After discussing some well-known examples in geometry and function theory, we study surfaces in space that are defined by the vanishing of the torsion of integral curves of a given vector field.
KeywordsVector Field Space Curve Ricci Flow Space Curf Lorenz Attractor
This is a record of a lecture given at the ESMA conference on mathematics and art at the Institut Henri Poincaré in Paris on 20 July 2010. The aim was to present a number of striking images and animations based on the application of techniques from both differential geometry and dynamical systems. This article retains the title of the lecture, even though it can only present still images; the true dynamic content can be found from links in the references at the end of the article.
The significant nature of the particular subject is not in question. The original method that the lecturer was describing formed the basis for the proof of the Poincaré conjecture that was confirmed by the offer of a Fields Medal to Gregori Perelman in 2006. It is based on the concept of Ricci flow, concerning trajectories like the looping black curve in Fig. 1 but in a much more abstract space of tensors. As for the sketches, the reader can judge to what extent these are artistic, but there is no doubt that they reflect the creative nature of the geometric arguments involved and the extent to which contemporary mathematics naturally blurs the distinction between creativity and discovery.
Experts will recognize other main themes in Fig. 2, namely (bottom left) a simple Riemannian metric ds 2 in two isothermal variables x,y that represents the geometry of a surface, and the concept of a discrete quotient ("this/G"), which is used by mathematicians to describe an everyday object like a torus. A more concrete version of the latter is the opaque doughnut-shaped surface visibile in Fig. 3 housing a Möbius band whose red boundary is a knot on the torus surface.
2 Creativity Versus Discovery
As its name suggests, differential geometry combines the use of differential calculus with the description of geometrical objects. It grew out of more conventional analysis in which geometrical ideas abound, for example recognizing a function by means of its graph. We shall illustrate this idea with the well-known example of iterated exponentiation in which it is hard to deny some element of creativity amidst the scientific journey of discovery that began with Euler in the 1700s, and continues to fascinate both amateur and professional mathematicians.
One can think of Fig. 7 as the floor plan associated to a video game; at the bifurcation point one is abandoning Euler’s interval of certain convergence for the duality inherent in the second stage that takes place in the first almost-circular chamber. The triple forking occurs at the next doorway at the centre of the overall black rectangle. Soon after that, quantum-like effects occur as gems that are to be plucked out of the air for extra points, though their presence goes unnoticed on the accompanying low-resolution images. Such gems include various spirals; the one in Fig. 6 could with hindsight be drawn with a simple mathematical formula using the techniques of , though it is actually the genuine object that was plotted by computer with exponential iteration.
3 Serret-Frenet Geometry
Whilst t can be thought of as time, it is better to make use of a different variable, denoted s, that represents arc-length measured along the curve. This is a natural parameter that is uniquely specified by the curve (independently of how it is traced) and a starting point (that we take to correspond to s = 0); if the curve were a piece of cotton, it would suffice to pull it tight and measure its length from the anchored starting point.
The curvature of a point in the plane measures the bending of the curve at that point. More precisely, it is equal to the rate of change of the angle between the tangent line to the curve and a fixed direction. When the curve is a circle of radius r, the bending is the same at any point of the circle, and the curvature assumes the constant value 1/r, In this sense, curvature equals the reciprocal of radius.
By taking plane sections of surfaces, one can discuss the curvature of a surface relative to any tangent direction and the maximum and minimum values of this function have special significance. Their product is the so-called Gaussian curvature that is generalized by the tensors of Ricci and Riemann, the basis for measuring the curvature of space-time in General Relativity. Positive Gaussian curvature at a point indicates that the surface is bending away entirely to one side of its tangent plane, and represents convexity at that point.
A similar definition applies to the curvature of a space curve. Once one has specified a point of the curve by assigning the value of s, then Κ (s) equals 1/r, where r is the radius of the circle that best fits the curve at that point. A circle is a very special sort of space curve not just because its curvature is constant but because (as a consequence) it always lies in some plane. In general, the fit of a circle to a space curve at a given point can only be approximate, though one can prove that there is a best fit (the "osculating" or "kissing" circle). In Fig. 8, the curve has alternating straight and curly segments since s cos s is oscillatory and keeps returning to zero. But the amplitude of its peaks becomes greater and greater, so the curly interludes become more pronounced as one moves to the right.
