Doing some laundry last night, I threw a duvet cover and nine pairs of socks into the dryer together. (Household hint: Don’t.) The duvet cover is a giant fabric pouch with a slit along one side; think of a queen-size pita pocket. Initially, all the socks were outside the pouch. When I pulled the load out of the dryer, all but three of the socks were inside the cover.

There’s nothing obvious about the geometry of this big, floppy bag that would suggest it has any special ability to capture socks. It’s not shaped like a fish trap with a funnel opening. In the random tumbling of the dryer, I would have thought that socks would move into or out of the opening with the same probability. But if that’s the case, then finding a 15–3 distribution is quite a fluke. There are 2¹⁸ = 262,144 ways of arranging the socks in two groups, and only 5,220 of them have three or fewer socks in one of the groups. That’s less than 2 percent and a little beyond the 2σ level of unlikelihood. Does Maxwell’s sock demon live in my dryer?

The spirals and whorls seen in sunflowers, pine cones and various other plant structures have long held a special fascination for mathematicians and for biologists with a mathematical bent. After all, you can find Fibonacci numbers in those natural patterns—who could resist? But it’s not just Golden Ratio mysticism that accounts for this interest. More important, I think, is the mere fact that these patterns are simple and orderly enough that we have some hope of understanding them at a deep level. If you want to build a general theory of plant form and growth, then the process that yields the spiral patterns—called phyllotaxis—is a good place to start.

The Joint Mathematics Meetings in New Orleans had an especially good session on phyllotaxis, organized by SIAM, the Society for Industrial and Applied Mathematics. I learned a lot.

Plant stems and roots grow mainly from the tip, from a region of rapid cell division called the apical meristem. It’s not hard to see how this causes elongation of a shoot, but what about branching? Buds and florets and various other structures do not just arise at random as a plant grows; they are spaced at regular intervals, often in a helical pattern. For example, if you number the branches from top to bottom along a plant stem, then as you visit the branches in numerical order, you’ll find you are also going around the stem repeatedly. The divergence angle between successive branches is a crucial factor in determining the overall geometry of the plant. If the angle is 90 degrees, say, then the plant will have fourfold symmetry, and branch n will always lie directly above branch n+4. Supposedly, the divergence angle is often near 137.5 degrees, which is the “golden angle,” dividing a circle into two pie slices whose central angles are in Fibonacci ratio (the limit of the series 1/1, 1/2. 2/3, 3/5, 5/8…).

How do plants accomplish this trick? A basic idea formulated in the 19th century (and perhaps glimpsed even earlier) is that any emerging branch inhibits the growth of other branches nearby; thus a new branch can begin developing only after expansion or elongation has created enough space to make room for it. Alan Turing, more than 50 years ago, suggested a chemical mechanism that might account for this effect.

This approach to understanding plant growth is by now textbook material, but there were some ideas presented in the six talks of the New Orleans session that came as news to me.

For starters, Turing got it backwards (which is not at all the same as getting it wrong). Turing’s model of biological development supposed that a few isolated hotspots produce a chemical growth factor, which then diffuses throughout the tissue; the gradient of concentration controls growth, with new buds appearing only where the concentration exceeds some threshold. The evidence now suggests that plant growth factors (called auxins) are produced by all cells at roughly the same level, and the concentration gradients arise not from passive diffusion but from active transport. Cells pump the auxins “uphill,” toward regions where their concentration is already elevated. Thus there is positive feedback: Abundant auxin attracts still more of it. If this mechanism operated without opposition, all the auxin would eventually accumulate in one place; the counterbalance is the continual creation of new cells, which has the effect of diluting concentration. In New Orleans Eric Mjolsness of the University of California Irvine presented these results, which have just been published in the Proceedings of the National Academy of Sciences. In a follow-on talk Przemyslaw Prusinkiewicz of the University of Calgary presented an algorithmic model based on the experimental results; this work too has recently appeared in PNAS.

Whereas the auxin-pump mechanism fills in some intricate biomolecular details, another line of work highlights a model of phyllotaxis that is simpler and than most others. Pau Atela and Christophe Golé of Smith College illustrated the idea with a penny game. Start with some pennies at the bottom of a sheet of paper, arrayed randomly but neither overlapping nor separated by more than one diameter. Now add pennies one at a time, always at the lowest available position on the paper. Each newly placed penny will be tangent to two others already present. (Tangency to three or four neighbors is possible but vanishingly rare.) A few stages of the process are illustrated below, where each newly added penny is shown in red.

If we interpret the rectangular sheet of paper as an unrolled cylinder, then the patterns produced in this way mimic phyllotaxis. The vertical position of a penny represents the height of a branching or budding point along a cylindrical stem; horizontal position corresponds to angle around the stem. (Note that on the unrolled cylinder left and right edges are identified, so that a penny going off the right side of the sheet comes back at the same height on the left.) Atela and Golé show by analysis and by numerical simulation that periodic branching patterns generally emerge even from random starting positions. If I understand correctly, they find that the famous golden angle is not very common when they measure the angle between individual successive branch points; on the other hand, the average angle does seem to converge on a value in the neighborhood of 137 degrees.

Although the penny model of phyllotaxis was new to me, it is not really real new at all. The model originated with work by Mary and Robert Snow in the 1930s and has been studied by several others since then, including Stéphane Douady, who also spoke at the New Orleans session. For more details see the excellent phyllotaxis web site assembled by Atela and Golé and their students in conjunction with an exhibition at Smith in 2002–2003.

