We investigate the light transmitted or rejected from nacre (mother of pearl) taken from the iridescent shell of the bivalve Pinctada fucata. These nacre surfaces have a rich structure, composed of aragonite crystals arranged as tablets or bricks, 5 m wide and 400-500 nm thick, surrounded by 30nm thick organic mortar. The light rejected from these shell surfaces, or transmitted through thin polished layers, is rich in its polarization content, exhibiting a space dependent variation in the state of polarization with a high density of polarization singularities. Our goal is to use the polarization information to infer the structure of the biominerals and the role of the organic layer in determining the orientation of the crystals. In the experiments we send the light from a laser with a uniform state of polarization onto the shell, and analyze the light that is either transmitted or rejected, depending on the type of experiment, imaging it after its passage through polarization filters. We use the images from distinct filters to obtain the Stokes parameters, and hence the state of polarization, of each image point. We also construct the Mueller matrix for each imaged point, via 36 measurements. We do this for distinct physical and chemical treatments of the shell sample. Preliminary data shows that the organic layer may be responsible for organizing a multi-crystalline arrangement of aragonite tablets.