Professional mathematicians often speak of the beauty of mathematics and the elegance of its solutions. Yet the esthetic appeal of math is rarely conveyed to students at the elementary, secondary, or even college level. Instead, most of us develop phobias in school about math's elusive logic and then pass these negative impressions on to our children. We should all be having fun with math and helping our kids to do better in life by encouraging them to appreciate not only its usefulness but especially its charm. That's just what veteran math educator Alfred Posamentier sets out to do in this...

The central theme of this monograph is that the cosine seminvariants are the key to crystal structures. The cosine seminvariants are the cosines of those linear combinations of the phases (the so-called structure seminvariants) whose values, for a given functional form for the geometric structure factor, are uniquely determined by the crystal structure alone and are there- fore independent of the choice of permissible origin. It follows that the cosine seminvariants themselves are uniquely determined, in general, by the observed magnitudes of the normalized structure factors. The values of...