\( \def\Fix{\mathrm{Fix}} \def\D{\mathsf{D}} \def\C{\mathsf{C}} \def\D{\mathsf{D}} \def\GL{\mathsf{GL}} \def\OO{\mathsf{O}} \def\SO{\mathsf{SO}} \def\RR{\mathbb{R}} \def\ZZ{\mathbb{Z}} \def\Aut{\mathrm{Aut}} \def\xx{\mathbf{x}} \)
jam - wiki

Web pages

  • Permutation group (wikipedia)
  • Lecture notes in Geometry (these notes contain accounts of much of the geometric side of our course)
  • Wallpaper groups on wikipedia
  • A host of useful short articles are available here (written by Prof. Keith Conrad at the University of Connecticut, USA). There is one on Group Actions which is directly useful, but there are many more which could be interesting.


Symmetric oscillations

In both of these, the oscillations are periodic orbits. Consider the triangular shape: here at every instant of time the drop has \(\D_3\) symmetry, while after half a period the drop is rotated by \(2\pi/6\). Together this gives the spatio-temporal symmetry: in the notation of Chapter 5, \(H=\D_6\) and \(\theta:H\to S^1\) has \(\theta(g)=0\) if \(g\in\D_3\), and \(\theta(g)=1/2\) if \(g\not\in\D_3\). Analysis of the other oscillations is similar.


  • Video about iOrnament. iOrnament is an impressive piece of software for generating wallpaper patterns, as well as similar patterns on the sphere and the hyperbolic plane. For iPad and other iThings only (not android unfortunately). This 35-minute video is by the developer of iOrnament, Jürgen Richter-Gebert - a Professor of Mathematics at University of Munich.