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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
˃ 2
˃ 3
˃ 4
of length 4. Then the sub-chain ˃ 1
˃ 2
˃ 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
˃ 1
˃ 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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