Let
C
be a closed simplicial cone in Euclidean space
R
n
. The Klein polyhedron of
C
is the convex hull of the non-zero points of
C
∩
Z
n
.
Suppose
α
>
0
is an irrational number. In
R
2
, the cones generated by
{
(
1
,
α
)
,
(
1
,
0
)
}
and by
{
(
1
,
α
)
,
(
0
,
1
)
}
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
Z
n
. Then the integer lengths of the edges of these two Klein polyhedra encode the continued-fraction expansion of
α
, one matching the even terms and the other matching the odd terms.
Suppose
C
is generated by a basis
(
a
i
)
of
R
n
(so that
C
=
{
∑
i
λ
i
a
i
:
(
∀
i
)
λ
i
≥
0
}
), and let
(
w
i
)
be the dual basis (so that
C
=
{
x
:
(
∀
i
)
⟨
w
i
,
x
⟩
≥
0
}
). Write
D
(
x
)
for the line generated by the vector
x
, and
H
(
x
)
for the hyperplane orthogonal to
x
.
Call the vector
x
∈
R
n
irrational if
H
(
x
)
∩
Q
n
=
{
0
}
; and call the cone
C
irrational if all the vectors
a
i
and
w
i
are irrational.
The boundary
V
of a Klein polyhedron is called a sail. Associated with the sail
V
of an irrational cone are two graphs:
the graph
Γ
e
(
V
)
whose vertices are vertices of
V
, two vertices being joined if they are endpoints of a (one-dimensional) edge of
V
;
the graph
Γ
f
(
V
)
whose vertices are
(
n
−
1
)
-dimensional faces (chambers) of
V
, two chambers being joined if they share an
(
n
−
2
)
-dimensional face.
Both of these graphs are structurally related to the directed graph
Υ
n
whose set of vertices is
G
L
n
(
Q
)
, where vertex
A
is joined to vertex
B
if and only if
A
−
1
B
is of the form
U
W
where
U
=
(
1
⋯
0
c
1
⋮
⋱
⋮
⋮
0
⋯
1
c
n
−
1
0
⋯
0
c
n
)
(with
c
i
∈
Q
,
c
n
≠
0
) and
W
is a permutation matrix. Assuming that
V
has been triangulated, the vertices of each of the graphs
Γ
e
(
V
)
and
Γ
f
(
V
)
can be described in terms of the graph
Υ
n
:
Given any path
(
x
0
,
x
1
,
…
)
in
Γ
e
(
V
)
, one can find a path
(
A
0
,
A
1
,
…
)
in
Υ
n
such that
x
k
=
A
k
(
e
)
, where
e
is the vector
(
1
,
…
,
1
)
∈
R
n
.
Given any path
(
σ
0
,
σ
1
,
…
)
in
Γ
f
(
V
)
, one can find a path
(
A
0
,
A
1
,
…
)
in
Υ
n
such that
σ
k
=
A
k
(
Δ
)
, where
Δ
is the
(
n
−
1
)
-dimensional standard simplex in
R
n
.
Lagrange proved that for an irrational real number
α
, the continued-fraction expansion of
α
is periodic if and only if
α
is a quadratic irrational. Klein polyhedra allow us to generalize this result.
Let
K
⊆
R
be a totally real algebraic number field of degree
n
, and let
α
i
:
K
→
R
be the
n
real embeddings of
K
. The simplicial cone
C
is said to be split over
K
if
C
=
{
x
∈
R
n
:
(
∀
i
)
α
i
(
ω
1
)
x
1
+
…
+
α
i
(
ω
n
)
x
n
≥
0
}
where
ω
1
,
…
,
ω
n
is a basis for
K
over
Q
.
Given a path
(
A
0
,
A
1
,
…
)
in
Υ
n
, let
R
k
=
A
k
+
1
A
k
−
1
. The path is called periodic, with period
m
, if
R
k
+
q
m
=
R
k
for all
k
,
q
≥
0
. The period matrix of such a path is defined to be
A
m
A
0
−
1
. A path in
Γ
e
(
V
)
or
Γ
f
(
V
)
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
C
⊆
R
n
, with generators
(
a
i
)
and
(
w
i
)
as above and with sail
V
, the following three conditions are equivalent:
C
is split over some totally real algebraic number field of degree
n
.
