group of a projective plane while the affine case is handled by the incidence structure method. The collineation group (order 23 38) of a "random" projective plane of order 81 supplied by Gordon Royle was found in 1 202 seconds. As with graphs and designs the G-set mechanism gives the action of the collineation group on any appropriate set.
Magma contains facilities for creating and computing with incidence geometries and coset geometries. These have been developed by Dimitri Leemans (Brussels).
The Magma Incidence Structure type comprises a set of points and a set of blocks together with an incidence relation. Following Bekenhout, we define a more general object as follows: An incidence geometry is a 4-tuple Γ = (X,*,t,I) where
X is a set of elements;
I is a non-empty set whose elements are called types;
t : X→I : x ↦ t(x) is a type function which maps an element to its type;
* is an incidence relation that is a reflexive and symmetric relation such that ∀x,y∈X, x*y and t(x)*t(y) ⇒ x = y.
We also introduce group-geometry pairs or coset geometries. Roughly speaking, these are geometries constructed from a group and some of its subgroups in the following way. Let I be a finite set and let G be a group together with a finite family of subgroups (Gi)i∈I. We define the incidence geometry Γ = Γ(G, (Gi)i∈I) as follows. The set X of elements or varieties of Γ consists of all cosets gGi, g∈G, i∈I. We define an incidence relation * on X by:
g1Gi * g2Gj iff g1Gi∩g2Gj is non-empty in G.
Γ(G, (Gi)i∈I) may also be called a group-geometry pair.
Creation of incidence geometries
Conversion of an incidence geometry to a coset geometry
Set of types, rank
Diagram, incidence graph, elements
Residue, truncation, shadow, shadowspace
Properties: flag-transitive geometry, residually connected, firm, thin, thick
Test if a geometry is a graph and conversion of such a geometry to a graph
Automorphism group
Correlation group
Creation of coset geometries
Conversion of an incidence geometry to a coset geometry
Set of types, rank
Diagram
Residue, truncation
Properties: flag-transitive geometry, residually connected, firm, thin, thick
Test if a geometry is a graph and conversion of such a geometry to a graph
Borel subgroup, group of the geometry
Maximal and minimal parabolic subgroups
Kernel of a geometry, i-kernels, quotient
Determine intersection properties
Test primitivity properties: primitive, weakly primitive, residually primitive, weakly residually primitive
Determine whether a coset geometry is locally 2-transitive
A cone (in a toric lattice L) is the convex hull of finitely many rays. A (rational) polytope is the convex hull of finitely many points of L. More generally a (rational) polyhedron is given by the Minkowski sum of a polytope and a cone. There exist dual definitions in terms of the intersection of finitely many half-spaces. There is no requirement that polyhedra are of maximum dimension in the ambient toric lattice. Important operations include:
Standard constructions of cones, half-spaces, cross polytopes, cyclic polytopes, etc.;
Creation of cones and polyhedrons via a system of inequalities;
The defining inequalities of a cone or polyhedron;
The dual cone or polyhedron;
Taking the (affine) linear space spanned by, or contained in, a polyhedron;
Standard properties such as dimension, volume, and boundary volume;
Triangulations of a polytope and of its boundary;
Information on whether a polyhedron is simplicial, reflexive, terminal, etc.;
Construction of the cone in L×ℤ spanned by a polyhedron P in L;
The polyhedron defined by taking the convex hull of P∩ℤn for any polyhedron P;
The infinite and compact parts of a polyhedron;
Calculation of intersections, Minkowski sums, translations, etc.;
Taking hyperplane slices;
The minimum ℚ-generators and ℤ-generators of a cone;
Point membership;
Point counting and enumeration of lattice points;
The Ehrhart series, Ehrhart polynomial, and δ-vector of a polytope;
Faces of a cone or polyhedron, the f-vector and h-vector;
Construction of the face-poset;
The group of lattice automorphisms fixing a polytope.