  We consider an infinite 3-dimensional elastic continuum whose material points
experience no displacements, only rotations. This framework is a special case
of the Cosserat theory of elasticity. Rotations of material points are
described mathematically by attaching to each geometric point an orthonormal
basis which gives a field of orthonormal bases called the coframe. As the
dynamical variables (unknowns) of our theory we choose the coframe and a
density. We write down the general dynamic variational functional for our
rotational theory of elasticity, assuming our material to be physically linear
but the kinematic model geometrically nonlinear. Allowing geometric
nonlinearity is natural when dealing with rotations because rotations in
dimension 3 are inherently nonlinear (rotations about different axes do not
commute) and because there is no reason to exclude from our study large
rotations such as full turns. The main result of the paper is an explicit
construction of a class of time-dependent solutions which we call plane wave
solutions; these are travelling waves of rotations. The existence of such
explicit closed form solutions is a nontrivial fact given that our system of
Euler-Lagrange equations is highly nonlinear. In the last section we consider a
special case of our rotational theory of elasticity which in the stationary
setting (harmonic time dependence and arbitrary dependence on spatial
coordinates) turns out to be equivalent to a pair of massless Dirac equations.
