In the geometry of numbers, the Klein polyhedron, named after Felix Klein, is used to generalize the concept of continued fractions to higher dimensions.
Let
be a closed simplicial cone in Euclidean space
. The Klein polyhedron of
is is the convex hull of the non-zero points of
.
Relation to continued fractions
[edit]
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
[edit]
Suppose
is generated by a basis
of
(so that
), and let
be the dual basis (so that
). 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 (one-dimensional) edge of
;
- the graph
whose vertices are
-dimensional faces (chambers) 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
(with
,
) 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
.
Generalization of Lagrange's theorem
[edit]
Lagrange 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:
is split over some totally-real algebraic number field of degree
.
- 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.
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 (with period one) in
, with period matrices
and
. We have
and
.
- The paths
and
are periodic (with period one) in
, with period matrices
and
. We have
and
.
Generalization of approximability
[edit]
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
.
- O. N. German, 2007, "Klein polyhedra and lattices with positive norm minima". Journal de Théorie des Nombres de Bordeaux 19: 175--190.
- E. I. Korkina, 1995, "Two-dimensional continued fractions. The simplest examples". Proc. Steklov Institute of Mathematics 209: 124–144.
- G. Lachaud, 1998, "Sails and Klein polyhedra" in Contemporary Mathematics 210. American Mathematical Society: 373–385.