The functions Κ (s) and τ(s) crop up in the so-called Serret-Frenet equations that enable one to define and differentiate a triad of unit mutually perpendicular unit vectors at each point of a space curve. This is a topic covered in every introductory textbook to differential geometry, and we refer the reader to, for example, [3, 4, 5].
4 The Torsion of a Vector Field
Starting from any point (apart from a black dot), one can in theory follow the arrows and generate a unique trajectory. It is more interesting to visualize families of trajectories, like the one in Fig. 10. In this way, the vector field gives rise to a family of spaces curves that never touch one another. (The vector field is allowed to be zero at one or more points; at such points the arrow has zero length, no curve passes through the point, and strictly speaking neither Κ nor τ have a value there.) Whilst the typical trajectory is infinite in length, it is known that certain trajectories are actually closed curves or knots of varying complexity .
One can think of the Fig. 14 as representing a "stand" that could be placed on an office desk in order to properly display the more famous Lorenz curves. The second "upside-down" view betters displays the self-intersections of the surface, and a key feature of it, namely that it incorporates two "leaves" that are roughly planar (in theory these extend to infinity but of course we are only plotting part of the surface). One can understand this feature as follows. Roughly speaking, the butterfly attractor does itself fit into two planes that intersect at an acute angle (this is best seen in Fig. 10). If this were exactly true then all the points on the corresponding planar trajectories would have zero torsion (recall that τ measures the extent to which a curve does not lie in a plane).
The colours of Figs. 14 and 15 have a precise mathematical meaning. The spectrum (Red–Orange–Yellow–Green–Blue–Indigo–Violet) is used to indicate the angle with which the trajectories following the vector field emerge from the surface. Red indicates that they are tangent, so if the two leaves were exact planes they would be painted red all over. At the opposite extreme, violet represents points where the trajectories emerge at right angles (visible on subsequent images). Green would represent roughly a 45° angle of incidence between surface and curve.
Another aspect of the surface, more evident in the versions of Figs. 1 and 15, is the presence of cusps that form at two of the three points where the vector field is zero (physically, these can be thought of as equilibrium points of the dynamical system). Near these two points, the surface resembles a double cone (whose vertex is the respective point) separated by a plane. Actually, the surface is singular at all three such points, although the nature of the singularity at the origin is more complicated. What is perhaps more surprising is the presence of external spikes, though these point to the behaviour of other trajectories that are not shown.
Although the blue-green surface of Fig. 15 is also defined by setting the polynomial in Fig. 13 equal to zero, its colouring is related to the non-negative curvature function Κ. The latter becomes infinite as one approaches the two equilibrium points.
5 A Gallery of Null-Torsion Surfaces
We shall briefly describe the images on the following page.
The vector fields giving rise to the previous images all possess a certain degree of symmetry, characterized by the action of some group of transformations that leave the equations invariant. For example, the Lorenz field V defined by (1) is unchanged when the signs of both x and y are reversed. This represents invariance by a 180° rotation around the z axis, which incidentally identifies two of the three equilibrium points. For the sinusoidal figures, the group consists of translations defined by a lattice in space.
A consequence of this is that the "tri-rotation" ρ maps each trajectory of Q a onto another trajectory.
I first started using iterated exponentiation to illustrate bifurcation in a project for a Maple course I taught in Oxford in the 1990s. Remnants of this course are available on my homepage.
The study of space curves with assigned torsion and curvature is described in , and I used it as a student competition to find an interesting space curve by guessing simple candidate functions in a differential geometry course in Turin.
The vector field Q a is the basis of the author’s animation
Figure 25 arises from the vector field (4) with cubic components. As personal computing power increases, it will be easier to study more and more complicated vector fields and their associated surfaces.
The ESMA conference taught me not just to better appreciate artistic aspects inherent in modern mathematics, and but also the importance of explaining to a wider public the underlying geometrical ideas I deal with on a daily basis. I found myself having to work on a number of deep mathematical problems to answer some of the questions that arose. In this respect, I wish to express gratitude to the participants, and in particular François Apery, Claude Bruter, Eugenia Emets, George Hart, and Jos Leys. Thanks are also due to Claude for his afternoon visits from Gometz to Bures encouraging me to finish the text.
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