I can help clarify the concept of Euler's rotation theorem for you. It seems there might be a misunderstanding regarding the theorem and the nature of rotations.
Euler's rotation theorem states that any arbitrary three-dimensional rotation can be described by a single rotation about a specific axis. This means that any rotation in three dimensions can be expressed as a rotation around a single axis by a certain angle. The theorem does not imply that there are always two rotation axes involved in every rotation.
In your example of a rotation of length 3: τ σ τ, it appears that you are describing a sequence of two rotations, τ and σ, followed by another rotation τ. Each rotation represents a transformation around a specific axis. However, it's important to note that the order and combination of rotations determine the resulting transformation.
If we take your example of τ σ τ, let's assume τ represents a rotation of 90 degrees around the x-axis, and σ represents a rotation of 90 degrees around the y-axis. Applying these rotations sequentially would result in a net rotation of 180 degrees around an axis determined by the combination of the two original axes (x and y).
However, it's incorrect to assume that any point would not move at the end of this rotation. In fact, under this specific rotation sequence, every point in space would be moved from its original position. The combined effect of the rotations would cause each point to undergo a transformation.
To summarize, Euler's rotation theorem does not suggest that any combination of rotation axes will result in some points remaining stationary after rotation. Rather, it states that any arbitrary three-dimensional rotation can be described by a single rotation around a specific axis. The specific combination of rotations and the resulting transformation depends on the angles and order of the rotations applied.