Klein polyhedron

In the geometry of numbers, the Klein polyhedron, named after Felix Klein, is used to generalize the concept of simple continued fractions to higher dimensions.

be a closed simplicial cone in Euclidean space

is the convex hull of the non-zero points of

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.

is generated by a basis

for the line generated by the vector

of a Klein polyhedron is called a sail.

of an irrational cone are two graphs: Both of these graphs are structurally related to the directed graph

has been triangulated, the vertices of each of the graphs

: Lagrange proved that for an irrational real number

Klein polyhedra allow us to generalize this result.

be a totally real algebraic number field of degree

The simplicial cone

The period matrix of such a path is defined to be

The generalized Lagrange theorem states that for an irrational simplicial cone

The vertices of the sail are the points

The path of vertices

in the positive quadrant starting at

and proceeding in a positive direction is

be the line segment joining

is called badly approximable if

An irrational number is badly approximable if and only if the partial quotients of its continued fraction are bounded.

[1] This fact admits of a generalization in terms of Klein polyhedra.

, define the norm minimum of

be the sail of an irrational simplicial cone

In two dimensions, with the cone generated by

, they are just the partial quotients of the continued fraction of