Undoing Rotations

Ray told me about an interesting new scientist article titled “Mathematicians have found a hidden 'reset button’ for undoing rotation”. It popularises this article by Jean-Pierre Eckmann and Tsvi Tlusty published in the Physical Review Letters last month. Here is the arxiv version. I'll explain here what the results of the paper are, and I give a different proof for a weaker theorem.

What is it about?

It is about sequences of rotations in 3-dimensional euclidian space. One describes such a rotation \(R\) using two pieces of information:

\[ \underbrace{\theta}_{\text{angle}}\qquad\qquad\qquad\underbrace{A}_{\text{axis of rotation}} \]

Composition of a sequence \(R_1,\ldots, R_k\) of rotations is again a rotation. This is known as Euler's rotation theorem, and shows that rotations have a group structure.

(One can also see rotations as orthogonal matrices with determinant \(+1\). Orthogonal here means that \(MM^T=I\). The group of these matrices is called the special orthogonal group and is denoted by \(SO(3)\). Its matrices have an eigenvalue \(e^{\theta2\pi\mathbf{i}}\), its complex conjugate, and \(1\). The eigenspace of \(1\) is the axis of rotation, while \(\theta\) is the angle. Composing rotations is just matrix multiplication. )

The article poses the question: given a sequence of rotations

\[ (\theta_1,A_1), (\theta_2, A_2), \ldots, (\theta_k, A_k) \]

is it possible to scale the angles such that the composition is equal to the identity. In other words, does there exist a real \(\lambda>0\) such that composing

\[ (\lambda\theta_1,A_1), (\lambda\theta_2, A_2), \ldots, (\lambda\theta_k, A_k) \]

yields the identity rotation, namely rotation by angle \(0\).

In the article the authors demonstrate that such a scaling exists rarely. Indeed, the existence of such \(\lambda>0\) implies that

\[ \frac {\theta_i}{\theta_j} \in \mathbb Q. \]

However - and this is the interesting phenomenon - if the sequence of rotations is done twice, such scaling can almost always be found. In other words, the question of whether there exists a \(\lambda>0\) such that

\[ (\lambda\theta_1,A_1), (\lambda\theta_2, A_2), \ldots, (\lambda\theta_k, A_k), (\lambda\theta_1,A_1), (\lambda\theta_2, A_2), \ldots, (\lambda\theta_k, A_k) \]

is the identity rotation, almost always has a positive answer. More precisely

Theorem. (Eckmann, Tlusty 2025) Unless for every \(i\) there exists a \(j\ne i\) such that

\[ \theta_i = \theta_j, \]

there always exists a \(\lambda>0\) such that the sequence of rotations above, performed twice, equals the identity rotation.

Another proof of a weaker theorem

The proof of the theorem above uses geometry of numbers. There very well may be an elementary proof, but I will provide one that uses diophantine approximations (indirectly also geometry of numbers) for a weaker theorem. On the positive side, it is rather short and illustrative. I have not put any effort in finding an elementary proof.

Theorem. Assume that there is some \(\theta_i\) which cannot be written as a \(\mathbb Q\)-linear combination of \(\theta_j\), \(j\ne i\). Then there exists \(\lambda>0\) that makes the sequence of rotations above done twice, equal to the identity rotation.

Proof.

Let the composition of

\[ (\lambda\theta_1,A_1), (\lambda\theta_2, A_2), \ldots, (\lambda\theta_k, A_k), \]

be the rotation

\[ (\theta, A). \]

Define the function \(F\ :\ \mathbb R_+\to [0, 2\pi)\) which maps

\[ \lambda\mapsto \theta. \]

So \(F\) is mapping the scaling factor \(\lambda\) to the resulting angle of rotation \(\theta\). To prove the theorem it suffices to show that there is some \(\lambda^*\in\mathbb R_+\) such that \(F(\lambda^*)=\pi\), (or 0), meaning that scaling by \(\lambda^*\) we end up doing a rotation with angle \(\pi\), doing this rotation twice results in the identity. Since \(F\) is a continuous function, by the intermediate value theorem, if there are \(\lambda_1,\lambda_2\), such that

\[ F(\lambda_1) \le \pi \le F(\lambda_2), \]

then either for some \(\lambda^*\) we have \(F(\lambda^*)=\pi\), or \(F(\lambda^*)=0\), in either case performing this rotation twice leads to the identity rotation and proves the theorem.

We will now show the existence of such \(\lambda_1,\lambda_2\). To this end, let

\[ \{\phi_1,\ldots, \phi_l\}\subseteq \{\theta_1,\ldots, \theta_k\}, \]

be a maximal \(\mathbb Q\)-linearly independent subset. Note that \(\theta_i\) in the assumption of the theorem belongs to this subset, and that \(l\ge 2\). Assume without loss of generality that \(\phi_1 = \theta_i\). Here we apply Kronecker's theorem on simultaneous diophantine approximation (we probably need something much weaker) which says that if \(\phi_1,\ldots, \phi_l\) are \(\mathbb Q\)-linearly independent then the set of values

\[ \left\{\left(n\phi_1\ \mathrm{mod}\ 2\pi, \ldots, n\phi_l\ \mathrm{mod}\ 2\pi\right)\ :\ n\in\mathbb N\right\} \]

is dense in \([0,2\pi)^l\). As a consequence, for any given \(x\in [0,2\pi)\), and \(\epsilon>0\), there exists some \(\lambda>0\) such that

\[ \left(\lambda\phi_1\ \mathrm{mod}\ 2\pi, \ldots, \lambda\phi_l\ \mathrm{mod}\ 2\pi\right) \]

is in the \(\epsilon\)-ball of \((x,0,\ldots,0)\). Since when the independent variables \(\lambda\phi_2,\ldots, \lambda\phi_l\) are close to 0, so are the dependent ones \(\lambda\theta_j\), it follows that the resulting angle of rotation, when angles are scaled by \(\lambda\), is close to \(x\). Since \(x\) was arbitrary the existence of \(\lambda_1\), \(\lambda_2\) above is proved, and with it the theorem.