Let φ 1 , φ 2 , φ 3 {\displaystyle {}\varphi _{1},\varphi _{2},\varphi _{3}} be rotations around the x {\displaystyle {}x} -axis, the y {\displaystyle {}y} -axis, and the z {\displaystyle {}z} -axis, with orders ℓ 1 , ℓ 2 , ℓ 3 {\displaystyle {}\ell _{1},\ell _{2},\ell _{3}} (that is, φ 1 {\displaystyle {}\varphi _{1}} is a rotation about the angle 360 / ℓ 1 {\displaystyle {}360/\ell _{1}} degree around the x {\displaystyle {}x} -axis, etc.). Let 1 ≤ ℓ 1 ≤ ℓ 2 ≤ ℓ 3 {\displaystyle {}1\leq \ell _{1}\leq \ell _{2}\leq \ell _{3}} . For which tuples ( ℓ 1 , ℓ 2 , ℓ 3 ) {\displaystyle {}(\ell _{1},\ell _{2},\ell _{3})} is the group generated by these three rotations finite?
be rotations around the 
-axis, the 
-axis, and the 
-axis, with
orders
(that is, 
is a rotation about the angle 
degree around the -axis, etc.).
Let
.
For which tuples 
is the
group
generated
by these three rotations finite?
