Came across this link recently. I’m a huge fan of John Baez. He writes often, about many subjects and in a particularly illiminating way allowing a broad audience to read and make the connections he’s aiming for. The link brings you to part one of a series which seems to be at least thirteen parts long. The goal is to look over the platonic solids and use them to develop Coxeter complexes, notation and groups in order to be able to study the Archimedean solids but eventually generalize to four dimensions.
I’m on a bit of a kick lately about polyhedra after spending a portion of this semester trying to develop some curricula for a liberal arts math course I’ve been teaching. As you might imagine, during the semester many questions of my own have arisen and have culminated recently in a desire to understand more at a deeper level.
Baez also writes about connections between the Quaternions, 4D platonic solids, the 3D platonic solids and why 4D has the largest number of platonic solids. The link to that article is http://math.ucr.edu/home/baez/platonic.html. What’s also of interest is how these ideas of symmetry apply to tilings and tessellations. See for example a wonderful colloquium given at the Harvard Physics Department by Peter Lu on Islamic geometric patterns and the mathematics they probably used to generate them (http://www.youtube.com/watch?v=rldnu9rNpH8&list=WLD2BABB65BCF7F811). This brings up Penrose tilings.
These topics (among others) feature several characteristics that drive me. First is the interplay between deep mathematical ideas and extremely visual (and beautifully visual at that) representations. Second is the interplay between pure investigation and imagination and the discovery, consequentially, of physical properties or theories. Thirdly, and this is related to the first, is to develop a set of craft based mathematical activities to create a summer camp (or series of camps) around. To be able to teach a craft and the corresponding mathematics and to enable the child to explore either by using either. Last and certainly not least is that much of this relates to the physics that I am interested in learning more about.
I’ll post (or try to remember) when I learn more. Stay tuned for my thoughts on braids, knots, geometrical EM, Coxeter stuff, 3d->4d and more.