A complex logarithm turns circles into straight lines, for example, so that if we stare at a light circle on a dark background, a straight line of neural activity will cut across V1. (If two lines meet at an angle, a complex logarithmic map will turn them into two other lines meeting at the same relative angle. This is a type of conformal map, which preserves angles locally but distorts shapes at larger scales.

Perhaps it’s just because I’ve been reading “The Call of Cthulhu” lately, but the parallel between this sort of distortion and Lovecraft’s description of the weird, non-Euclidian angles of R’lyeh and the city of the Elder Things is pretty striking,

Given that Lovecraft was very clearly influenced by dreams, and that the Klüwer form constants are visible in hypnogogic and hypnopompic states, I guess it’s no surprise.

The back story to this post is a little funny:

You can tell from the ID number (302) that I started this post a good, longish while ago. In fact, I wrote it last September, when the features editor of New Scientist showed up and said, essentially, “Well why don’t you write a popular science story, Mr. Critic-Pants.” Given that Greg Egan and I had been complaining pretty vocally, he had every reason to be nettled; reining in my instinct for snarky replies, I took the next block of free time I had and wrote the above post.

At that, it’s shorter than I’d thought it would be: 1100 words instead of 2400 or thereabouts.

I had it almost finished when the folks at the n-Category Café said, “No Man but a blockhead ever did anything involving being peppered with buckshot, except for money,” so I stuck it on the drafts pile while I tried to figure out what to do with it, and then I got busy with other things.

Yesterday, I got Pharyngulated, and I realized I should put up something for people coming by, something perhaps of more general interest than my forthcoming essay on the Dirac Equation. I looked on ye olde draft pile, saw this item and realized that it read fine. I tweaked the beginning a little and pushed it out into the ‘tubes.

Now, I’ll have to figure out how I should handle the next installments. This area of research touches upon several ideas which have wide applicability: mean-field theories, phase transitions, critical points, etc. My plan at the moment is to explore each of those general topics and connect them to this specific example, rather than to give a blow-by-blow exposition of a technical paper.

Finding that sort of basic exposition online can be harder than one might expect. . . .

Bressloff, P.C., Cowan, J.D., Golubitsky, M., Thomas, P.J., Wiener, M.C. (2002). What Geometric Visual Hallucinations Tell Us about the Visual Cortex. Neural Computation, 14(3), 473-491. DOI: 10.1162/089976602317250861