how (some) fireflies do it

Fireflies in Borneo have a wonderfully simple and distributed way of synchronizing their blinks.

If I had the skill I’d make a screensaver of it. A couple of ways to play with the concept:

I wouldn’t expect all bugs to have *exactly* the same period, but how much variation is tolerable? What if there are two populations, indistinguishable except that their periods differ by an irrational factor?

What if each bug has a different hue, and responds only to others that are near on the color wheel (perhaps only in one direction)? Might a stable cycle result, rather than synchrony of all?

what, more links?

Hm, the first two links here have been lying around for five years; guess I ought to shove them out.

Pascal-Emmanuel Gobry on restructuring the banks

“Zomia”, a large region in Asia that was effectively stateless until recently

James Leroy Wilson on The Limits of Utilitarianism. The payoff is near the bottom.

points problems pointers problems

Two-fifths of the links in my sphere arrangements page were broken. Ouch! I found new addresses for a few of them, and used the Wayback Machine for the rest.

adaptive sampling

I got an interesting idea today.

As you may already know, I’ve been making models of Klein bottles an’ stuff; heretofore they’ve all been in the form of bent rods, but where possible I’d prefer a continuous surface. (A hollow body must have holes so that unused powder can be shaken out; but not all of my designs have enclosed spaces.) How to place a minimum number of vertices so that deviations from the abstract shape are within the resolution of the process? That’s less obvious with more degrees of freedom.

So, today’s idea. Start with an arbitrary set of sample nodes (in the abstract space of the parametric variables, rather than on the target surface itself), and their Delaunay triangulation. Along each edge of the triangulation, measure the deviation of the surface from a straight line; this gives the edge a weight. Move each node to the weighted average of its neighbors (with a bit of noise); thus, an edge whose image is strongly curved gets shorter.

After the movement phase, each edge ought to be checked, whether it’s still a Delaunay edge or needs to be replaced by the other diagonal of the quadrilateral formed by its two triangles. I don’t yet have criteria for adding nodes where existing nodes are too far apart, or merging them if they become redundant.

Scribbles: The Ensmoothening, Part II

One thing I noticed in that last series of charts is that more than one degree of discontinuity doesn’t help: the best-looking curves are mostly on the diagonal, where only the last nonzero derivative is discontinuous. Here, therefore, are those curves all together.

In column zero, the tangent angle is piecewise constant; in column one, it is a piecewise linear function of path length, resulting in six circular arcs; in column two it is piecewise quadratic, resulting in six clothoid arcs with continuous curvature; and so on.

Of course the arcs are approximated by cubics; to improve the match, I put a knot wherever any derivative crosses zero, as well as at the discontinuities. (See the knots.)

ensmoothening scribbles

Presented for your consideration: the somewhat disappointing results of an experiment in using piecewise polynomial spirals, of varying degrees of continuity, to fit the Takana — disappointing in that few if any of the curves are as pretty as I hoped.

I treat here only those that can be drawn with a single stroke. (The others can be built by combining subsets of these strokes.) In each chart, the degree of continuity increases downward, and the degree of the polynomials increases to the right.

A polynomial spiral is a curve whose tangent angle is a polynomial function of arc length; it has the form `integral(exp(i*f(t)))`. (I implement it as a Taylor series.) In principle, `f` could be any real-valued function. If `f` is constant, you get a straight line; if `f` is linear (leftmost column in these small charts), you get a circle; if `f` is quadratic, you get an *Euler spiral* or *Cornu spiral* or *clothoid*, which is much used in railroads and highways to avoid sudden changes in lateral acceleration.

Here `f` is a least-squares fit to the step function which is the direction of the squared stroke. The top row of the chart shows continuity of degree zero: the component arcs meet, but that’s all; `f` is discontinuous. Degree one: the tangent angle is a continuous function of arc length. Degree two: the first derivative of tangent angle with respect to arc length, i.e. the curvature, is continuous. Degree *n*: the (n-1)th derivative of tangent angle, i.e. the (n-2)th derivative of curvature, is continuous.

Click each chart to extend it.

**Later:** I have come to a couple of conclusions. In most of these charts, the best entry to my eye is where `f` is piecewise quadratic with one continuous derivative. More than one degree of polynomial above the continuous degree adds little fidelity and detracts from beauty.

( . . more . . )