For each of the
a
i
there is periodic path of vertices
x
0
,
x
1
,
…
in
Γ
e
(
V
)
such that the
x
k
asymptotically approach the line
D
(
a
i
)
; and the period matrices of these paths all commute.
For each of the
w
i
there is periodic path of chambers
σ
0
,
σ
1
,
…
in
Γ
f
(
V
)
such that the
σ
k
asymptotically approach the hyperplane
H
(
w
i
)
; and the period matrices of these paths all commute.
Take
n
=
2
and
K
=
Q
(
2
)
. Then the simplicial cone
{
(
x
,
y
)
:
x
≥
0
,
|
y
|
≤
x
/
2
}
is split over
K
. The vertices of the sail are the points
(
p
k
,
±
q
k
)
corresponding to the even convergents
p
k
/
q
k
of the continued fraction for
2
. The path of vertices
(
x
k
)
in the positive quadrant starting at
(
1
,
0
)
and proceeding in a positive direction is
(
(
1
,
0
)
,
(
3
,
2
)
,
(
17
,
12
)
,
(
99
,
70
)
,
…
)
. Let
σ
k
be the line segment joining
x
k
to
x
k
+
1
. Write
x
¯
k
and
σ
¯
k
for the reflections of
x
k
and
σ
k
in the
x
-axis. Let
T
=
(
3
4
2
3
)
, so that
x
k
+
1
=
T
x
k
, and let
R
=
(
6
1
−
1
0
)
=
(
1
6
0
−
1
)
(
0
1
1
0
)
.
Let
M
e
=
(
1
2
1
2
1
4
−
1
4
)
,
M
¯
e
=
(
1
2
1
2
−
1
4
1
4
)
,
M
f
=
(
3
1
2
0
)
, and
M
¯
f
=
(
3
1
−
2
0
)
.
The paths
(
M
e
R
k
)
and
(
M
¯
e
R
k
)
are periodic (with period one) in
Υ
2
, with period matrices
M
e
R
M
e
−
1
=
T
and
M
¯
e
R
M
¯
e
−
1
=
T
−
1
. We have
x
k
=
M
e
R
k
(
e
)
and
x
¯
k
=
M
¯
e
R
k
(
e
)
.
The paths
(
M
f
R
k
)
and
(
M
¯
f
R
k
)
are periodic (with period one) in
Υ
2
, with period matrices
M
f
R
M
f
−
1
=
T
and
M
¯
f
R
M
¯
f
−
1
=
T
−
1
. We have
σ
k
=
M
f
R
k
(
Δ
)
and
σ
¯
k
=
M
¯
f
R
k
(
Δ
)
.
A real number
α
>
0
is called badly approximable if
{
q
(
p
α
−
q
)
:
p
,
q
∈
Z
,
q
>
0
}
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
C
=
{
x
:
(
∀
i
)
⟨
w
i
,
x
⟩
≥
0
}
in
R
n
, where
⟨
w
i
,
w
i
⟩
=
1
, define the norm minimum of
C
as
N
(
C
)
=
inf
{
∏
i
⟨
w
i
,
x
⟩
:
x
∈
Z
n
∩
C
∖
{
0
}
}
.
Given vectors
v
1
,
…
,
v
m
∈
Z
n
, let
[
v
1
,
…
,
v
m
]
=
∑
i
1
<
⋯
<
i
n
|
det
(
v
i
1
⋯
v
i
n
)
|
. This is the Euclidean volume of
{
∑
i
λ
i
v
i
:
(
∀
i
)
0
≤
λ
i
≤
1
}
.
Let
V
be the sail of an irrational simplicial cone
C
.
For a vertex
x
of
Γ
e
(
V
)
, define
[
x
]
=
[
v
1
,
…
,
v
m
]
where
v
1
,
…
,
v
m
are primitive vectors in
Z
n
generating the edges emanating from
x
.
For a vertex
σ
of
Γ
f
(
V
)
, define
[
σ
]
=
[
v
1
,
…
,
v
m
]
where
v
1
,
…
,
v
m
are the extreme points of
σ
.
Then
N
(
C
)
>
0
if and only if
{
[
x
]
:
x
∈
Γ
e
(
V
)
}
and
{
[
σ
]
:
σ
∈
Γ
f
(
V
)
}
are both bounded.
The quantities
[
x
]
and
[
σ
]
are called determinants. In two dimensions, with the cone generated by
{
(
1
,
α
)
,
(
1
,
0
)
}
, they are just the partial quotients of the continued fraction of
α
.
