O. R. Norton, Science Graphics, Bend, Oregon 97708 (personal communication, 1996).

T. Kobayashi, K. Higuchi, “On the pyramidal faces of ice crystals,” Contrib. Inst. Low Temp. Sci. Hokkaido Univ. Ser. A, No. 12, 43–54 and 13 plates (1957).

If there are at least three linearly independent vectors among Pv1, … , Pvk, then there is at most one induced pole symmetry w* for the given w, P, and V. Otherwise there may be more than one w*.

Half of SO(3) and the partition can be visualized as follows. The orientations in SO(3) whose (one-point) zenith loci are on the upper hemisphere are pictured as the upper hemisphere with a line segment 0 ≤ ϕ < 360 attached at each point and pointing radially outward—a boy’s head with a brush haircut. The segment—strand of hair—attached to the sphere at Y is the coset Zu = {zrot(ϕ) · u}, where u, corresponding to ϕ = 0 and thought of as a point on the sphere, is the rotation that takes Y to k and that has horizontal rotation axis. The halo-making sets are the sets consisting of entire segments—no hairs are split. The zenith locus of a halo-making set is the portion of scalp from which its hair is growing. (The brush cut picture can be extended to most of SO(3) by extending the hemisphere downward toward the South Pole -k. But the South Pole cannot be added to complete the picture, since there the above description of u is not enough.)

S. W. Visser, “Die Halo-Erscheinungen,” in Handbuch der Geophysik, F. Linke, F. Möller, eds. (Gebrüder Borntraeger, Berlin-Nikolassee, 1942–1961), Vol. 8, pp. 1027–1081.

The group W of orthogonal transformations w satisfying wk = ±k is a subgroup of the group O(3) of all orthogonal transformations, and Z is a normal subgroup of W. The quotient group W/Z consists of the four cosets Z · e, Z · xrot, Z · yref, and Z · zref, which correspond to the four lines of Table 1. The homomorphism of W onto {e, -e, yref, -yref} that is implicit in the theorem can be expressed as w = zrot(ϕ) · yrefi · zrefj → Z · yrefi · zrefj → (-1)jyrefi+j, the first mapping being the quotient map onto W/Z. The quotient group formally captures our intuition that we often do not worry about zrot(ϕ).

A subtle and logically important point that can nevertheless be ignored on a first reading is that whether a given rotation u is a plate (Parry, alternate Parry, etc.,) orientation depends on the wedge under consideration. Technically, this is because the vector N1(u) depends on the wedge as well as on u, since u starts with the crystal oriented so that the specified wedge is in standard orientation. For example, the plate orientations for wedge 1 6 have the form u = zrot(ϕ)yrot(-45), whereas the plate orientations for wedge 3 1 have the form u = zrot(ϕ)yrot(45). They are not the same.

J. Moilanen, M. Pekkola, M. Riikonen, Finnish Halo Observers Network, URSA, Raatimiehenkatu 3 A 2, 00140, Helsinki, Finland (personal communication, 1994).

Thus Par ij = Pu, where P = N3 and where the matrix u gives the orientation, or frame, of wedge i j. In Section 1 we said that since the spin vector P is a wedge vector, then Pu is independent of u. That, however, assumed there was only one wedge under consideration (and one spin vector). Now Pu depends on u. The technical explanation is that P(u) [e.g., Eq. (24)] depends not only on u but on the wedge under consideration. But it is probably best to think less technically: At any moment there is the vector P, and there are the wedge frame vectors A, B, C for the wedge under consideration. Then Pu is given by Eq. (21) as always. Equation (23) is also correct, with A, B, C being the columns of the matrix u. Of course Pu depends on the choice of spin vector as well as on the frame.

W. Tape, Atmospheric Halos (American Geophysical Union, Washington, D.C., 1994).

[CrossRef]

R. Greenler, Rainbows, Halos, and Glories (Cambridge U. Press, New York, 1980).