Suppose is an irrational number. In, the cones generated by and by give rise to two Klein polyhedra, each of which is bounded by a sequence of adjoining line segments. Define the integer length of a line segment to be one less than the size of its intersection with. Then the integer lengths of the edges of these two Klein polyhedra encode the continued-fraction expansion of, one matching the even terms and the other matching the odd terms.
Graphs associated with the Klein polyhedron
Suppose is generated by a basis of , and let be the dual basis. Write for the line generated by the vector, and for the hyperplane orthogonal to. Call the vector irrational if ; and call the cone irrational if all the vectors and are irrational. The boundary of a Klein polyhedron is called a sail. Associated with the sail of an irrational cone are two graphs:
the graph whose vertices are vertices of, two vertices being joined if they are endpoints of a edge of ;
the graph whose vertices are -dimensional faces of, two chambers being joined if they share an -dimensional face.
Both of these graphs are structurally related to the directed graph whose set of vertices is, where vertex is joined to vertex if and only if is of the form where and is a permutation matrix. Assuming that has been triangulated, the vertices of each of the graphs and can be described in terms of the graph :
Given any path in, one can find a path in such that, where is the vector.
Given any path in, one can find a path in such that, where is the -dimensional standard simplex in.
proved that for an irrational real number, the continued-fraction expansion of is periodic if and only if is a quadratic irrational. Klein polyhedra allow us to generalize this result. Let be a totally real algebraic number field of degree, and let be the real embeddings of. The simplicial cone is said to be split over if where is a basis for over. Given a path in, let. The path is called periodic, with period, if for all. The period matrix of such a path is defined to be. A path in or associated with such a path is also said to be periodic, with the same period matrix. The generalized Lagrange theorem states that for an irrational simplicial cone, with generators and as above and with sail, the following three conditions are equivalent:
For each of the there is periodic path of vertices in such that the asymptotically approach the line ; and the period matrices of these paths all commute.
For each of the there is periodic path of chambers in such that the asymptotically approach the hyperplane ; and the period matrices of these paths all commute.
Example
Take and. Then the simplicial cone is split over. The vertices of the sail are the points corresponding to the even convergents of the continued fraction for. The path of vertices in the positive quadrant starting at and proceeding in a positive direction is. Let be the line segment joining to. Write and for the reflections of and in the -axis. Let, so that, and let. Let,,, and.
The paths and are periodic in, with period matrices and. We have and .
The paths and are periodic in, with period matrices and. We have and .
Generalization of approximability
A real number is called badly approximable if is bounded away from zero. An irrational number is badly approximable if and only if the partial quotients of its continued fraction are bounded. This fact admits of a generalization in terms of Klein polyhedra. Given a simplicial cone in, where, define the norm minimum of as. Given vectors, let. This is the Euclidean volume of. Let be the sail of an irrational simplicial cone.
For a vertex of, define where are primitive vectors in generating the edges emanating from.
For a vertex of, define where are the extreme points of.
Then if and only if and are both bounded. The quantities and are called determinants. In two dimensions, with the cone generated by, they are just the partial quotients of the continued fraction of.
