This is actually a presentation that crosses the disciplines of chemistry, mathematics, biochemistry, and structural biology. "Quaternions" (qua-TURN-ee-ons) are four-dimensional mathematical objects that have been used to describe orientations in space. This talk presents work that we have done here at St. Olaf College over the past two years (and that was an outgrowth of discussions at the 2007 Gordon Research Conference on Visualization) in which we have explored the use of quaternions for quantifying and identifying protein and nucleic acid secondary structure. In conjunction with this research, being the current principal developer of the open source Jmol molecular visualization project, I've been able to enhance Jmol so that it can display many aspects of protein and nucleic acid structure in ways that have not previously been possible, including representations of a novel mathematically sound definition of "straightness" we have developed here at St. Olaf College.