Siliceous and Delicious

I left off yesterday wondering about the problem of how to describe straight-edged forms. Wikipedia helped me out by defining curvature of a circle for me as the reciprocal of it’s radius. This is neat because as the radius of the circle goes to infinity, i.e. the line approaches being straight, the curvature goes to zero. So I could simply use this formulation of curvature, K=1/R, as my parameter for the model. If we make h from yesterday (the height from top to bottom of the two-sided form) 2, then K would range from zero for straight edges (unfeasible for two-sided forms, of course) to a maximum of 1, when the radius of the circle equals h/2.

I could also extend this range to negative values of K representing curvature in the opposite direction:

That sort of covers the straight-line problem from yesterday, I think. It still leaves unanswered the question of how non-circular one-segment forms can be represented (ellipses and ovals). My first instinct is to just apply the same approach as for the two-segment forms—just add a sine curve. But I’m not sure what this will actually yield. An ellipse?

I’m not sure how this works out. I have a feeling that adding a complete sine wave is just going to shift the shape, since the sine wave is based on the circle. But maybe not. I’m getting stuck thinking about it, and I think I have to do one of two things (or both)—talk to someone who understands the math better, or has a better feel for it (like Jacques, for example), and to start messing around with this on the computer, maybe write some code so I can actually toy around with these parameters. Jacques may be able to help with that, too.

Decided to email Jacques and ask for some time to chat with him rather than running my head against the proverbial wall and taking the wind out of my proverbial motivational sails. In the meantime, I decided to print out and read the paper Jacques had sent me in reply to my latest progress report—it’s about the morphogenesis of the diatom frustule, i.e. how the cellular mechanics of building the shell actually works, a subject in which he professed to having a rekindled interest.

The paper he sent is mostly just a series of very nice SEMs of frustules in statu nascendi. I didn’t get much out of it at first, but then consulted the Sumper paper in science to which Jacques also referred, saying the observations in the new paper were at odds with Sumper’s model. Sumper suggests a morphogenetic model in which the arrangement of lipid vesicles in the SDV by simply physical processes (hexagonal packing) and fragmentation of those vesicles as the lipid concentration drops can explain the self-similar structure of areolae, cribra, and cribrella. The morphogenetic steps documented in this new paper, however, are at odds with this—rather than a series of self-similar hexagons growing ever inwards, it seems that what’s going on is quite a bit more complicated.