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Remark 3.1.
If
ʱ<
3
and sin
2
, then (
a
) is indeed a generating chain,
see Laczkovich
[
7
, Theorem 2.4]. Notice that there are infinitely many such
ʱ
.
If
ʱ<
3
∈
Q
and
√
3sin
ʱ,
cos
ʱ
, then (
b
)
,
(
c
)
,
(
d
) are all generating chains, see
Laczkovich [
7
, Theorem 2.5], and Theorem 2.2(ii). If
we
let
ʱ
be the smallest
angle
of
the right triangle with sides
n
2
+3
,n
2
∈
Q
3
,
2
√
3
n
(
n>
4), then
ʱ<
3
−
and
√
3sin
ʱ,
cos
ʱ
∈
Q
. Hence there are also infinitely many
ʱ
for generating
chains (
b
)
,
(
c
)
,
(
d
).
(
6
,
6
,
2
3
)
are only of the
Theorem 3.5.
Intermediate triangles other than
following types:
(
ʲ,
3
,
2
3
ʲ
)
,
where
ʱ, ʲ
take suitable irrational multiples of
ˀ
, and infinitely many different
values are possible for
ʱ, ʲ
.
(
ʱ,
2
ʱ, ˀ
−
3
ʱ
)
,
−
Proof.
If an intermediate triangle has commensurable angles, then, by Corollary
3.2, it is
(
6
,
6
,
2
3
). If it has non-commensurable angles, then, by Theorem
3.4, it must be one of the types
(
ʲ,
3
,
2
3
(
ʱ,
2
ʱ, ˀ
−
3
ʱ
)
,
−
ʲ
), and infinitely
many values are possible for
ʱ, ʲ
by Remark 3.1.
Problem 3.1.
Determine all possible values of
ʱ
and
ʲ
of the above theorem.
Problem 3.2.
Determine the original triangles, terminal triangles, and inter-
mediate triangles completely.
Theorem 3.6.
The length of a generating chain of triangles is at most
3
.
Proof.
Suppose that there exists a generating chain
˃
1
ₒ
˃
2
ₒ
˃
3
ₒ
˃
4
of length 4. Then the sub-chain
˃
1
ₒ
˃
2
ₒ
˃
3
is either (
∗
)oratypeof
(
a
)
,
(
b
)
,
(
c
)
,
(
d
) of Theorem 3.4. Hence
˃
3
is an isosceles triangle other than
(
6
,
6
,
2
3
), and hence
˃
3
is terminal by Theorem 3.3 (ii). Therefore
˃
4
cannot
exist, a contradiction.
4
In Higher Dimensions
Let us state here some known results in dimension
3. A
d
-dimensional
Hill-
simplex
[
6
] (or Hadwiger-Hill simplex [
5
]) of angle
ʸ
, denoted by
Q
d
(
ʸ
), is defined
as the convex hull of the vectors 0
,v
1
,v
1
+
v
2
,...,v
1
+
v
2
+
≥
+
v
d
, where
v
1
,...,v
d
are linearly independent unit vectors such that the angle between every
two of them is equal to
ʸ
. A Hill-simplex of angle
ˀ/
2 is called an
orthogonal
Hill-
simplex, and a 3-dimensional Hill-simplex is called a Hill-tetrahedron. Figure
3
shows the orthogonal Hill-tetrahedron
Q
3
(
ˀ/
2).
For every
d
···
3, by bisecting
Q
d
(
ˀ/
2) successively, we can get a generating
chain of
d
-simplices
≥
˃
d
such that
˃
d
=
Q
d
(
ˀ/
2) and
˃
0
is similar to
˃
d
, see for the details, Maehara [
9
].
Since
˃
0
˃
0
ₒ
˃
1
ₒ
˃
2
ₒ ··· ₒ
˃
d
, we can extend this generating chain to both direction, and we
can get arbitrarily long generating chain. (Cf. Theorem 3.6.)
∼
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