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A finite sequence of simplices
˃
1
˃
2
...˃
k
(or written as
˃
1
ₒ
˃
2
ₒ ··· ₒ
˃
k
)
is called a
generating chain of length
k
if
˃
i
ₒ
˃
i
+1
and
˃
i
∼
˃
i
+1
for
i
=
1
,
2
,...,k
−
1. For example, in the two-dimensional case,
(
6
,
3
,
2
)
(
6
,
6
,
2
3
)
(
3
,
3
,
3
)
ₒ
ₒ
(
∗
)
is a generating chain of length 3 (see Fig.
1
), where
(
ʱ, ʲ, ʳ
) denotes a triangle
with interior angles
ʱ, ʲ, ʳ
. Note that, as far as generating relations concern, we
need not distinguish two similar triangles. So, we may regard
(
ʱ, ʲ, ʳ
)asany
representative of the triangles with interior angles
ʱ, ʲ, ʳ
.
(
6
,
3
,
2
(
3
,
3
,
3
(
6
,
6
,
2
3
)
)
)
Fig. 1.
A generating chain
In this paper, we consider mainly the 2-dimensional case. Our results strongly
depend on the works of Laczkovich [
7
,
8
]. We show the following.
•
e
(
{
3
}
,
{
n
}
) = 2 for every
n>
3
,n
=6.
e
(
{
4
}
,
{
n
}
) = 2 for every
n>
4.
e
(
{
3
}
,
{
4
}
,
{
6
}
,
{
12
}
)=2.
If
m, n
(
m
=
n
) are su
ciently large, then
e
(
{
m
}
,
{
n
}
)=2.
•
All right triangles are original.
Isosceles triangles other than
(
6
,
6
,
2
3
) are all terminal.
Intermediate triangles are only the following types:
(
6
,
6
,
2
3
)
,
(
ʲ,
3
,
2
ˀ−
3
ʲ
(
ʱ,
2
ʱ, ˀ
−
3
ʱ
)
,
)
,
3
where
ʱ, ʲ
take suitable irrational multiples of
ˀ
(infinitely many values are
possible for
ʱ, ʲ
).
•
The generating chain (
∗
) is the only chain of length
≥
3 that contains the
(
6
,
6
,
2
3
).
The length of every generating chain of triangles is at most 3.
intermediate triangle
2 Polygons
Theorem 2.1.
If a polygon
P
has
n
sides whose lengths are linearly independent
over the rational field
Q
, then
e
(
P
)
≥
n/
3
.
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