From Computed to Crocheted MeshMarch 1, 2005
Hinke Osinga and Bernd Krauskopf of the University of Bristol sent the following responses to questions from SIAM News about their crocheted Lorenz manifold (shown at right, photographed against a white background).
For readers outside the dynamical systems community, could you briefly describe the active research area of computing invariant manifolds?
The behavior of a dynamical system is determined by its "skeleton," which consists of the different attractors (steady states, periodic solutions, or more complicated sets), as well as saddle-type objects with their global stable and unstable manifolds. Global manifolds are complicated objects that must be found numerically. They are hypersurfaces consisting of infinitely many trajectories that end up (or come from) a saddle-type object. All other trajectories qualitatively follow the dynamics given by the "nearest" global manifolds. This feature was recently utilized in the Genesis Mission (http://www.genesismission.org), which sent a spacecraft to a saddle-type periodic orbit around the Lagrange point between the earth and the sun to collect solar dust particles. The spacecraft traveled on global manifolds to its destination and back to earth virtually without any fuel.
How did the crochet instructions come from your algorithm for computing manifolds?
Our algorithm builds up the surface from a central point (the saddle equilibrium at the origin in the Lorenz system) by adding rings of mesh points that lie at equal geodesic distance from this point. (Geodesic distance is the arclength of the shortest path on the surface.) Each new ring (a geodesic level set) is initially built up by finding the mesh points closest to the mesh points on the last ring at a slightly larger geodesic distance. Extra mesh points are then added where appropriate to maintain the quality of the mesh.
We realized that the distance between rings translates to a particular type of crochet stitch (crochet stitches vary in length). Adding a mesh point simply means to add a new stitch.
While this interpretation is fairly straightforward, the information about the three-dimensional coordinates of each mesh point is lost. However, the information on the local curvature of the surface is encoded by how stitches are added (or sometimes removed).
This obviously isn't a case of a vague artistic impression of a scientific phenomenon, but how close is the correspondence?
The correspondence is one-to-one in terms of the number of mesh points (stitches) and where points are added or deleted. The mounted object is
amazingly close to the computer version when mounted properly (as described in The Mathematical Intelligencer).
Mainstream stories about this work mentioned that in crocheting, you gained insight into how chaos arises in the Lorenz system. Is that true?
It is more a case of illustrating and communicating the complicated nature of the system even to nonspecialists. In particular, sensitive dependence on initial conditions (the hallmark of chaotic dynamics) can be explained as the difficulty of deciding on which side of the surface one starts. It must be mentioned that the surface is, in fact, infinitely large and only the first part was crocheted! On the other hand, seeing the object in three dimensions does give a much better impression of its geometry, even for the specialists.
Did you know how great it would turn out as soon as you realized the crochet-ability, or did you have to finish the whole thing?
As soon as we had the idea, we were convinced that the result would be something special. It was not clear from the outset what the object would really look like when crocheted. In fact, as the crocheting progressed, we got more and more worried about how to get it into the desired shape.
The key is to fix the z-axis (which is invariant and part of the manifold) and to insert a stiff flexible wire around the rim of exactly the right length (as determined by the algorithm). To our relief, it turned out that the manifold essentially falls into its desired shape with minimal tweaking. We further stabilized it by weaving a wire along the so-called strong stable manifold. We were delighted at how well everything worked.
As mathematicians, you can't have all that much experience with mainstream media. How have you worked the media into your schedules?
We were really amazed at the international media interest generated by
the press release of 15 December. We certainly did not expect to be on the main news on national TV in both the UK and Russia! We found ourselves spending quite a lot of time answering e-mails and phone calls. Fortunately, the initial media storm happened out of term time. Even now, we are kept quite busy as the interest is shifting to the more scientific press, such as SIAM News.
At the same time, we are also answering many e-mails from the general public, mathematical and crochet experts included. We have already seen pictures of three additional crocheted Lorenz manifolds!