After a two-month hiatus I've put some new photos up, of Timothy's 8th birthday party. We hired a local company to come to our house and do a reptile show, which seemed to go over well with the boys.

My new computer and new version of photoshop have left me unconfident of the color calibration on the photos, though. I ended up calibrating the monitor merely by going with the built-in calibration and setting gamma to 2.2, since that made various internet test charts look closer than I was getting with more sophisticated calibration procedures. And after some effort I've figured out how to get Photoshop, Preview, and Safari to all agree with each other on what my images look like, an essential first step in getting consistent color for my photos. But everything's slightly more washed out and desaturated when I view it in Camino; other people's images, too, not just mine.

One of the things I discovered I needed for consistent cross-application color was to embed a color profile tagging my jpeg files as sRGB. My older images, which are untagged, are still coming out consistently washed out and desaturated in Safari as well as Camino. So I suspect what's happening in Camino is that it's stripping the color profile information before it passes the image to the OS display routines, and that the OS on this machine, if it sees untagged color, just sends it straight to the monitor without conversion rather than as in previous versions assuming sRGB. If so, I may have to switch to Safari for my more serious web photography as well as going back and adding profiles to all my old images (fortunately easy with exiftool).

On the other hand, maybe I will hear back that my new images are seriously blackened, that my color calibration is all screwed up, and that I will have to process this batch of photos over again after completely redoing my color setup.

A corollary to the observation that printed conference proceedings are obsolete: it is preferable to use color illustrations than black and white, even for proceedings that will be printed black and white.

To expand a little on the reasoning for this, the audience for your paper consists of three classes of people: those who view it on-screen, those who print it themselves from an on-line version, and those who read the printed proceedings. Color helps convey information more effectively people in the first two classes, the third class is rapidly diminishing, and a careful choice of colors can ensure that even the people who only see the black and white version (or who are color blind) get an understandable figure. So supplying the illustrations in color helps more people than it hurts.

A few weeks ago I asked whether periodic tilings of the plane have periodic colorings with four colors, and if so what the relation is between the periodicity of the coloring and of the tiling. As discussed in the comments there, another way of asking the same question is, if a graph is embedded in the torus, is there a finite cover of the torus such that the lift of the graph to the cover needs only four colors?

I still don't know the answer to the question for four colors, but the answer turns out to be yes for five colors. Albertson and Stromquist [PAMS 84:449-456, 1972] showed that if a torus graph has no noncontractable cycles of length less than eight, it's five-colorable. And it's easy to find a finite cover of a torus in which any noncontractable cycle has to pass across eight copies of the original torus, hence have length at least eight. Or, put another way, for any periodic tiling of the plane, if you make the pattern of repetitions big enough so that each tile in the tiling is eight or more tiles away from its nearest repeated copy, you can use only five colors in a periodic coloring.

The floret pentagonal tiling is shown below with two colorings. On the left is an optimal coloring: it uses the fewest colors possible, six, among colorings that respect all translational symmetries of the tiling. The quotient of the plane tiling by these symmetries is a tiling of the torus by six tiles, in which each tile touches each other, so six colors is optimal. But on the right, we have a periodic tiling with only three colors! The trick is that it has fewer symmetries: only 1/3 of the symmetries of the uncolored tiling preserve the colors in the coloring on the right.



What can be said about this kind of problem more generally? If one has a periodic tiling of the plane, how few colors are required to color it periodically, and what can be said about the index of the coloring's symmetry group as a subgroup of the uncolored tiling's symmetries? The four-color theorem together with a standard application of König's lemma shows that any tiling has a four-coloring, but I don't see how to force the coloring to be periodic and still use only four colors in general.

ETA: Five colors suffice. Still unclear whether four colors are always possible.

Choosing Colors for Geometric Graphs via Color Space Embeddings, the third of my Graph Drawing papers, is now up on arXiv. This is the source of the multicolored checkerboard, or baseball diamond, or whatever you want to call it, that I posted about a few months back.

After giving my last two lectures of the school year today, I'm not up for much more than posting pretty pictures, so:



( The explanation )

Here's an illusion to prove you wrong: Big Spanish Castle. Stare at the dot in the color-reversed image, then without moving your eyes, move your mouse over the image. You should see it as a vivid color photo, until you start moving your eyes at which point it will be revealed as black and white. Via Metafilter.

With feeling. So we'll wait for it to come around on the guitar, here and
sing it when it does. Here it comes.

You can do anything you want, with s2flexisquares.
You can do anything you want, with s2flexisquares.
Walk right in it's around the back
Just a half a mile from the railroad track
You can do anything you want, with s2flexisquares.

Which is to say, I found the loophole that lets you use your favorite CSS styling for livejournal, instead of being stuck applying minor color and font tweaks to the pre-existing styles.

Except for one small detail that I'm still unable to get right (aka Alice): the html for the friends page still hardcodes a white background and black text around the userpic. Grr....so close. Thanks to anchan218 for helping me fix that last small detail!

Apparently the color of geometry is a light pinkish beige. As opposed to math, which is more of a warm medium gray. So now we know.