Through the Eyes of Galileo's Heirs: Deciphering the Laws of Nature Through the Language of Mathematics

Abstract

When I was a graduate student at Yale, back in what seems like the Paleozoic era, one of the courses that I took was called “Statistics for Morphometrics.” It was designed for those graduate students, like me, who were planning to eventually use some assortment of measures and statistics in our budding morphology-based dissertation work. The course was taught by a very bright fellow named Ray Heimbuch, an advanced graduate student in physical anthropology who already had a reputation for his exceptional skills in these areas. His presentations were extraordinary, and took his pupils deep into the continually evolving science of how one extracts meaningful data from the geometry of form. As he became more and more granular, we grasped that he saw the world in mathematical and geometric images beyond what most of the overwhelmed and struggling students, particularly me, could envision, now or, probably ever. With each presentation, it became clearer to me that I would have to accrue his incredible abilities and mathematical “eyes” if I was to undertake the research I wished to pursue. Or, I could become his really good friend. A few years later (with my successful Dissertation completed, and many papers hatched; e.g., see Laitman et al. 1978, 1979; Laitman and Heimbuch, 1982), I was the best man at Ray's wedding. Some weeks back as I penned this essay, I had the privilege of listening to Mike Hausman, our Professor and Chair of Orthopedics at Mount Sinai, and an internationally renown surgeon, as he taught my first-year anatomy students about the hand. I met Mike when I was his Teaching Assistant at Yale, also back in the Paleozoic, and noticed even then his extraordinary mind and special ability to see structure through mathematics and geometry. I again sat in awe in my lecture hall as this physician-surgeon-mathematician taught my class, not solely about the muscles or bones of the hand, but about the extraordinary patterns and geometric codes buried within the structure of nature's form. My young physicians-to-be sat enraptured as he guided them to the intersection of geometric shape and regularity of form, of Fibonacci numbers and patterns appearing from feathers to the shells of a nautilis to the fingers of the hand. And he took them back to the words of insight from one of the greatest minds of all time, Galileo Galilei, and his guiding principle that the laws of nature can only be understood through the language of mathematics itself (Galilei, 1623 1960). One of the observations that life has shown me is that some people's minds are wired very differently from others. No great or unique insight here, but it's what I've seen. In my inch of the scientific world—working on head and neck structures such as the larynx or ear—I've been amazed by how many of the surgeons, for example, who approach vocal folds are also extraordinary singers, or those who meticulously seek to reconstruct one's hearing are also talented musicians. No group, however, has impressed—awed might be a better term—me more than those who see the world through mathematical equations or through the ever-morphing lenses of geometry. Their eyes and brain are clearly wired differently than my plebian connections, and try as I have, I just cannot ingest or digest that which they see. I can remember so acutely working with my above-mentioned colleague, Ray Heimbuch, at the dawn of the computer-era at Yale as he devised coordinate data plots of our cranial base studies that afforded previously unassailable information. What struck me then, as it does now as I look back, was how he could “see” things in geometric shapes that were obviously constantly morphing and changing in his mind that took me forever to grasp. At first, I felt that I was just a tad slow (no comments here, please) but came to realize that what was different was not me, but him. He had some inner gift that set him apart, allowing him to see what could not be seen. This special issue of The Anatomical Record, “Assessing function via shape: What is the place of Geometric Morphometrics in functional morphology?” Guest Edited by Siobhan B. Cooke and Claire E. Terhune (Cooke and Terhune, 2015), two outstanding and creative young thinkers in the field of functional primate anatomy, bring together a like-minded cohort of mathematically blessed thinkers. Their collective efforts put forward a state-of-the art smorgasbord of the status of the ever-growing field of “Geometric” morphometrics, and how it is aggressively developing and, indeed, changing the way the larger world of evolutionary biology understands the nexus of shape and pattern. Indeed, while the global questions that these “geo-morphometricians” are investigating are not new, advances in the geometric morphometric (GM) “tool-kit” as they call it (the advances in technologies and statistical applications and their interface) have enabled new questions to be asked on areas and regions previously outside the range of reliable quantifications and analyses. For those who do not follow the morphing and changing world of form and pattern analysis on a daily basis, a few words of simple description are in order. As noted, those with the gifted eyes of mathematical thinkers interested in deciphering the meaning of form have been at this a long while. Many notable researchers have entered the intellectual fray, ranging from the classic work by D'Arcy Thompson On Growth and Form in 1917 (Thompson, 1917) to Fred Bookstein's Measuring and Reasoning: Numerical Inference in the Sciences, earlier this year (Bookstein, 2014).“Geometric” morphometrics, as highlighted in this issue, focuses on an advanced form of metrical/mathematical study. Earlier attempts to derive an understanding of the shape/function nexus were based upon what is generally called “Traditional” morphometrics (TM) or traditional morphometric analyses. This approach, still a major component of many studies, largely employs use of linear distances such as length, width or height, often with assorted ratios, or areas and angles as well. Multivariate analyses are often employed, offering, at times most valuable insights. The main value of TM is that measures are fairly easy to obtain, and thus considerable amounts can be retrieved. One major disadvantage of TM, however, is that linear measures are highly correlated with size, thus making true shape assessments very difficult and, oft-times, unreliable. Also, visualizations of shape differences, graphical representations, are largely not possible via the classic approach. GM has taken assessments to the next level. It relies largely on the use of precise landmarks to describe shape, but now in a two- or three-dimensional space. When combined with the evolving “tool-kit” of the GM workers, such as employment of powerful analyses such as Generalized Procrustes Analysis (GPA), new and robust views of the interface of measurement and graphical representation are possible. This, in turn, has allowed for new approaches and insights to assess the shape/form/function nexus. Cooke and Terhune have presented herein an extraordinary view of GM studies of the primate world, the home base of their own interests and expertise and the center of much study on form/function morphometrics. One will find within great diversity in both the species that come under the caliper as well as the anatomical regions explored. Species studied range from New World monkeys to Old World monkeys such as macaques to our closest relatives the chimpanzees and gorillas. Fossil relatives put under the spotlight extend back to our earliest australopith ancestors from millions of years before the present to those more-recent, ever-pesky Neanderthal cousins. As for anatomical regions investigated, studies span the body literally from head to foot, from a number on the skull, mandible and mastication; to the pelvis; to bones of the hand; down to those of the ankle. There are also some most insightful discussions of the strengths and weaknesses of GM, how it interfaces with other approaches such as finite element analysis (FEA) and biomechanical analyses, and its place in bringing the collective field forward. In the introductory overview for the issue, Cooke and Terhune note, almost apologetically, that they have approached the material from an applied, not a mathematical approach, so that all, young and old, traditionalists and newcomers can understand and use geometric morphometrics. Sorry, guys, you can't escape who you are that easily; you can't hide that you and your ilk can see what others can't. You may have thrown away your pocket protectors (I bet if I show these folks my old slide rule they'll hug it like it was the Holy Grail!), but you're still the smart kids in the class who understand the equation on the chalkboard (remember those?) that the rest of us just don't get. It's ok to have a special gift, as you all do, even if you don't realize it. You're the heirs of Galileo's eyes, eyes that saw the world in mathematical shades of triangles and rectangles, proportions and angles, and all in vibrant three-dimensions. Your special issue gives us all a chance to peek into a world most of us can't normally capture, a chance to see what the kids with the pocket protectors know. It's really cool. By the way, I can probably beat all of you in stickball, so there.