Clockface polygons and the collective joy of making mathematics together

The social and embodied nature at the heart of all knowing, doing, and learning contrasts with the images that pervade our cultural imagination of mathematical work as a solitary, cognitive activity. This article describes a playful experiment by the author group to do collective mathematics, in an extended effort to construct alternative images, instincts, and practices for ourselves. We present a pair of episodes of mathematical exploration that come from our work together and that we have seen as an early success, intimating features of a stabilized collective mathematics that we hope to continue pursuing. Coming from a single investigation of our group, these episodes offer narrative accounts of the parallel inquiries of subgroups, working to define and characterize a mathematical space we had collectively identified, and then to formulate and investigate conjectures about that space. The narratives are followed by a discussion of themes within and across them and reflections on their significance as a step toward self-organized collective mathematics

Simple Radiometric Method for Accurately Quantitating Epitope Densities of Hapten–Protein Conjugates with Sulfhydryl Linkages

Control of small molecule hapten epitope densities on antigenic carrier proteins is essential for development and testing of optimal conditions for vaccines. Yet, accurate determination of epitope density can be extremely difficult to accomplish, especially with the use of small haptens, large molecular weight carrier proteins, and limited amounts of protein. Here we report a simple radiometric method that uses ¹⁴C-labeled cystine to measure hapten epitope densities during sulfhydryl conjugation of haptens to maleimide activated carrier proteins. The method was developed using a (+)-methamphetamine (METH)-like hapten with a sulfhydryl terminus, and two prototype maleimide activated carrier proteins, bovine serum albumin (BSA) and immunocyanin monomers of keyhole limpet hemocyanin. The method was validated by immunochemical analysis of the hapten–BSA conjugates, and least-squares linear regression analysis of epitope density values determined by the new radiometric method versus values determined by matrix-assisted laser desorption/ionization mass spectrometry. Results showed that radiometric epitope density values correlated extremely well with the mass spectrometrically derived values (<i>r</i>² = 0.98, <i>y</i> = 0.98<i>x</i> + 0.91). This convenient and simple method could be useful during several stages of vaccine development including the optimization and monitoring of conditions for hapten–protein conjugations, and choosing the most effective epitope densities for conjugate vaccines