DISCUSSION PAPER: A GEOMETRY FOR BIOLOGY

This paper starts with an introductory rationale (polemic?) for a geometry in tune with biology. It is followed by an overview of a new geometry intended to begin meeting that need. This geometry is elaborated more extensively elsewhere,’ but toward a different end, visual science. Our concepts of space are deeply rooted in surveying. One need only look at the derivation of the word “geometry” to verify this. The first postulate of euclidean geometry is: “A straight line can be drawn from any point to any other point.” One already sees the primitive act of surveying. With the line, the simplest “objects” do not yet appear. In 2-space. three lines are required; in 3-sp:ice, four triangles. The inclusion of the interior requires still other steps. Euclid goes from triangles to more complex rectilinear objects, polygons. The only seriously considered nonpolygon is the circle. Where are the objects of biology? Where is the kidney bean, the tadpole‘? Note that the latter wiggles and is not congruent with o r similar to even itself. Despite the intervening millennia and many subsequent geometries, this situation has not been remedied. Projective geometry expanded the range of interest, but its content is still limited to lines and conic sections. Projections are still a generalization of the surveying viewpoint. With the advent of coordinate geometry. the range of specifiable curves expanded tremendously. But the curves are still rigidly defined and still by surveying. In invoking the immense power of algebra and analysis, it initiated the process by which geometry has been replaced by algebra. What resulted had extreme power for physics. It has the flavor of trajectories and only occasionally do objects (closed forms) appear. Curves of vastly different functional specification may be arbitrarily close. Hidden algebraic constraints have been added. In addition, they are still described by their boundaries. It is implicitly assumed that objects are generated by cutting them out with a scissors or milling machine, since no descriptive distinction is made between objects and thcir boundaries. Despite D’Arcy Thompson’s elegant and heroic effort to apply analytical transformations to biological shape, a natural geometry for biological shape did not evolve. Except for the equiangular spiral, his successes were primarily in shapes determined by surface forces. More recently, the non-euclidean (Gauss used “anti-euclidean”) geometries have not helped the problem much and have even led i t i n a wrong direction, since organisms live in an effectively euclidean space. These geometries extend the notion of “line” to “geodesic.” I n that development, first Gauss and later Rieniann ~ i s e “intrinsic coordinates.” a descriptive system that avoids the arbitrariness of an external coordinate system by always staying within the curve or manifold. Curves and higher dimensional spaces and manifolds are described entirely by specification of curvature within the space as one traverses it. Since an object is described by its boundary, i t means that the boundary is not en-