During the summer of 2005, the Visual and Infrared Mapping Spectrometer onboard the Cassini spacecraft observed a series of occultations of the star o Ceti (Mira) by Saturn's rings. These observations revealed pronounced variations in the optical depth of the A ring with longitude, which can be attributed to oriented structures in the rings known as self-gravity wakes. While the wakes themselves are only tens of meters across and below the resolution of the measurements, we are able to obtain information about the orientation and shapes of these structures by comparing the observed transmission at different longitudes with predictions from a simple model. Our findings include the following: (1) The orientation of the wakes varies systematically with radius, trailing by between 64?? and 72?? relative to the local radial direction. (2) The maximum transmission peaks at roughly 8% for B = 3.45?? in the middle A ring (???129,000 km). (3) Both the wake orientation and maximum transmission vary anomalously in the vicinity of two strong density waves (Janus 5:4 and Mimas 5:3). (4) The ratio of the wake vertical thickness H to the wake pattern wavelength ?? (assuming infinite, straight, regularly-spaced wake structures) varies from 0.12 to 0.09 across the A ring. Gravitational instability theory predicts ?? ??? 60 m, which suggests that the wake structures in the A ring are only ???6 m thick. ?? 2007. The American Astronomical Society. All rights reserved.