Last week, the theme for my idle thoughts seemed to be symmetry.
I should have written about it last week while it was still all fresh in my mind. I can no longer recall all of the facets of symmetry that I had floating in my head last week.
I first noticed symmetry becoming a theme for the week on Tuesday while sitting at the bus stop. I had been pondering several software schemes to aid in crossword puzzle design. Every newspaper-quality crossword puzzle has 180-degree rotational symmetry (with respect to where the letters go and where the letters cannot go). All of them.
I'd known about that symmetry for some time. But, sitting at the bus stop last week, there was much more depth to it. Blank crossword puzzles spun in my minds eye in front of me.
The symmetry played into a symmetry I had just learned during a math lecture. A Dutch auction[*] mirrors a sealed-bid, winner-pays-highest-bid auction. An English auction[**] mirros a sealed-bid, winner-pays-the-second-highest-bid auction.
Other things which played into the symmetry theme for the week was the idea of a "diameter of a group with respect to a set of generators". Without getting into a bunch of group theory here, I was particularly curious about this with respect to the set of 1/4 turns on a Rubik's cube. Needless to say, symmetries abound.
I also spent some time thinking about symmetric encryption schemes. It was interesting to me that most schemes are based on mirror symmetry, but some are based on processes which are their own inverse. In some cases, those are both the same thing.
And, somehow, my mind had some symmetry (not just duality) between offense and defense in volleyball.
Bleh, this has been a really incoherent entry.
* A Dutch auction is one where the price lowers steadily over time until someone yells "Stop!"
** An English auction is the kind which you see in the movies.