For each irrational number
α
\alpha
, with continued fraction expansion
[
0
;
a
1
,
a
2
,
a
3
,
…
]
[0; a_1, a_2,a_3, \dots ]
, we classify, up to translation, the one dimensional almost periodic tilings which can be constructed by the projection method starting with a line of slope
α
\alpha
. The invariant is a sequence of integers in the space
X
α
=
{
(
x
i
)
i
=
1
∞
∣
x
i
∈
{
0
,
1
,
2
,
…
,
a
i
}
X_\alpha = \{(x_i)_{i=1}^\infty \mid x_i \in \{0,1,2, \dots ,a_i\}
and
x
i
+
1
=
0
x_{i+1} = 0
whenever
x
i
=
a
i
}
x_i = a_i\}
modulo the equivalence relation generated by tail equivalence and
(
a
1
,
0
,
a
3
,
0
,
…
)
∼
(
0
,
a
2
,
0
,
a
4
,
…
)
∼
(
a
1
−
1
,
a
2
−
1
,
a
3
−
1
,
…
)
(a_1, 0, a_3, 0, \dots ) \sim (0, a_2, 0, a_4, \dots ) \sim (a_1 -1, a_2 - 1, a_3 - 1, \dots )
. Each tile in a tiling
T
\textsf {T}
, of slope
α
\alpha
, is coded by an integer
0
≤
x
≤
[
α
]
0 \leq x \leq [\alpha ]
. Using a composition operation, we produce a sequence of tilings
T
1
=
T
,
T
2
,
T
3
,
…
\textsf {T}_1 = \textsf {T}{}, \textsf {T}_2, \textsf {T}_3, \dots
. Each tile in
T
i
\textsf {T}_i
gets absorbed into a tile in
T
i
+
1
\textsf {T}_{i+1}
. A choice of a starting tile in
T
1
\textsf {T}_1
will thus produce a sequence in
X
α
X_\alpha
. This is the invariant.