The Growth and Curvature of Surfaces C Goodman-Strauss We are surrounded by surfaces that grow and develop, are shaped and sculpted through the control of curvature. Though tremendous advances have been made in the study of genetics of living organisms, we still are far from understanding how genes control geometry. In the talk, we’ll discuss a mathematical model of the growth and control of the curvature of surfaces. First we’ll sketch out a general discussion of curvature. How does an ear, a leaf, or a cluster of blossoms grow, consistently and reliably?

Curvature The curvature of a surface describes, in essence, how “roomy” a surface is, how much “stuff” there is. Curvature does not depend on how the surface is placed in space. Surfaces we might usually think of as “curved” might well be flat: For example, a cylinder is perfectly “flat”, because it can be made from a flat piece of paper. Gauss first gave the notion of curvature of a surface; though mathematicians usually describe curvature using the tools of differential geometry, there is an amazingly simple (and more general!) definition:

Definition: The total curvature of any topological disk is the turning deficit around its boundary.

Total turning = 360° so turning deficit = 0°Every planar disk has 0° curvature

A path around this cone has 270° total turning; the turning deficit is thus 90° The total curvature of this cone is 90° On the other hand, this cone has total curvature 270°

And this ‘cone’ has total curvature –90° We can measure the total curvature of all kinds of regions in this way... Lets try some examples!

In essence, regions of positive curvature tend to be “bulgy” and regions of negative curvature tend to be “crinkly” This lettuce has astoundingly negative curvature

The Gauss-Bonet theorem makes this precise. An Aside: Curvature and topology are tightly linked: Negatively curved surfaces ‘like’ to have genus if possible, and topology determines curvature. Gauss-Bonet Theorem: The total curvature of a closed surface of genus n is (1 – n) 720°

Curvature and growth How can curvature be controlled while a surface is growing along a ‘front’?

Consider fitting together puzzle pieces to make a surface, “growing” the surface outwards along some boundary. Here for example, the “puzzle pieces” are individual units of shell

Locally the curvature of the resulting surface depends on how fast this boundary is expanding. If the boundary is expanding rapidly with each step, then the curvature will be negative. If the boundary is contracting, or growing very slowly, the curvature of the surface will be positive. negative triangles meet in 7’s flat triangles meet in 6’s positive triangles meet in 5’s

This is precisely the effect we see here: as the shell accretes, the boundary is growing extremely rapidly and we have high negative curvature.

Crocheting patterns perfectly illustrate this

Post Tag systems and growth In general though, ANY BEHAVIOUR CAN BE CONTROLLED: To demonstrate this we’ll use a universal computer due to Emil Post, called “Post Tag Productions” One has an alphabet, a set of rules and a starting letter: a->abc b->aabbca c->ba At each step, one has a word. You cross off the first two letters, and depending on the original first letter, add a word to the back. abbca bcaabc aabca bcaabc aabca etc. This particular system repeats forever In general however, it is undecidable if a given system will grow forever, “repeat” or “crash” In particular, rates of growth are deterministic but wild.“repeat”“crash”

Indeed, this is a model of the control of complicated but determined control of curvature in living surfaces.

Biology Computes However, quite unlike the classical substitution systems, it is likely that it is undecidable, for example, whether one may repeatedly substitute ad infinitum; it is quite likely that it is undecidable whether a given rule will be needed; it is likely that it is undecidable how frequently a given rule will be applied. In particular, it is certainly undecidable, given a particular regular substitution system, what the curvature of the corresponding surface will be. Or to put it another way, any desired behaviour can be attained.