Geometry of Biological Time, Chapt 2.

Co-tidal map from NASA via Wikicommons. The points of intersection are the "phase singularities" where the tidal phase is undefined.

Slowly making my way through this book. Chapter 2 is about phase singularities -- places where the phase of some oscillation is undefined. The coolest example is the earth's tides. The surface of the earth is a sphere ("S2" in topology speak) and the tides are defined by a phase (S1). So for each point on earth at any given moment there's a tidal phase. But S2->S1 mappings (with certain continuity assumptions) must contain phase singularities -- there must be places where you can't define the phase. Above is a map from NASA showing these places as the intersections of the co-tidal lines. You can think of the tides as sloshing around those points where the sea level doesn't change.

The chapter is mostly about biological versions of such phase singularities. Detailed examples are given from fruit fly circadian rhythms, but the technical details of the experiments were overwhelming so I didn't fully follow and decided, perhaps unwisely, to plod forward without complete understanding.

Geometry of Biological Time, Chapt 1.

I've started reading A. T. Winfree's book (father of Erik): "Geometry of Biological Time". Sometimes one finds just the right book that fills in the gaps of one's knowledge; this book is just right for me at this moment, as if I was fated to read it.

It begins with an excellent introduction to topology mapping. I had picked up some of the ideas by osmosis, but the first 20 pages were an excellent and helpful series of discussions that help solidify my understanding of this subject. He lucidly expands on abut 15 topological mappings in increasing complexity. For each, he provides intuitive examples with lovely side discussions such as relating the S1 -> I1 mapping to the international-date-line problem and the astonishment of Magellan's expedition to the loss of a day upon the first round-the-world trip. (I first heard of this idea as the climax of the plot of "around the world in 80 days"). He introduced the idea of all such mapping problems as singularities in the mapping functions. Again, this was something that I half-understood intuitively and thus it was very helpful to have it articulated clearly.

I now realize that in previous amorphous computing experiments described in this blog, I had been exploring S1 x S1 -> S1 mappings (circular space by oscillator space mapping to a visible phase). This S1xS1->S1 mapping is exactly where he heads after his introduction as a place of interest. In other words, I had ended up by intuition exactly where he did.

It's a very long and dense book, if I can maintain my way through it, it may generate a lot of blog entries!