Observed maximum rotation (Ωe) of schistosity and compositional banding (fiducial surface) encapsulated in and passing through the center of a garnet relative to the same surface away from the garnet (fiducial plane = S) commonly results from a sequence of velocity gradients ([∂u̇ i /∂x j ] or [u̇ i , j ]) that have affected the surrounding rock. The circumstances under which Ωe represents the cumulative rotational motion, ∫ ω̇dt = Ω, resulting from the antisymmetric part of [u̇ i , j ] are described, assuming the garnet to have the idealized shape of a rigid sphere. For the example of simple shear of a mass containing an isolated garnet, theory and experiment show that Ω = Ω e = |½γ|, where γ = the “amount of the shear,” if S parallels the shearing planes, Ω is also equal to Ω e if two of the principal axes of the symmetric part of [u̇ i , j ] remained in S throughout the deformation and S did not rotate relative to the same geographic frame of reference in which Ω takes place. Interpretation of Ω in terms of Ωe is complicated where there has been rotation of S, Ω s , because of the effect on Ω s of both symmetric and antisymmetric parts of [u̇ i , j ]. Ω is affected only by the antisymmetric part. Thus Ω e does not uniquely measure Ω. The primary utility of Ω e lies in the testing of hypothesized models.
The geometric arrangement of a fiducial surface passing through the center of a “rotated garnet” (≡ garnet showing nonzero Ω e ), the central surface, is readily described using a set of nested rings sharing an axis containing a diameter of each (suggestion of J. B. Thompson, Jr.). For rotation about a single axis in the fiducial plane during growth, the included and immediately peripheral fiducial surface has symmetry C 2 h (= 2/m) and consists of a smooth surface having two synclastic regions of opposite curvatures tangent to one another at the center of nucleation, all surrounded by an anticlastic region that extends outward to the bounding fiducial plane. Included surfaces that do not pass through the center of nucleation (noncentral surfaces) are similar except for deterioration of symmetry to C s (= m). The unique twofold axis of symmetry within the fiducial surface is a line of striction and is the observed rotational axis. It is the only straight line within any of the included surfaces. Growth and rotation about a second axis not parallel to the first is recognizable and causes the symmetry of the included central surface to become S2 (= 1̄). Methods of determination of axial orientations and amounts of rotation (Ωe) for both stages of doubly rotated garnets are applied to examples from southeastern Vermont. These methods take advantage of the features of symmetry and differential geometry of the included surfaces.
Presence of a plane of symmetry for the set of surfaces included in a garnet indicates that the velocity gradient, referred to Cartesian axes having a principal axis normal to the fiducial plane, shared the same plane of symmetry. A common type of flow in stratified rocks probably consists of superposition of (1) dilatational motion along three mutually perpendicular principal axes and (2) simple shearing motion having the axis of shear parallel to one of these axes and shearing planes parallel to the axis of shear and one of the other principal axes of (1). The shearing planes of (2), which are not the slip surfaces of the superposed deformation, commonly parallel the planes of stratification. If Ω s is due to an additional superposed rigid-body rotational motion on a velocity gradient of the above type, then Ω e can be considered to equal an internal “physical” rotation Ω i for that part of the deformation after nucleation of the garnet. In that case the relationship of Ω s and Ω e to Ω for the same interval would not be simple, except where Ω s took place about the axis of shear of (2). In the latter case Ω = Ω s + Ω i , with due allowance for association of sign of quantities with sense of rotation.
Other uses of rotated garnets relate to determination of fold kinematics, shear stress, deformational heating, angular velocity, strain rates, paragenetic sequence and correlation, nature of reactions in the rock, location of their crystallization nuclei, variations of flow laws with time, repeated episodes of tectono-metamorphism, fabric sequence, and variations in growth rates of garnets.