Imagine hauling water out of a well in a bucket with a long rope handle, using a second rope looped through that handle. The two ropes are clasped to each other; the ends of one rope are pulled upward while the ends of the other are pulled downward by the water's weight. One might expect that, under tension, the shapes of the ropes would be built from circular arcs (where the ropes loop around each other) and long straight segments to the ends.
Using a simple mathematical model for the behavior of linked ropes of fixed thickness, a team of researchers from Georgia, Illinois, Massachusetts and Berlin have discovered that this simple picture does not describe the true shape of the clasped ropes. Instead, there is a surprise: when the ropes are pulled tight against each other, their tips pull away from one another at the center of the clasp, creating a small enclosed gap.
The researchers' model is easy to describe: If each piece of rope in a link has diameter 1, that means the core curves (along the center of each rope) must stay distance 1 from each other. What the team calls the "Gehring ropelength problem" is to minimize the total length of a given link (meaning the length of those core curves) subject to this constraint that curves stay distance 1 apart; this means the ropes, modeled as diameter-1 tubes around the curves, stay disjoint.
The simplest version of the problem, posed by Gehring in the 1970s and solved soon thereafter, asks for the tight shape of the simplest link, what mathematicians call the Hopf link: two circles (closed curves) linked as in a chain. There were no surprises here: the tight configuration uses two round circles, lying in perpendicular planes and each passing through the other's center. In 2002, several members of the current team (Cantarella, Kusner and Sullivan [CKS02]) extended this result to longer chains. When a simple chain of many components is pulled tight, the components at the end are still round circles. However, the components in the middle (each linked to two neighbors) are elongated like the track in a stadium, consisting of two semicircles connected by two straight segments. Since many real-life chains are already designed this way, the shapes of these tight chains again came as no surprise.
|The tight configuration of a simple chain is built from circles at the ends and stadium curves in the middle.|
However, the new results of the full team surprised even the researchers involved. Their new paper "Criticality for the Gehring Link Problem" (1MB PDF) introduces a "Balance Criterion" which describes not only minimizing configurations for the ropelength problem, but all equilibrium configurations. And the semicircular configuration of the clasped ropes described above is not balanced: the entire weight of the bucket would be transmitted, through the circular arc of its rope handle, onto just the single lowest point of the second rope.
The researchers describe instead a balanced configuration in which the tips are at distance 1.06. The slight extra length needed for this increase in tip separation is more than offset by now being able to narrow the tips; this equilibrium configuration is about half a percent shorter than the original clasp. Another surprising feature is that the curvature of each core curve is unbounded at the tip.
[Sullivan had run numerical simulations [Sul03] suggesting that there should be such a gap in the clasp. Inspired by these, Eugene Starostin has recently announced [Sta03] an independent derivation of the equations for the clasp, without a proof of criticality.]
|Left, the setup for the clasp problem, and the naively expected configuration with semicircular arcs. Right, the actual shape of the Gehring clasp.|
|Below, some views of the gap chamber between the components of the clasp. For more details and high-resolution images, see the paper.|
The team has also found an equilibrium configuration for the Borromean rings. This link of three components has the interesting property that if any one component were removed the other two could fall apart. For this reason it was used by the dukes of the Italian island of Borromeo on their family crest as a symbol of strength in unity. (The same pattern appears in the logo of Ballantine beer.) In the ropelength-critical configuration, the three components lie in three perpendicular planes. Each one is a nonconvex curve which is smooth except at 14 points. The curves include arcs from the critical clasp described above; here again, in the six regions where a pair of components clasp each other, there is a small gap between them.
Except for the family of minimizers described above [CKS02] which are built in a simple way from circles and straight lines, these Borromean rings are the first explicit example of a ropelength-critical link.
|The tight configuration of the Borromean rings. Right, with thick tubes of diameter 1 just touching each other; left, with thinner tubes of diameter 2/5.|
The researchers plan to extend their work to an even more realistic model of thick ropes, incorporating a stiffness parameter which bounds how sharply the core curve can bend. For this clasp problem, the unbounded curvature at the tip is no longer allowed, so a more intricate shape will be needed.
The mathematicians on the team are
[CKS02] J.Cantarella, R.Kusner, J.Sullivan: "On the Minimum Ropelength of Knots and Links". Inventiones Math. 150:2, 2002, pp 257-286. www.arxiv.org/math.GT/0103224 .
[Sul03] J.Sullivan: "The tight clasp", Electronic Geometry Models 2001.11.001, May 2003. www.eg-models.de/2001.11.001 .
[Sta03] E.Starostin: "A constructive approach to modelling the tight shapes of some linked structures". Proc. Appl. Math. Mech. 3:1, pp 479-480.