About the pretentiously literary title of this post: I know it’s right on the tip of your tongue…. Yes, that’s right, it’s Dylan Thomas:

The force that through the green fuse drives the flower
Drives my green age; that blasts the roots of trees
Is my destroyer.
And I am dumb to tell the crooked rose
My youth is bent by the same wintry fever.

The mathematicians—5,130 of them, at last report—are in New Orleans this weekend. The occasion is the annual joint meeting of the American Mathematical Society and the Mathematical Association of America (with participation from several other organizations). When the same group met here six years ago, the influx of 5,000 mathematicians added 1 percent to the city’s population. Now it’s probably about 2 percent.

Over the next few days I’m going to write up some brief notes about a few of the talks I’ve heard here, and post some observations on the city itself. I’ve also set up a Flickr stream for photographs made on this trip, both in New Orleans and on the road to and from. See the pictures here.

For now, just one quick note on city geography.

The founders of New Orleans laid out a rectilinear grid of streets aligned parallel and perpendicular to the Mississippi River; in the map above the original grid is the section marked Vieux Carré. As the city expanded both upriver and downriver, the planners had a problem: The river is anything but straight. The solution they chose was a patchwork of locally rectilinear lattices with discontinuities between them. I am particularly conscious of the discontinuities because the route from my hotel to the meeting site crosses one of them, between the Vieux Carré and the Faubourg Marigny. (In the map above, my hotel is at the position of the green marker, and most of the meeting events are at the red marker; the route shown is the one suggested by Google Maps.) When I first looked at the map, I thought the street pattern offered me a stroke of good fortune. If the entire city had been ruled according to the grid in the neighborhood of my hotel, I would have had a much longer walk; the 45-degree bend at Kerlerec Street allows me to take a diagonal shortcut for much of the trip. But then I reflected that if the Vieux Carré grid had been adopted everywhere, my walk would have been even shorter.

In this particular case, the route proposed by Google looks like an optimal one; that is, if we ignore certain oddities induced by one-way streets, there is no other path that would be significantly shorter. But, more generally, how does one find the best route through a grid with discontinuities? Within any one section of the grid, distance is measured by the Manhattan metric (s = x + y), and all reasonable paths are equivalent. (A path is reasonable if it stays within the smallest rectangle enclosing the origin and destination and if it never moves away from the destination.) Thus the problem comes down to choosing the point along the line of discontinuity where you cross over from one grid to the other. What is the rule for choosing the best crossover point? Is it unique or are there multiple equivalent points?

Update 2007-01-09. This is in response to Stephan Mertens’ suggestion in the comment below. His proposal to choose the crossing point nearest to where a straight line intersects the boundary is doubtless excellent practical advice, useful for navigating in the real world. But we’re talking mathematics here, and usefulness in the real world has nothing to do with it. Does Mertens’ method always produce the mathematically optimal path?

Consider the continuous case—the limit you reach as the lattice spacing goes to zero. What you have then are two anisotropic media where speed varies with direction; in the worst case you are slowed down by a factor of √2. Does the straight-line heuristic work in this case? I think not. The situation is similar to (but a little more complicated than) the description of a light ray traversing a boundary between two media that differ in index of refraction. Light bends at such a boundary, following the path that minimizes the total travel time.

Going back to the discrete-lattice case, it’s not quite clear (to me, at least) how to apply the continuum solution, because the problem itself is not very clearly stated. If the two lattices have the same spacing but different orientations, then they are incommensurable, and adjustments of some kind have to be made all along the boundary. In the real New Orleans street grid, this is done in an ad hoc way; note, for example, that Bourbon Street disappears when you leave the Vieux Carré. Any precise statement of the problem would have to say just how the grids are joined, and that gets messy. Nevertheless, I offer the example below for consideration.

Here we want to navigate between the red points. The straight-line heuristic appears to recommend crossing the boundary at the blue point, but there are shorter paths via the green point.

Update 2007-01-12. After further forthing and backing, Stephan Mertens and I have agreed not to disagree about this issue. In order to avoid the messy problem of drawing streets to connect unaligned lattices, Mertens makes the civilized suggestion that we build a park in the interstices, like so:

Within the park, a walker is freed from the constraints of the lattice and can follow any Euclidean path. In the case shown above, either the green or the yellow path (or many others) would be allowed. Mertens writes:

For this simple model one can write down the distance between a point in the left and a point in the right grid, as a function of the transition points where the walker enters or leaves the transition region. It is most convenient to use different coordinate system in the grids: Both have (0,0) at the point where the grids touch and increase y moving upward and x moving away from the crack. Note that the y coordinate of the transition point can vary continously since the walker may decide to leave the grid everywhere on the street that runs along the crack.

The resulting function can be plotted with Mathematica; apparently the minimum is unique, and a strategy that seems to be not too far off (this time I am cautious with my claims!) is to pass the through the park using the two gateways that have the y coordinates of source and target. In other words: Run all the way straight in each grid to reach the transition region.

There are cases where the run-straight-to-the-park strategy differs substantially from the procedure of drawing a straight line from origin to destination. The example below (where the yellow path is shorter than the red one) offers a template for generating street grids in which the penalty for choosing the straight-line path becomes arbitrarily large.