I don’t know why it took so long for me to realize that having a star on a tree could be a mathematical statement and not Christian. I’ve been meaning to learn more about the Kepler-Poinsot solids for a while and this was as good an excuse as any. George Hart describes it well though I can’t open the *wrl files on some of my machines. Kepler-Poinsot solids are the 3D analogue of regular star polygons. Both generalize our notion of ‘regular’ to allow for intersecting edges and faces. For example, the regular pentagram,
has five vertices and five sides. Where we see the lines cross do not count as vertices. There are five equal angles (the angles at the pointy vertices) and all five sides are equilateral. In the same way the small stellated dodecahedron has pentagrams for faces, but they intersect. I found a really nice video on YouTube which shows the face of each Kepler-Poinsot solid and how the faces fit together. It’s really helpful. Note: the words ‘dodecahedron’ refer to the fact that these solids have twelve faces not because they are derived from the Platonic solid.
Given that brief lesson I now am curious how I would go about creating an accurate model of these star polyhedra? I’ve long obtained my print outs for polyhedra nets but I realize that in the case of the Kepler-Poinsot solids that these are not faithful because the faces don’t intersect. I am puzzled at the moment.
It’s worth noting as well that these solids fail to satisfy Euler’s formula for polyhedra, and that there have been generalizations of this formula to fit these solids. This reminds me that I need/want to reread by Imre Lakatos. Accessible to a wide audience, it is written as a story or dialogue between a number of characters, all students of varying intelligence, about Euler’s formula and to what it applies. It presents a view of mathematics that every student should experience. Math is not handed down from on high, it is discovered through work and mistakes. Definitions are creative works and they are worked on over and over again until they are just right. Too often students only see the polished end product and not the rough and messy process.
Speaking of books, Stu and I swung by the library after the morning final yesterday for the end-o-semester stocking up of break reading material. Some of the books picked up were just to flesh out the office reference section but several others are related to some topics I’d like to understand better or present on for a general audience.
(1) by Coxeter. After reading about half of Baez’s series on Platonic solids in 3D and 4D and seeing (and almost understanding) the Coxeter notation, I really need to digest some of this stuff. Who better to learn from than the man who saved geometry? This also fits a larger theme of becoming better at geometry in dimensions more than three and developing a better understanding for why (if it is the case) 4D seems to be the very best ‘D’ of them all.
(2) by Alain Connes. Easily one of my new heroes given his interests in physics and philosophy (picked up a couple of books on such essays as well, more later). Overall I’m unlikely to make any swift progress in NCG as the prerequisites are steep. It’s impressively abstract and difficult but I have hope. The space of Penrose tilings are studied and techniques from NCG are applied to it. There will be a future series of posts on that space if nothing else. I’d like to be able to understand another example as well (something reportedly simple like the noncommutative torus) but we’ll see.
(3) Some books on mechanics. I’ve been in love with geometry and topology in physics for some time now and would like to start giving a talk or two per semester on the ideas you find therein. My thought for the spring would be a talk introducing the basics through a couple of examples, a simple pendulum with and without gravity and a compound pendulum with and without gravity. Stay tuned for progress on that end. Some of it will just be recasting ideas that students are familiar with in the context of differential geometry. It would be nice to begin to set the foundations to eventually talk about geometry in statistical mechanics (dimensions much much higher than three!), quantum mechanics (NCG) and electromagnetism (fiber bundles and gauge theories!).
For now, however, I should return to grading finals. Happy Holidays.